An overview of generalised Kac-Moody algebras on compact real manifolds July 28, 2022 Rutwig Campoamor-Stursberg1∗, Marc de Montigny2†, Michel Rausch de Traubenberg3‡ 1 Instituto de Matemática Interdisciplinar and Dpto. Geometŕıa y Topoloǵıa, UCM,E-28040 Madrid, Spain 2 Faculté Saint-Jean, University of Alberta, 8406 91 Street, Edmonton, Alberta T6B 0M9, Canada 3 Université de Strasbourg, CNRS, IPHC UMR7178, F-67037 Strasbourg Cedex, France ∗ Email: rutwig@ucm.es † Email: mdemonti@ualberta.ca ‡ Email: Michel.Rausch@iphc.cnrs.fr Abstract A generalised notion of Kac-Moody algebra is defined using smooth maps from a compact real manifold M to a finite-dimensional Lie group, by means of complete orthonormal bases for a Hermitian inner product on the manifold and a Fourier expan- sion. The Peter–Weyl theorem for the case of manifolds related to compact Lie groups and coset spaces is discussed, and appropriate Hilbert bases for the space L2(M) of square-integrable functions are constructed. It is shown that such bases are charac- terised by the representation theory of the compact Lie group, from which a complete set of labelling operator is obtained. The existence of central extensions of generalised Kac-Moody algebras is analysed using a duality property of Hermitian operators on the manifold, and the corresponding root systems are constructed. Several applications of physically relevant compact groups and coset spaces are discussed. 1 Introduction Kac-Moody algebras have been used in theoretical physics from the beginning 1980s onwards in various different contexts, such as string theory, the study of critical phenomena in two- dimensional statistical systems, Yang–Mills theory as well as in applications to exact solvable models (see e.g. [1] and references therein). Besides the axiomatic construction, Kac-Moody algebras (or more precisely, affine Lie algebras) can be obtained from affine extensions of the loop algebra of smooth maps from the unit circle S1 into a simple Lie group [2, 3, 4, 5, 6, 7]. 1 Another infinite-dimensional Lie algebra widely encountered in two-dimensional Conformal Field Theory, as well as in string theory, is the Virasoro algebra, the central extension of the Witt algebra, that is, the centrally extended Lie algebra of polynomial vector fields on the circle S1 [8] (and e.g. [9] and references therein). Various types of generalisations of Kac- Moody algebras respectively affine Lie algebras 1 have been proposed in the literature, such as the so-called quasi-simple Lie algebras in Ref. [10], the generalised Kac-Moody algebras in Ref. [11], the Borcherds algebra [12], as well as related structures like the Monster algebra [13, 14] and the Monstrous Moonshine [15, 16, 17]. As a matter of fact, Kac-Moody and Virasoro algebras are deeply related to the one- dimensional compact manifold S1. In this context, it is natural to expect that physical theories with more than two dimensions involve richer structures. It is with such possibilities in mind that we discuss hereafter a generalisation of the notion of Kac-Moody algebras associated to compact manifolds M of dimension higher than one. In particular, we shall restrict ourselves to certain type of manifolds, namely compact Lie groups M = Gc or coset spaces M = Gc/H , where H ⊂ Gc is a closed subgroup. The reason for these choices lies on the fact that the harmonic functions on the corresponding manifold M can be classified in terms of the representation theory of the Lie group Gc. The algebras described in this paper do not belong to the general classification of Kac-Moody algebras given by Kac [3]; rather, they represent generalisations of affine Lie algebras which, as we will observe below, admit roots but not simple roots and thus no Cartan matrices (except, as we will prove in Section 4, when the number of central charges, or ‘order of centrality’, is equal to one). Moreover, unlike the usual Kac-Moody algebras, we can construct all the generators of our generalised algebras. The notion of generalised Kac-Moody algebras is motivated by various phenomena in higher-dimensional physics, and possess the salient feature of being fully specified by har- monic expansions on M. These algebras are potentially of use in the Kaluza-Klein theory (see e.g. [18, 19, 20], with the latter reference being motivated in the supergravity context), where the space-time takes the form K = R1,3 × M. Symmetries in K, in particular the Nœther theorem, lead naturally to such generalised Kac-Moody algebras. Similarly, this type of structure emerges naturally through the consideration of current algebras [21, 22]. For instance, the authors of reference [23] analysed the symmetries corresponding to the massive states in the Fourier expansion for a Kaluza-Klein compactification in five dimensions, with these symmetries involving Kac-Moody and Virasoro algebras without central extensions. These authors suggested that similar infinite-dimensional symmetries should also appear in more complicated higher-dimensional theories with non-Abelian symmetry of the extra di- mensions. The relevant point is that this type of algebras admits central extensions. These central extensions can be introduced in two different but related ways, either by introducing two-cocycles in their Lie brackets or, more physically, adding Schwinger terms [24] to the current algebra. The structure of the paper is the following: In Section 2 we define generalised Kac- Moody algebras by means of the set of smooth maps from a compact real manifold to a real or complex finite-dimensional Lie group, in terms of a complete orthonormal basis for 1In the following (unless otherwise stated) we will always refer to Kac-Moody algebras instead of affine Lie algebras in order to be coherent with the physical literature. 2 the Hermitian scalar product on the manifold with a Fourier expansion. In the following, we shall restrict our discussion to manifolds related to compact Lie groups, mainly due to technical reasons. Although there is no doubt that the case of non-compact Lie groups is full of interest, with potential applications to non-Euclidean spaces and general manifolds, their study require techniques somewhat different from those used in this work. The main difference between compact and non-compact Lie groups is that unitary representations of the former are finite dimensional, whilst those of the latter are infinite dimensional. In addition, non-compact Lie groups exhibit irreducible unitary representations occurring outside the space of square integrable functions on the group, implying that a more general positive measure on the space of irreducible unitary representations must be defined, thus leading to more general integral formulae as the Plancherel formula instead of the Peter–Weyl theorem of the compact case [25]. Another technical difficulty resides in the division into discrete and continuous series, specifically in the context of the normalisation problem for discrete and the continuous spectra. For these reasons, in this paper we shall only offer a glimpse of the (rather different) constructions based on non-compact groups, the general analysis of which would be beyond the scope of our work. In Section 3, we discuss the Fourier expansion on manifolds taken as Lie groups and coset spaces, and discuss the Peter-Weyl theorem in this context. The corresponding Hilbert basis B of L2(M) is appropriately identified. As elements of B are characterised by the representation theory of Gc, we consider the labelling problem and identify a minimal set of operators, beyond the usual Casimir operators and Cartan subalgebra of Gc, to identify unambiguously all elements of B. With these considerations, in Section 4, we construct the generalised Kac-Moody algebras for the case where the underlying manifold is a compact Lie group and the coset space is a factor space of a compact Lie group by a closed subgroup. It is shown that these algebras admit central extensions related by some kind of duality to certain Hermitian operators of M. The root system of the centrally extended algebra is identified and some elements of the representation theory are given, at least for the simplest case, corresponding to the n−dimensional tori M = T n. In Section 5, some applications of the construction are presented in detail. Finally, in Section 6, some conclusions are drawn and potential generalisations of the approach discussed. 2 Algebras associated to compact manifolds In the following we shall assume that M is a compact real manifold. Let L2(M) denote the space of square integrable functions on M and dµ(M) the integration measure on M. If B = {ρI(m), I ∈ I} is a complete orthonormal basis for the Hermitian scalar product on M, with I a countable set, the identity (ρI , ρJ) = ∫ M dµ(M)ρI(m)ρJ (m) = δIJ , m ∈ M , is satisfied. As a consequence, a function Φ ∈ L2(M) can be described in terms of the basis B as Φ(m) = ∑ I∈I ΦIρI(m) ≡ ΦIρI(m) , 3 where ΦI = (ρI ,Φ) correspond to the expansion coefficients. An important question concerns the problem whether, given two elements ρI , ρJ ∈ B, the product still belongs to the space L2(M). In this work, this will be the case, as we assume that all functions are bounded, i.e., |ρI | < MI for some MI , so that ρIρJ ∈ L2(M). This enables us to consider the Fourier expansion of product of elements of B ρI(m)ρJ (m) = cIJ KρK(m) , (1) with cIJ K ∈ C. In general, it is difficult to derive precise formulae for the coefficients cIJ K , but we shall discuss some examples where they can be explicitly computed, at least partially. In this paper we will restrict our analysis to manifolds associated to compact Lie groups Gc. Therefore, the functions ρI of the orthonormal basis B can be organised using the representation theory of Gc, i.e., each ρI belongs to a given representation of Gc. As Gc is compact, the functions ρI are automatically bounded. In addition, the product (1) can be evaluated using representations of Gc and the corresponding Clebsch-Gordan coefficients. It is important to observe that for ρ (resp. ρ′) belonging to a representation D (resp. D′), the fact that ρ and ρ′ are commuting functions implies that the product ρρ′ belongs to S(D⊗D′), where S denotes the symmetric tensor product of D and D′. Consider now a simple complex or real finite-dimensional Lie group G with Lie algebra g. We denote its basis elements by Ta with a = 1, · · · , dim g. The Lie bracket is given by [Ta, Tb] = ifab cTc . It is well known that a Kac-Moody algebra can be associated to the Lie algebra g via the set of smooth maps from the circle S1 to G [2, 3, 4, 6, 7]. Similarly, the notion of generalised Kac-Moody algebra associated to the manifold M, denoted by g(M), can be defined by using the set of smooth maps from M to G as described hereafter [10, 11, 26]. Let G(M) denote the group of smooth maps from M to G and let g ∈ G, so that g = ei θ aTa (2) holds if G is compact. In these conditions, any element of G can be represented by the exponential of an appropriate element of g, while for the non-compact case, we have to replace it by a finite product of exponentials. The element in G(M) associated to (2) is given by ĝ(m) = ei θ a(m)Ta , where now θa(m) are square-integrable functions of M. In a neighbourhood of the identity the following approximation holds ĝ(m) ∼ 1 + i θa(m)Ta = 1 + i θaIρI(m)Ta , 4 where ρI(m) ∈ B. In particular, the set of functions from M → G leads, at the infinitesimal level, to the Lie algebra g ( M ) with basis g ( M ) = { TaI(m) = TaρI(m), a = 1, · · · , dim g, I ∈ I } , and Lie brackets [TaI , TbJ ] = i fab ccIJ KTcK . (3) If g is a real Lie algebra, then the generalised Kac-Moody algebra, denoted by g(M), will be real, because the manifold M is real. Clearly g(M) constitutes a generalisation of the usual notion of Kac-Moody algebras, but restricted hereafter to the context of compact manifolds M. The algebra (3) can be further enlarged introducing central charges and additional operators. Actually, the possible central extensions of (3) were fully classified in [6]. It is worthy to be mentioned that the construction can be naturally adapted to Lie supergroups and Lie superalgebras, resulting in the notion of generalised super-Kac-Moody algebras [27, 28, 29, 30]. An alternative physical motivation for considering generalised Kac-Moody algebras is related to Kaluza-Klein theories [18, 19, 20] (and references therein) and current algebras [21, 22]. Indeed if we consider a (4 + n)-dimensional compactified space-time of the form K = R 1,3 ×M , where R1,3 is the four-dimensional space-time and M a compact n−dimensional real man- ifold, it follows from the Nœther theorem that the conserved charges can be expressed in terms of the fields belonging to the (4 + n)−dimensional space-time. If we denote by Ta the conserved charge associated to a Lie algebra g, and by yA (A = 1 · · · , n) the coordinates on M, then integration over the space part of R1,3 but not over the internal space M and the equal-time commutation relations lead to the current algebra [ Ta(y), Ta′(y ′) ] = i faa′ bTb(y)δ n(y − y′) , (4) where the δ−distribution is defined in Appendix A, equation (76). Now consider Gc, a compact Lie group and H ⊂ Gc. Let us introduce a Hilbert basis of L2(M) as above, and set Ta(y) = TaI ρ̄ I(y) , then upon integration by ∫ dny ∫ dny′ (see Appendix A) where M = Gc or M = Gc/H , gives rise to [ TaI , Ta′I ] = i faa′ bcII′ JTbJ . (5) We thus obtain a generalised Kac-Moody algebra as defined in (3). If we add a Schwinger term to (4), we can define possible central extensions in close analogy with the Pressley–Segal analysis of central extensions of generalised Kac-Moody algebras [6, 24]. Now, considering 5 the Lie algebra of vector fields on M generated by LAI = −i ρI∂A (where ∂A = ∂ ∂yA ), the algebra (5) extends to [ TaI , TbJ ] = i fab ccIJ KTcK , [ LAI , LBJ ] = −i ( (∂AρJ)LBI − (∂BρI)LAJ ) , (6) [ LAI , TaJ ] = ρI∂AρJTaJ = dAI,J KTaK , where the summation over repeated indices is implicit and ρI∂AρJ = dAI,J KρK . The authors of Ref. [23] analysed the symmetries induced by the massive modes appear- ing in the Fourier expansion for a five dimensional compactified space-time R1,3×S1 by using an algebra of the type (6); however in the context of centreless (usual) Kac-Moody and Vi- rasoro algebras. It was further mentioned that these results can potentially be extrapolated to higher dimensional space-times. 3 Fourier expansion on compact manifolds In this section we briefly discuss the Fourier expansion on compact manifolds. Indeed, the usual Fourier analysis on the circle S1 can be extended to compact manifolds. Specifically, we consider two types of manifolds: compact Lie groups and coset spaces of compact Lie groups. For the two situations, we obtain the basic functions appearing in the Fourier analysis by group theoretical arguments. 3.1 The compact manifold as a Lie group Let Gc be a simple real compact Lie group and R̂ = {Rk, k ∈ Ĝc} be the set of all ir- reducible unitary representations of Gc, and Ĝc the set of labels of such representations.2 As Gc is compact, each of such representations is finite-dimensional; we denote the corre- sponding dimension of Rk by dk. For a matrix representation D(k)(g) ∈ Rk, g ∈ Gc, we denote the matrix elements by D(k) i j(g). As Gc is a group, for two matrix representations D(k)(g), D(k)(g ′) ∈ Rk, the matrix product D(k)(g ′′) = D(k)(g ′g) = D(k)(g)D(k)(g ′) is again a representation and therefore belongs to Rk. In other words, each column (resp. each line) of the matrix elements D(k) i j(g), j = 1 · · · , dk (resp. D(k) i j(g), i = 1 · · · , dk) is a Gc−representation. Let dµ(Gc) be the Haar measure of Gc and consider the space of square integrable func- tions, L2(Gc), defined on the manifold Gc and normalised as ∫ Gc dµ(Gc) = 1 . This allows us to state the following theorem. 2The notation Ĝc used here for the set of labels should not be confused with Ĝ, which is often used in the literature to denote the affine algebras. 6 Theorem 3.1 (Peter-Weyl [31]) Let R̂ = {Rk, k ∈ Ĝc} be the set of all unitary irre- ducible representations of Gc, and D(k)(g) ∈ Rk, for g ∈ Gc. Then the set of functions on Gc, ψ(k) i j(g) = √ dkD(k) i j(g), k ∈ Ĝc , i, j = 1, · · · , dk, g ∈ Gc, (7) forms a complete Hilbert basis of L2(Gc) with inner product (ψ(k) i j , ψ ′ (k′) i′ j′) = ∫ Gc dµ(Gc)ψ (k) i j(g)ψ(k′) i′ j′(g) = δkk′δ i′ i δ j j′ . For any function Φ in L2(Gc), we have Φ(g) = ∑ k∈Ĝc dk∑ i,j=1 φki jψ(k) i j(g) ≡ φki jψ(k) i j(g) , where the coefficients φki j are given by φki j = ∫ Gc dµ(Gc)ψ (k) i j(g)Φ(g) . As the representation is unitary, it follows that ψ (k) i j(g) = ψ(k) i j(g −1) . In the following, we describe left-coset manifolds M = Gc/H for compact groups Gc. As mentioned earlier, we do not discuss manifolds based on non-compact groups in detail. Albeit formally feasible, as we shall illustrate with some examples, the construction involves various subtleties that go beyond the scope of this paper. Essentially, instead of using the Peter-Weyl theorem for the particular compact groups discussed here, the construction in the non-compact case requires the more general Plancherel theorem (see, e.g. [25, 32, 33]), compounded by the infinite-dimensional unitary representations of non-compact groups. In- stead of the direct sums used in the context of square-integrable functions, the corresponding expression in the non-compact case uses the Plancherel integral. 3.2 Compact manifolds as a coset space Gc/H In this section, we consider the manifold to be a left coset M = Gc/H with respect to a closed subgroup H of Gc. Denote by h the Lie algebra associated to H . In general, Gc/H , which is the set of equivalence classes g1 ∼ g2 iff ∃ h ∈ H such that g2 = g1h , does not form a group unless H is normal in Gc. The elements of the coset space Gc/H are denoted by [g] and we have [g1] = [g2] iff ∃ h ∈ H such that g2 = g1h. Thus if r ∈ [r] then [gr] = [r′] ⇔ gr = r′h ⇔ r′ = grh−1 , 7 which defines the left action of Gc on Gc/H . At the Lie algebra level, we write the generators of gc, namely Ta (with a = 1, · · · , dim gc), as follows: Ui with i = 1, · · · , dim h, and Vp with p = 1, · · · , dim gc−dim h. The elements Vp belong to the space gc/h, which is not generally a Lie algebra. The commutations relations take the form (a) [Uj , Uk] = i gjk ℓUℓ , (b) [Uj , Vp] = i (Rj)p qVq , (c) [Vp, Vq] = i gpq jUj + i gpq rVr . The relations (a) are trivially satisfied as h is a Lie subalgebra of gc, whereas the relations (b) imply that gc/h is a representation of gc. We remark that if gpq r = 0 holds in (c), then the manifold Gc/H is said to be a symmetric space. Now let us extend the harmonic expansion of Theorem 3.1 to a coset space Gc/H seen as a manifold [18]. As the elements of Gc/H are equivalence classes, we consider now a function Φ ∈ L2(Gc/H), with components Φi, which belongs to a certain representation RH of H and, for a given h ∈ H , we denote by Di j(h) its matrix elements in RH . Thus, we have Φi(gh) = Di j(h)Φ j(g) , h ∈ H, g ∈ Gc/H . In order to apply the harmonic expansion to a coset manifold Gc/H , we do not need to consider all the representations of Gc. Let us consider the representation R(k) ∈ R, k ∈ Ĝc with matrix representative D(k)(g) for g ∈ Gc, such that D(k)(gh) = D(k)(g)D(h) . This relation is possible if and only if, in the embedding H ⊂ Gc, we have R(k) = mkRH ⊕ · · · . In other words, RH is containedmk times (withmk > 0) inR(k), so thatmk is the multiplicity of RH in the decomposition H ⊂ Gc. We denote by R̂|RH the set of representations of Gc satisfying this property, while Ĝc|RH denotes the set of corresponding labels. The harmonic expansion takes the form Φi(g) = ∑ k∈Ĝc|RH mk∑ n=1 dk∑ j=1 √ dk dD Φ(k)jnD(k) i j,n(g) , where dD is the dimension of the representation RH . Let g = rh, with r a representative of the equivalence class [r] ∈ Gc/H . From the identities 8 ∫ dµ(Gc) √ dk dD D(k) j i,n(g −1)Φi(g) = ∫ dµ(Gc) √ dk dD D(k) j i,n(h −1r−1)Φi(rh) = ∫ dµ(Gc) √ dk dD D(k) j ℓ′,n(r −1)Dℓ′ i(h −1)Di ℓ(h)Φ ℓ(r) = ∫ dµ(H)Dℓ′ i(h −1)Di ℓ(h) × ∫ dµ(Gc/H) √ dk dD D(k) j ℓ′(r −1)Φℓ(r) = ∫ dµ(Gc/H) √ dk dD D(k) j ℓ′,n(r −1)Φℓ(r) , we conclude that the coefficients of the expansion are given by Φ(k)jn = ∫ dµ(Gc/H) √ dk dD D(k) j i,n(r −1)Φi(r) . Here, we shall be interested only in functions Φ in the trivial representation of H , so that the expansion simplifies to Φ(g) = ∑ k∈Ĝc|R0 mk∑ n=1 dk∑ j=1 √ dk dD Φ(k)jnD(k) i0 j,n(g) , withR0 andD(k) i0 j,n(g) the trivial representation ofH . In all examples that will be presented in Section 5, we have mk = 1. Unitary representations of compact Lie algebras are classified either by their Dynkin labels or their Young tableaux which correspond to tensors of a certain type, notably for the classical series an = su(n + 1), bn = so(2n + 1), cn = usp(2n), dn = so(2n). The rep- resentations which contain the scalar representation can then be deduced from either the Dynkin label or from the Young tableau. For instance, the only representations that lead to a scalar representation for the embedding SO(n− 1) ⊂ SO(n) are the representations with dominant weight |n, 0, · · · , 0〉 corresponding to n-th order symmetric traceless tensors, i.e., the nth-order symmetric power of the fundamental representation [1, 0n−1]. 3.3 Labeling functions in the Peter-Weyl theorem The functions appearing in harmonic analysis on M = Gc or M = Gc/H are associated to all the finite-dimensional unitary representations of Gc. It is however well known that, within a given representation, a weight vector is generally not uniquely defined by its eigenvalues with respect to a given Cartan subalgebra. The purpose of this section is to identify a minimal set of operators to characterise unambiguously all weight vectors in an arbitrary representation. In essence, the main properties of the labelling problem for semisimple Lie algebras are deduced from a theorem due to Racah [34], which in modern terminology can be stated as follows: 9 Proposition 3.2 Let g be a simple (compact) Lie algebra of rank ℓ. Then the following conditions hold: 1. g admits ℓ independent primitive Casimir operators {Cd1 , · · · , Cdℓ}.3 2. Each Cdk can be represented as a homogeneous polynomial of degree dk in the genera- tors. 3. The degrees dk of the invariants satisfy the following numerical identity: ℓ∑ k=1 dk = dim g + ℓ 2 . 4. Any irreducible representation D of g is completely determined by dim g+ℓ 2 labels, from which (a) ℓ labels characterise the representation D as eigenvalues of the Casimir operators {Cd1 , · · · , Cdℓ}. (b) A number of dim g− ℓ 2 internal labels are required to separate the states within the multiplet D. Although D can be distinguished from other non-equivalent representations by means of the eigenvalues of the Casimir operators or, alternatively, the highest weight with respect to a given Cartan subalgebra h, the choice of internal labels is far from being unique, and usually depends on a specific chain of proper subalgebras g1 ⊂ g2 ⊂ · · · ⊂ g such that, in each step, the Casimir operators of the subalgebra are used to separate states [36, 37]. We give in Appendix B some details on the construction of internal labels beyond the Cartan subalgebra. In fact, when M = Gc, we understand the action of the group Gc on a matrix M in a representation D as a right action and left action, i.e., considering g ∈ Gc and its corresponding matrix D(g) in the representation D we have M →M ′ = D(g)MD(g)t . This means that the matrix elements of M are labeled by three types of indices associated to their corresponding operators: 1. ℓ labels which specify the representation: they can be the eigenvalues of the Casimir operators or the eigenvalues of the Cartan generators on the highest weight; 2. dim g−ℓ 2 labels which characterise the lines and are associated to internal labels and Cartan subalgebra for the right action. 3By primitive Casimir operators we mean those of minimal degree in the generators [35, 34]. 10 3. dim g−ℓ 2 labels which characterise the columns and are associated to internal labels and Cartan subalgebra for the left action. Therefore dim g operators are needed to label unambiguously all states when M = Gc, whilst for M = Gc/H the number of operators needed is ℓ to label all representations and dim g−ℓ 2 internal labels.4 4 Generalised Kac-Moody algebras In this section we build explicitly the generalised Kac-Moody algebra associated to the manifold M, where M is either a compact Lie group Gc or a coset space Gc/H with respect to a closed subgroup H ⊂ Gc. This construction proceeds in several steps. 4.1 Construction of the algebra Let g be a simple real (or complex) Lie algebra with basis {Ta, a = 1, · · · , dim g}. The Lie brackets take the form [ Ta, Tb ] = i fab cTc . Further, denote the Killing form by 〈 Ta, Tb 〉 0 = gab ≡ Tr ( ad(Ta)ad(Tb) ) . Let M be a compact n = (p + q)−dimensional manifold of volume V , isomorphic to either Gc or Gc/H , and suppose that we have a parameterisation yA = (ϕi, ur) = (ϕ1, · · · , ϕp, u1, · · · , uq) such that ∫ M dµ(M) = 1 V ∫ M dpϕ dqu = 1 . Consider the set of square integrable functions on M which are periodic in all ϕ−directions, but not in the u−directions. As done in Sections 3.1 and 3.2, we introduce a Hilbert basis of L2(M) identified with a minimal set of labels (see Section 3.3) B = { ρI(ϕ, u) , I ∈ I } , where I denotes the set of all labels needed to identify the states unambiguously. Let g(M) be the set of smooth maps from M into g: g(M) = { TaI = TaρI(ϕ, u) , a = 1, . . . , dim g , I ∈ I } . In this case (see Appendix A), the Lie brackets take the form [ TaI , TbJ ] = i fab ccIJ KTcK . (8) 4Actually the full set of labels is only necessary for the generic case. For representations of Gc exhibiting some kind of symmetry, the number of internal labels needed is usually smaller. 11 The precise form of the coefficients cIJ K defined in (1) is irrelevant at this stage. We shall present several explicit examples in Section 5. Finally, the Killing form in g(M) is given by 〈 X, Y 〉 1 = ∫ M dµ(M) 〈 X, Y 〉 0 , (9) for X, Y ∈ g(M). It follows (see Appendix A) that ρI(ϕ, u) = ηIJρ J(ϕ, u) , so that 〈 TaI , TbJ 〉 1 = gabηIJ . Central extensions of the generalised algebra. One natural question is whether the algebra determined by (8) admits central extensions. This problem was completely solved by Pressley and Segal in [6] (see Proposition 4.28 therein). Given a one-chain C (i.e., a closed one-dimensional piecewise smooth curve), the central extension is given by the two-cocycle ωC(X, Y ) = ∮ C 〈 X, dY 〉 0 , (10) where dY = ∂AY dyA = ∂iY dϕi+∂sY dus is the exterior derivative of Y . We observe that the two-cocycle ωC is non-trivial even if the manifold M has a trivial fundamental group. In fact this result extends also for non-compact differentiable manifolds. Furthermore, as pointed out by Pressley and Segal, this result is somewhat disappointing, as central extensions are characterised by maps from C → M, where C is one-dimensional. Stated differently, there is no central extension built up from maps N → M when dimN > 1. The two-cocycle can be written in alternative form. Indeed we have [38] ωC(X, Y ) = ∫ M 〈 X, dY 〉 0 ∧ γ , (11) where γ is a closed (n− 1)−current (a distribution) associated to C. In particular if γ = n∑ A=1 (−1)AγA dy1 ∧ · · · ∧ dyA−1 ∧ dyA+1 ∧ · · · ∧ dyn, then dγ = n∑ A=1 ∂AγAdy 1 ∧ · · · ∧ dyn = 0 . Using the topological properties of the manifolds S2 and S1 ×S1, the authors of Ref. [11] classified all possible central extensions for the case in which γ is defined only by functions and not distributions. For our purposes, in order to have some contact with the current algebra (13), hereafter, we will consider n specific (n− 1)−forms : γ(A) = (−1)AkAdy 1 ∧ · · · ∧ dyA−1 ∧ dyA+1 ∧ · · ·dyn , A = 1, · · · , n 12 where kA ∈ R. Thus ω(A)(TaI , TbJ) = kAgab ∫ M dµ(M) ρI(ϕ, u)∂AρJ(ϕ, u) = kAgabdAIJ . The brackets of the centrally extended algebra g(M) take the form [ TaI , TbJ ] = i fab ccIJ KTcK + gab n∑ A=1 kAdAIJ . (12) The algebra constructed by this procedure is closely related to a current algebra with Schwinger terms [ Ta(y), Ta′(y ′) ] = i faa′ bTb(y)δ n(y − y′)− i n∑ A=1 kA∂Aδ n(y − y′) . (13) Indeed, upon integration by ∫ dny ∫ dny′ (see Appendix A, in particular, equation (76)) equation (13) leads to equation (12). In Ref. [26], Bars constructed centrally extended extensions of the generalised Kac-Moody algebras g(S2) and g(S1 × S1) by using a current algebra approach. As already mentioned, a generalisation of Kac-Moody algebras to the case of non-compact manifolds can be considered, hence the natural question whether these algebras admit central extensions arises. This question was briefly studied in [30], where it was shown that (10) still defines a two-cocycle as C has no boundary, but that the cocycle reformulation (11) must be treated with care because the manifold M is non-compact and divergence problems for the integrals may appear. A generic ansatz to circumvent this technical difficulty has not yet been found. Derivations of the generalised algebra. The last step in the construction of the generalised Kac-Moody algebra associated to the manifold M is to introduce the derivations ∂A. However, due to the specific parametrisation of M, the variables ϕ and u have different periodicity properties. The former are periodic whereas the latter are not. This in particular means that the operators dj = −i∂ϕj associated to the variables ϕj are Hermitian whilst, due to the boundary term in the integration by parts, the operators ds = −i∂us associated to the variables us are not Hermitian. However, as we shall see, additional Hermitian operators beyond dj , j = 1, · · · , p can be considered. The existence of these additional operators follows from the relation between the manifold M and the Lie group Gc. We can thus identify a maximal set of commuting Hermitian operators. Of course, the operators dj , j = 1, · · · , p are commuting Hermitian operators. As just mentioned, since the manifolds that we consider are of the form Gc or Gc/H , there exists a largest Lie algebra gm such that gc ⊆ gm, with gc the Lie algebra of Gc (see examples below), such that the basic functions ρI belong to some unitary irreducible representation of gm. Furthermore the generators of the Lie algebra gm can be realised as differential Hermitian operators acting on M. Thus, among those generators we can extract the generators of the Cartan subalgebra H1, · · · , Hk, where k is the rank of gm. We express these operators as Hj = −ifAj (y)∂A . 13 The Hermicity condition translates into ∂Af A j (y) = 0 and f rj | = 0 , r = 1, · · · , p, (14) where f rj | = 0 means that the boundary term associated to all u−directions vanishes. Now we identify among the generators d1, · · · , dp, H1, · · · , Hk the maximal set of commuting op- erators that we denote D1, · · · , Dr. These generators are easily seen to adopt the form Dj = −ifAj (y)∂A , j = 1, · · · , r and satisfy (14). Naturally, the functions ρI are eigenfunctions of Hj and we note Hj(ρI(y)) = I(j)ρI(y) , with I(j) the corresponding eigenvalue. It is worthy to be observed that there exists some kind of duality between the Hermitian operators Dj and central extensions. Indeed, one can easily show that the (n− 1)−forms γj = kj n∑ A=1 (−1)AfAj (y) dy 1 ∧ · · · ∧ dyA−1 ∧ dyA+1 ∧ · · · ∧ dyn , j = 1, . . . , r (15) kj ∈ R are closed because of the condition (14), and the corresponding two-cocycles are given by ωk(TaI , TbJ) = kkJ(k)gabηIJ . (16) The generalised Kac-Moody algebra is thus generated by 1. TaI which belong to g(M); 2. the Hermitian operators D1, · · · , Dr; 3. the central charges k1, · · · , kr associated to the Hermitian operators. The non-vanishing brackets of the generalised Kac-Moody algebra associated to M have the form [ TaI , TbJ ] = i fab ccIJ KTcK + gabηIJ r∑ j=1 kjI(j) , [ Dj, TaI ] = I(j)TaI , (17) where I(j) is the eigenvalue of Dj. The authors of Ref. [10] defined generalised Kac-Moody algebras associated to the torus Tn, which coincides with our construction for Gc = U(1)n, and showed that these algebras correspond to specific examples of what they called ‘quasi- simple Lie algebras’. It should be observed that all their operators dA are Hermitian, hence they did not encounter the problem mentioned above. 14 4.2 Root system of generalised Kac-Moody algebras The purpose of this section is to identify a root structure for the generalised algebras defined by equation (17). We begin with the roots of the finite-dimensional simple Lie algebra g. Suppose that g is of rank ℓ. Let H i, i = 1, · · · , ℓ, be the generators of the Cartan subalgebra of g and let Σ be the root system of g. We consider the corresponding operators Eα, α ∈ Σ, in the usual Cartan-Weyl basis. If we introduce ĝ(M) = Span { TaI , Dj , kj, a = 1, · · · , dim g, I ∈ I, j = 1, · · · , r } , (18) then we observe from the algebra (17) that, in addition to the elements of the Cartan subalgebra of g, the operators Dj and kj commute with each other for j = 1, . . . , r. Thus the Cartan subalgebra of ĝ(M) is then generated by H i, Dj and kj , where i = 1, . . . , ℓ, j = 1, · · · , r, and the Cartan-Weyl basis takes the form H i I and EαI , where the non-vanishing brackets read [ H i I , H i′ I′ ] = ηII′h ii′ r∑ p=1 I ′(k)kp , [ H i I , EαJ ] = cIJ Kαi EαK , [ EαI , EβJ ] =    Nα,β cIJ K Eα+βK , α + β ∈ Σ, cIJ K α ·HK + ηIJ r∑ p=1 J(k)kp , α + β = 0, 0, { α + β 6= 0 , α + β 6∈ Σ, (19) [ Di, EαJ ] = J(i)EαJ , [ Di, H j J ] = J(i)H i J , where hij = 〈 H i, Hj 〉 0 , with the Killing form 〈 ·, · 〉 0 defined at the beginning of Section 4, and the operators associated to roots of g are normalised as 〈 Eα, Eβ 〉 0 = δα,−β . Proceeding along the same lines as for usual Kac-Moody algebras (see e.g. [7], p. 343-344), we have for the Killing form of ĝ(M) 〈 TaI , TbJ 〉 = ηIJgab , 〈 Dj, TaI 〉 = 〈 kj , TaI 〉 = 0 , (20) 〈 ki, kj 〉 = 〈 Di, Dj 〉 = 0 , 〈 Di, kj 〉 = δij . 15 The root spaces are given by g(α,n1,··· ,nr) = { EαI with I(1) = n1, · · · , I(r) = nr } , α ∈ Σ, n1, · · · , nr ∈ Z , g(0,n1,··· ,nr) = { H i I with I(1) = n1, · · · , I(r) = nr } , n1, · · · , nr ∈ Z . (21) Unlike the usual Kac-Moody algebras, the root spaces associated to roots are infinite dimen- sional and we have [ g(0,n), g(α,m) ] ⊂ g(α,m+n),[ g(α,m), g(β,n) ] ⊂ g(α+β,m+n), α + β ∈ Σ with n = (n1, · · · , nr). Introduce also 0 = (0, · · · , 0). It is important to observe that the Lie bracket between two elements involves not only the root structure, but also the representa- tion theory of Gc, in the form of the Clebsch-Gordan coefficients cIJ K (see (19)). To define the set of positive roots, we use the lexicographic order: (α, 0, · · · , 0, n1, · · · , nr) > 0 if    either { ∃ k ∈ {1, · · · , r} s.t. nr = · · · = nk+1 = 0 and nk > 0 or nr = · · · = n1 = 0 and α > 0 . (22) By (20), we can endow the weight space with a scalar product. Indeed (α, c1, · · · , cr, n1, · · · , nr) · (α′, c′1, · · · , c′r, n′ 1, · · · , n′ r) = α · α′ + r∑ j=1 ( njc ′ j + n′ Jcj ) . We further observe that, as happens for usual Kac-Moody algebras [3], we have two types of roots. The set of roots (α, 0,n) of g(α,n) with α ∈ Σ,n ∈ Zr satisfy (α, 0,n) · (α, 0,n) = α · α > 0 , and are called real roots, whilst the set (0, 0,n) of g(0,n) with n ∈ Zr and satisfying (0, 0,n) · (0, 0,n′) = 0 , is called the set of imaginary roots. Recall that r denotes the number of central charges (see (18)), that we called the order of centrality. We now show that unless r = 1, we cannot find a system of simple roots for ĝ. To this extent, introduce αi, i = 1, · · · , ℓ the simple roots of g. If r = 1, and we denote by ψ the highest root of g, it is easy to see that α̂i = (αi, 0, 0) , i = 1, · · · , ℓ , α̂ℓ+1 = (−ψ, 0, 1) (23) is a system of simple roots of ĝ. Now, if we suppose that r = 2, as the positive roots are given by (i) (α, 0, 0, 0, 0) with α > 0, or (ii) (α, 0, 0, n1, 0) with α ∈ Σ, n1 > 0, or (iii) (α, 0, 0, n1, n2), α ∈ Σ, n1 ∈ Z, n2 > 0 and since the roots (α, 0, 0, n1, 0) are neither 16 bounded from below nor from above because n1 ∈ Z, we cannot define a simple root of the form (−ψ, 0, 0,−nmax, 1), where nmax corresponds to the highest possible value of n1 (or −nmax the lowest possible value of n1). This means that for r ≥ 2 we cannot construct a system of simple roots. In other words, the only generalised Kac-Moody algebras that admit simple roots are (obviously) the usual Kac-Moody algebras, but also the Kac-Moody algebras associated to SU(2)/U(1) studied in Section 5.2. In the latter case, the Dynkin diagram of ĝ(SU(2)/U(1)) is analogous to the Dynkin diagram of the corresponding usual Kac-Moody algebra ĝ(U(1)), but is dressed by the representation theory of SO(3). Indeed, in this case the root space is infinite dimensional (see (21)). We have seen that for a generalised Kac-Moody algebra of centrality order r > 1, the set of imaginary roots is r-dimensional. We may then wonder whether the algebra associated to the manifold M has some relationship with a degenerate Kac-Moody algebra with Cartan matrix of co-rank r. In fact, the algebra associated to the manifold M does not belong to the general classification of Kac-Moody algebras as given by Kac in [3]. Indeed, Kac-Moody algebras are defined by a (symmetrisable) Cartan matrix and thus admit a Chevalley-Serre presentation. Differently, the algebras considered in this paper represent generalisations of affine Lie algebras. In particular, we have seen that for centrality orders strictly higher than one, there does not exist a system of simple roots, and hence no Cartan matrix or Chevalley-Serre basis exist. Moreover, we can construct all the generators of our generalised algebra, whilst this is not the case for the Kac-Moody algebras (different from affine Lie algebras). Observe moreover that for a centrality order r = 1 corresponding to the algebra ĝ(U(2)/U(1)), we have a system of simple roots and a Chevalley-Serre presentation of the algebra. Even if the Dynkin diagram of ĝ(U(2)/U(1)) coincides with the Dynkin diagram of ĝ(U(1)), the former is dressed by the representation theory of SO(3). To finish this section, we briefly show that the generalised Kac-Moody algebras con- structed so far share some properties with the so-called Lorentzian Kac-Moody algebras [39, 40]. Lorentzian Kac-Moody algebras appear in M−theory or in eleven-dimensional su- pergravity compactifed on tori. Such algebras are defined by a Cartan matrix (or a Dynkin diagram) subjected to some constraints. To this extent, introduce the root-lattice of ĝ(M). We begin with the root lattice of g ΛR(g) = { ℓ∑ i=1 niαi , n i ∈ Z } , with α1, · · · , αℓ the simple roots of g, supposed of rank ℓ. We then introduce the two- dimensional Lorentzian even self-dual lattice [39, 40] Π1,1 = { (m,n) , m, n ∈ Z } , endowed with the Lorentzian scalar product (m,n) · (m′, n′) = mn′ + nm′ . Let (e = (0, 1), ē = (1, 0)) satisfying e·ē = 1, e·e = ē·ē = 0 be a basis of Π1,1. Now assume that M = Tr and introduce r−copies of Π1,1, and the corresponding basis (ei, ēi), i = 1, · · · , r. 17 Therefore we have (see (21)) ΛR(ĝ(T r)) ⊂ ΛR(g)⊕ Π1,1 ⊕ · · · ⊕ Π1,1 ︸ ︷︷ ︸ r−times , i.e., the root lattice of ĝ(Tr) is a sublattice of ΛR(g) ⊕ Π1,1 ⊕ · · · ⊕ Π1,1. More precisely, α ∈ ΛR(ĝ(M)) if α · ei = 0, i = 1, · · · , r. Thus the root system, as well as all generators of ĝ(Tr), are known whereas, as we have seen previously, for r > 1 it is not possible to identify a system of simple roots. As a consequence, a Chevalley-Serre basis is not available. In the same manner, some types of Lorentzian Kac-Moody algebras can be obtained from any semisimple Lie algebra g. For instance, the so-called ‘very extended Lie algebra’ g+++ is a rank ℓ + 3 Lorentzian Lie algebra associated to the semisimple Lie algebra g (of rank ℓ), where the simple roots are constructed from the simple roots of g and two copies of Π1,1 [39, 40]. Therefore, ΛR(g +++) ⊂ ΛR(g)⊕ Π1,1 ⊕Π1,1 . The simple roots of the very extended Lie algebra g+++ are known. This means that one can introduce a Chevalley-Serre basis for these algebras. However, in this case explicit formulae for all generators of g+++ are not available. Thus even if the root lattices of ĝ(T2) and g+++ are both sub-lattices of ΛR(g)⊕Π1,1⊕Π1,1 those two Lie algebras have different properties. 4.3 Representations of generalised Kac-Moody algebras In this paragraph we outline some relevant points concerning the representations of gen- eralised Kac-Moody algebras, with special emphasis on the existence of central charges in connection with the unitarity of representations. In the following, we consider the quasi- simple Lie algebras as introduced in [10], and assume that g is a compact real Lie algebra. In the Cartan-Weyl basis, the algebra is generated by ĝ(U(1)r) = { H i m , Eα,m , α ∈ Σ, m ∈ Z n, di, ki, i = 1, · · · , r } , and the Lie brackets are given by (m = (m1, · · · , mr)) [ H i m , H i′ m′ ] = δm+m′hii ′ r∑ i=1 miki , [ H i m , Eαn ] = αi Eαm+n , [ Eαm, Eβn ] =    Nα,β Eα+βm+n , α+ β ∈ Σ, α ·Hm+n + δm+m′ r∑ i=1 miki, α+ β = 0, 0, { α + β 6= 0 , α + β 6∈ Σ, (24) [ di, Eαm ] = miEαm ,[ di, H j m ] = miH i m . 18 In the following, in order to increase the readability of some long formulae, we sometimes use the convention that δa+b = δ0a+b = δ−ba . As before, we consider the set αi, i = 1, · · · , ℓ of simple roots and ψ the highest root of g. Consider also the fundamental weights µi, i = 1, · · · , ℓ of g satisfying 2µi · αj αj · αj = δij . We also suppose to be given a representation of ĝ(U(1)r) with highest weight |µ̂0〉 = |µ0, c,m〉 , where µ0 = piµ i , (p1, · · · , pr) ∈ N r such that the relations H i |µ̂0〉 = µi0 |µ̂0〉 , ki |µ̂0〉 = ci |µ̂0〉 , di |µ̂0〉 = mi |µ̂0〉 , and Eαm |µ̂0〉 = 0 , (α, 0,m) > 0 , H i m |µ̂0〉 = 0 , m > 0 . are satisfied. For any positive real root α̂ = (−α, 0,m), the generators X± α,m = √ 2 α · αE∓α,±m , hα = 2 α · α ( − αiH i 0 + r∑ i=1 miki ) , (25) span an su(2)−subalgebra. The unitarity condition implies the constraints 2 α·α ( − α · µ0 + r∑ i=1 cimi ) ∈ Z , r∑ i=1 cimi ≥ α · µ0 , (26) with the latter identity being a consequence of the relation ||X− α,m |µ̂〉||2 = 〈µ̂ | X+ α,mX − α,m|µ̂〉 = 〈µ̂ | [ X+ α,m, X − α,m ] |µ̂〉 = 〈µ̂ | hα | µ̂〉 = 2 α · α ( − α · µ0 + r∑ i=1 cimi ) ≥ 0 . If α > 0 then α · µ0 > 0, whereas for α < 0 we obtain α · µ0 < 0. We thus suppose that α > 0. In this case the second relation in (34) is very strong. Indeed if (−α, 0,m) > 0, this means that m = (m1, · · · , mk−1, mk, 0, · · · , 0) with mk > 0 and m1, · · · , mk−1 ∈ Z. The second condition of (34), which must be satisfied for any k = 1, · · · , r, is equivalent to impose that only one central charge is non-vanishing. Therefore, without loss of generality we can suppose cr = c 6= 0 and ci = 0, i = 1, · · · , r − 1. 19 Next, prior to analyse unitary representations, we observe that, since c1 = · · · = cr−1 = 0, the algebra ĝ(U(1)r) does not admit highest weight unitary representations. For this purpose, consider now α̂ = (−α, 0, m1, . . . , mk, 0, . . . 0) > 0 with 1 ≤ k ≤ r, mk > 0 and (m1, . . . , mk−1) ∈ Zk−1. The operators (with m = (m1, · · · , mk, 0, · · · , 0)) Y ± α̂ = √ 2 α · αE∓α,±m , h′α = 2 α · α ( − αiH i 0 ) , (27) also generate an su(2)−subalgebra. As before, the condition ||Y ± α̂ |µ̂〉||2 ≥ 0 holds, from which we deduce that k∑ p=1 mici ≥ α · µ . (28) However, as c1 = · · · = cr−1 = 0, this cannot be satisfied. This contradiction arises from the fact that the central charges vanish for k1, . . . , kr−1, and is in direct agreement with the well known result of classical Kac-Moody algebras that states that the unique highest weight unitary representation for k = 0 is the trivial one (the adjoint representation, for which the central charge vanishes, is not a highest weight representation). With these preliminaries, we will construct unitary representations in two steps. As there is only one non-vanishing central charge kr, in the first step we consider the usual Kac-Moody algebra g̃ = g (U(1)) = {Tam, kr, dr, α ∈ Σ, m ∈ Z} . (29) Representations of the latter are well known and correspond to the unitary representations of g̃ with highest weight |µ̃〉 = |µ, cr, mr〉. In this case, with ψ being the highest root of g, we obtain: ψ ψ · ψ = ℓ∑ i=1 qi αi αi · αi , qi ∈ N, (30) and the second condition of (34) translates to x ≥ piq i , (31) where x = 2 c ψ·ψ is the level of the representation. We now recall some known results on unitary representations of g̃ (see e.g. [7]). The simple roots of g̃ are given by α̂0 = (−ψ, 0, 1) , α̂i = (αi, 0, 0) , i = 1, · · · , ℓ where the second entry corresponds to the eigenvalue of the non-vanishing central charge and the third one to the eigenvalue of the corresponding Hermitean operator, say kr and dr respectively. Introduce also Σ̂ = { α̂ = (α, 0, n) , (0, 0, n) , α ∈ Σ, n ∈ Z } . 20 The fundamental weights are defined by µ̂0 = (0, 1 2 q0ψ · ψ, 0) with q0 = 1 , µ̂i = (µi, 1 2 qiψ · ψ, 0) , i = 1, · · · , ℓ , with the qi, i = 1, · · · , r defined by (30). We obviously have 2µ̂i · α̂j α̂j · α̂j = δij . A highest weight is then specified by µ̂0 = piµ̂ i, with pi ∈ N , i = 0, · · · , ℓ , and the level of the representation is then given by x = ℓ∑ i=0 piq i ≥ ℓ∑ i=1 piq i . We denote the corresponding representation space as Dµ̂0 . In a second step, let ĝ = g̃ (U(1)r−1), which corresponds to the set of smooth maps from U(1)r−1 into g̃. We deduce that (see (29)) ĝ(U(1)r) = g̃(U(1)r−1) (32) = { Tamr e i n−1∑ k=1 mkϕk , m ∈ Z r−1 , dr , kr and dj = −i∂j , j = 1, · · · , n− 1 } . As seen in (34), the central charges associated to dj, j = 1, · · · , r − 1 vanish. Consider now R = { |m〉,m ∈ Z r−1 } , (33) the set of all unitary representations of U(1)r−1. We have dj|m〉 = mj |m〉 . Using the harmonic expansion on U(1)r−1 〈(ϕ1, · · · , ϕr−1)|m〉 = ei(m1ϕ1+...mr−1ϕr−1) , unitary representations of ĝ are given by the tensor product D̂µ̃ = Dµ̃ ⊗R (34) and correspond to a harmonic expansion of the unitary representation Dµ̃ of g̃ on the manifold U(1)r−1. A unitary representation of ĝ(U(1)r) follows directly from unitary representations of g̃. As the space (33) is neither bounded from above nor bounded from below, unitary representations of ĝ(U(1)r) are not highest weight representations. As we have just seen, 21 unitarity of representations implies only one non-vanishing central charge. This result seems to be contradictory at a first sight, in particular when considering the tensor product of two- representations for two different non-vanishing central charges. As an illustration, consider for instance the generalised Kac-Moody algebra associated to the manifold M = T2. Denote with k = (k1, k2) the central charges (k1, k2 6= 0) of g12 = ĝ(U(1)2), without any unitarity constraints. From the previous result, two cases can be considered if one wants unitary highest weight representations: (1) k = (k1, 0) or (2) k = (0, k2). These two choices lead to two possible isomorphic (but different) algebras that we denote g1 respectively g2 (see Eq.[32]). We can now consider unitary representations of the first or the second algebra. Let D1 ⊗R2 (resp. R1 ⊗D2) be a unitary representation of g1 (resp. g2) with the notations of (34). While ( D1 ⊗ R2 ) ⊗ ( R1 ⊗ D2 ) is certainly a representation of the algebra g1 × g2, it is not a representation of the algebra g12, which has two non-vanishing central charges. This shows that the contradiction is only apparent, with no conflict emerging from the construction. Now we can extend part of the results to the general case, i.e., when M = Gc or Gc/H . Let α be a root of the compact Lie algebra g. Next, introduce X+ α,m ∈ g(−α,m) , X− α,m ∈ g(α,−m) , hα ∈ g(0,0) , with m = (m1, · · · , mr) > 0. Note that in this case, with the notations of equation (19), for X+ α,m ∈ g(−α,m), X − α,m ∈ g(α,−m) we have m(k) = mk and −m(k) = −mk respectively. Furthermore, the coefficient ηmn appearing in the bracket [ X+ α,m, X − α,m ] simplifies in this case to ηm, which is a sign (see the examples given in the next section). Assume further that the operators X± α,m, hα − 2 α·αηm r∑ i=1 miki are chosen in such way that they generate an su(2)−subalgebra: [ hα, X ± α,m ] = ±X± α,m , [ X+ α,m, X − α,m ] = hα − 2 α · αηm r∑ i=1 miki . (35) Then, in analogy to the previous discussion of the unitarity of representations, it follows that all central charges except one must be equal to zero. It is important to observe that, in absence of symmetries between the generators Di, we can have r different possibilities given by (eventually reordering the eigenvalues of the operators Di to define positive roots, see equation (22)) c = (0, · · · , 0, cp, 0, · · · , 0) or k = (0, · · · , 0, kp, 0, · · · , 0) p ∈ {1, · · · , r} . In this situation, it remains to identify the precise form of the operators in (35) and to es- tablish a condition analogous to (31) to characterise unitary representations. Independently of these conditions, unitarity leads to only one non-vanishing central charge. Consequently, there are no obstructions to introduce a system of simple roots, as seen in Section 4.2. The resulting Dynkin diagram of ĝ(M) is analogous to the Dynkin diagram of the corresponding usual Kac-Moody algebra ĝ(U(1)), but dressed with the representation theory of Gc. Let us emphasise again that, given the Lie algebra ĝ(M), all central charges except one must be equal to zero in order to guarantee the unitarity of a representation. This result 22 can be compared to that of the previous section, where we proved that only when the order of centrality, i.e., the number of non-vanishing central charges is equal to one, the algebra admits a system of simple roots. 5 Explicit construction in low rank In this section, we illustrate the procedure previously developed with some physically rele- vant examples in which we shall identify the coefficients cIJ K together with the Hermitian operators Di and their corresponding central extensions ki. The brackets will always be given by equation (17). We begin naturally with the Lie group SU(2), then we turn to the coset spaces SU(2)/U(1), SO(4)/SO(3), SU(3)/SU(2) and G2/SU(3). For elementary definitions see e.g. [41]. 5.1 Real Lie group SU(2) The group SU(2) is defined as the set of special unitary 2× 2 complex matrices: SU(2) = { U ∈ M2(C) such that U †U = 1, detU = 1 } , that can be written in the form U = ( α β −β α ) , α, β ∈ C , |α|2 + |β|2 = 1 , showing that the isomorphism of manifolds SU(2) ∼= S3 holds. Indeed, setting (see [41]) { α = cos θeiϕ1 , β = sin θeiϕ2 , 0 ≤ θ ≤ π 2 , 0 ≤ ϕ1 < 2π , 0 ≤ ϕ2 < 2π (36) we obtain a parameterisation of the sphere S3. In the language of Appendix A, this leads to the parameterisation of S3 given by 0 ≤ ϕ1, ϕ2 ≤ 2π , 0 ≤ u = 1 2 sin2 θ ≤ 1 2 . (37) This parameterisation is a bijection on a dense subset of SU(2), namely when θ 6= { 0, π 2 } . We observe that this parameterisation is not a homeomorphism from [0, π 2 ]× [0, 2π)× [0, 2π) onto S3, as the interval is non-compact. The scalar product on SU(2) is given by (see [41]) (f, g) = 1 2π2 π 2∫ 0 sin θ cos θdθ 2π∫ 0 dϕ1 2π∫ 0 dϕ2 f(θ, ϕ1, ϕ2) g(θ, ϕ1, ϕ2) . It is easy to observe that the functions ψa,b(θ, ϕ1, ϕ2) = √ (a + b+ 1)! a! b! αa(−β)b , a, b ∈ N , 23 satisfy the relations (ψa,b, ψa′,b′) = δaa′δ b b′ . (38) Furthermore, we can see from equation (38) that the functions α, β ∈ S 3 defined in equation (36) enable us to obtain all the matrix elements introduced in Section 3.1 in a ready manner. To this extent, we introduce a differential realisation of the generators of the Lie algebra su(2) in the above parameterisation of SU(2): J+ = J1 + iJ2 = 1 2 ei(ϕ1−ϕ2) ( − i tan θ ∂ ∂ϕ1 − i cot θ ∂ ∂ϕ2 + ∂ ∂θ ) , J− = J1 − iJ2 = 1 2 e−i(ϕ1−ϕ2) ( − i tan θ ∂ ∂ϕ1 − i cot θ ∂ ∂ϕ2 − ∂ ∂θ ) , (39) J3 = − i 2 ( ∂ ∂ϕ1 − ∂ ∂ϕ2 ) , with Lie brackets [ J3, J± ] = ±J± , [ J+, J− ] = 2J3 . This differential realisation acts on the rows of matrices and thus corresponds to a right action. Similarly we can define a left action acting on the columns. The generators are the same as in (39), except that we have to replace ϕ2 by −ϕ2. We do not give the form of the generators except for the last one J ′ 3 = − i 2 ( ∂ ∂ϕ1 + ∂ ∂ϕ2 ) , (40) but it can be explicitly checked that right and left actions commute. The space of irreducible unitary representations is given by R̂ = { Rℓ, ℓ ∈ 1 2 N } with the representation Rℓ of dimension dℓ = 2ℓ + 1. Thus, for each ℓ, we have to identify 2ℓ+1 equivalent representations associated to the right action. The key observation for this identification is given by the two complex-conjugate two-dimensional spinor representations defined as D 1 2 , 1 2 = { Φ 1 2 , 1 2 , 1 2 = √ 2α ,Φ 1 2 , 1 2 ,− 1 2 = √ 2β } , D− 1 2 , 1 2 = { Φ− 1 2 , 1 2 , 1 2 = − √ 2β ,Φ− 1 2 , 1 2 ,− 1 2 = √ 2α } , such that D− 1 2 , 1 2 = D 1 2 , 1 2 ∼= D 1 2 , 1 2 , as expected. In the notation above, the first index corre- sponds to the eigenvalue of the Cartan generator of the left action or J ′ 3 (see (40)), the last index to the eigenvalue of the Cartan generator of the right action or J3 (see (39)), whereas the second index corresponds to the eigenvalue of the Casimir operator of the spinor repre- sentation. This identification is in accordance with Section 3.3 and the labelling problem. This can be extended easily to an arbitrary representation. Indeed, for any ℓ ∈ 1 2 N, define the 2ℓ + 1 equivalent representation spaces corresponding to the right action Dm′,ℓ, 24 −ℓ ≤ m′ ≤ ℓ. Each space admits the highest weight vector Φm′,ℓ,ℓ = √ (2ℓ+ 1)! (ℓ+m′)!(ℓ−m′)! αℓ+m ′ (−β)ℓ−m′ (41) = (−1)ℓ−m ′ √ (2ℓ+ 1)! (ℓ+m′)!(ℓ−m′)! ei(ℓ+m ′)ϕ1−i(ℓ−m′)ϕ2 cosℓ+m ′ (θ) sinℓ−m ′ (θ) . In order to obtain the remaining vectors of the representation space Dm′,ℓ we use the relation Lk± ( eim1ϕ1−im2ϕ2F (θ) ) = (∓)k 2k ei(m1±k)ϕ1−i(m2±k)ϕ2 sink±m2 θ cos±m1 θ dk d(cos θ)k [ sin∓m2 θ cos∓m1 θ F (θ) ] , which can be proved by induction. It follows that Φm′,ℓ,m = √ (ℓ+m)! (2ℓ)!(ℓ−m)! ( J− )ℓ−m [ (−1)ℓ−m ′ √ (2ℓ+ 1)! (ℓ+m′)!(ℓ−m′)! ×ei(ℓ+m′)ϕ1−i(ℓ−m′)ϕ2 cosℓ+m θ sinℓ−m ′ θ ] = (−1)ℓ−m ′ 1 2ℓ−m √ 2ℓ+ 1 (ℓ+m′)!(ℓ−m′)! (ℓ+m)! (ℓ−m)! ei(m+m′)ϕ1−i(m−m′)ϕ2 × sin−m+m′ θ cos−ℓ−m ′ θ d d(cos θ)ℓ−m [ sin2ℓ−2m′ θ cos2ℓ+2m′ θ ] . When ℓ is an integer number and m′ ≥ 0, the formula simplifies. If we define u = cos 2θ and the polynomials Pm′,ℓ,m(u) = (−1)ℓ−m ′ 1 2ℓ √ 1 (ℓ+m′)!(ℓ−m′)! dℓ−m duℓ−m [ (1− u2)ℓ−m ′ (1 + u)2m ′ ] , we have Φm′,l,m = √ (ℓ+m)! (ℓ−m)! ei(m+m′)ϕ1−i(m−m′)ϕ2 sin−m+m′ 2θ(1 + cos 2θ)−m ′ Pm′,ℓ,m(u) . In particular Φm′,ℓ,0 = eim ′(ϕ1−ϕ2) sinm ′ 2θ(1 + cos 2θ)−m ′ Pm′,ℓ,0(u) . There are analogous formulæ for m′ ≤ 0. We conclude that the Φ-functions are orthonormal (Φm′ 1 ,ℓ1,m1 ,Φm′ 2 ,ℓ2,m2 ) = δ m′ 1 m′ 2 δℓ1ℓ2δ m1 m2 , 25 and constitute an orthonormal Hilbert basis which is well adapted to the Peter-Weyl theorem applied to SU(2): B = { Φm′,ℓ,m, ℓ ∈ 1 2 N,−ℓ ≤ m,m′ ≤ ℓ } . (42) From the highest weight (41) we have the conjugacy property Φ̄m ′,ℓ,m(θ, ϕ1, ϕ2) = (−1)m−m′ Φ−m′,ℓ,−m(θ, ϕ1, ϕ2) . (43) In order to define the Lie algebra ĝ(SU(2)), we introduce Ta,m′,ℓ,m = TaΦm′,ℓ,m(θ, ϕ1, ϕ2), the two Hermitian operators J ′ 3, J3 and their associated central charges k, k′ (see (15) and (16)) . The Lie brackets take then the form (see (17)) [ Ta1,m′ 1 ,ℓ1,m1 , Ta2,m′ 2 ,ℓ2,m2 ] = ifa1a2 a3cm′ 1 ,ℓ1,m1,m ′ 2 ,ℓ2,m2 m′ 3 ,ℓ3,m3Ta3,m′ 3 ,ℓ3,m3 +(−1)m1−m′ 1ga1a2δℓ1,ℓ2δm1+m2 δm′ 1 +m′ 2 ,(km2 + k′m′ 2) , (44)[ J ′ 3, Ta,m′,ℓ1,m ] = m′Ta,m′,ℓ1,m ,[ J3, Ta,m′,ℓ1,m ] = mTa,m′,ℓ1,m . We now proceed with the evaluation of the cIJ K coefficients. By using well-known results from the coupling theory of angular momenta (see e.g. [42, 43]) we obtain Φm′ 1 ,ℓ1,m1 (θ, ϕ1, ϕ2)Φm′ 2 ,ℓ2,m2 (θ, ϕ1, ϕ2) = ℓ1+ℓ2∑ ℓ=|ℓ1−ℓ2| λ(m′ 1, m ′ 2, ℓ, ℓ1, ℓ2) ( ℓ1 m1 ℓ2 m2 ∣∣∣ ℓ m1+m2 ) Φm′ 1 +m′ 2 ,ℓ,m1+m2 (θ, ϕ1, ϕ2), (45) where ( ℓ1 m1 ℓ2 m2 ∣∣∣ ℓ m1+m2 ) are the Clebsch-Gordan coefficients associated to the right action. In this expansion, we consider only the allowed values of m′ 1+m ′ 2, such that −ℓ ≤ m′ 1+m ′ 2 ≤ ℓ. There is no generic closed expression to compute the coefficients λ(m′ 1, m ′ 2, ℓ, ℓ1, ℓ2), although they can be computed recursively. For instance, for the highest value of m′ 1, m ′ 2, ℓ, ℓ1, ℓ2 we obtain : λ(m′ 1, m ′ 2, ℓ1 + ℓ2, ℓ1, ℓ2) = √ (2ℓ1 + 1)!(2ℓ2 + 1)! (2(ℓ1 + ℓ2) + 1)! (ℓ1 + ℓ2 +m′ 1 +m′ 2)!(ℓ1 + ℓ2 − (m′ 1 +m′ 2))! (ℓ1 +m′ 1)!(ℓ1 −m′ 1)!(ℓ2 +m′ 2)!(ℓ2 −m′ 2)! . 5.2 Coset space SU(2)/U(1) Assume that Q ∈ U(1) ⊂ SU(2) is given by Q = e2i θJ ′ 3 = ( eiθ 0 0 e−iθ ) . This means that α, β have a U(1)−charge equal to 1 and α, β have a U(1)−charge −1. More generally, the functions Φm′,ℓ,m(θ, ϕ1, ϕ2) have a charge 2m′. From Section 3.2, we just need 26 to consider functions that are neutral, i.e, Φ0,ℓ,m. This is possible if ℓ is an integer number. For such functions, we have Φ0,ℓ,m = (−1)ℓ 2ℓℓ! √ (2ℓ+ 1) (ℓ+m)! (ℓ−m)! eim(ϕ1−ϕ2) sin−m 2θ dℓ−m d(cos 2θ)ℓ−m ( (1− cos2 2θ)ℓ ) = Yℓm(2θ, ϕ1 − ϕ2) , where Yℓm are the usual spherical harmonics on the sphere S2. (Note, however, the uncon- ventional normalisation factor for the Y -functions.) If we perform the change of coordinates 0 ≤ ψ = 2θ ≤ π , 0 ≤ ϕ = ϕ1 − ϕ2 < 2π , 0 ≤ ϕ̃ = ϕ1 + ϕ2 < 4π , then the pointsm = (ψ, ϕ, ϕ̃ = cons.) parameterise points on the manifold S 2 ∼= SU(2)/U(1) ⊂ S3 ∼= SU(2). With this parameterisation, for the level surfaces ϕ = cons. we have, on the one hand L± = e±iϕ ( i cotψ ∂ ∂ϕ ± ∂ ∂ψ − i 1 sinψ ∂ ∂ϕ̃ ) = ∣∣∣ ϕ̃=cons. e±iϕ ( i cotψ ∂ ∂ϕ ± ∂ ∂ψ ) , L3 = −i ∂ ∂ϕ , as well as the relation (f, g) = 1 2π2 π 2∫ 0 sin θ cos θ dθ 2π∫ 0 dϕ1 2π∫ 0 dϕ2 f(θ, ϕ1, ϕ2) g(θ, ϕ1, ϕ2) = ∣∣∣ ϕ̃=cons. 1 4π π∫ 0 sinψ dψ 2π∫ 0 dϕf(ψ, ϕ, cte) g(ψ, ϕ, cons.) , thus reducing to the generators of so(3) (resp. to the Hilbert scalar product on S2). In the language of Appendix A, S2 is parameterised by 0 ≤ ϕ ≤ 2π , −1 ≤ u = cosψ ≤ 1 . It follows that the adapted Hilbert basis for SU(2)/U(1) is given by the usual spherical harmonics, B = { Yℓm, ℓ ∈ N, −ℓ ≤ m ≤ ℓ } , with the conjugacy relation Ȳ ℓm(ψ, ϕ) = (−1)mYℓ,−m(ψ, ϕ) . 27 In order to define the Lie algebra ĝ ( SU(2)/U(1) ) we introduce Ta,ℓ,m = TaYℓm(ψ, ϕ), the Hermitian operator L3 and its associated central charge k (see (15) and (16)). The Lie brackets take the form (see (17)) [ Ta1,ℓ1,m1 , Ta2,ℓ2,m2 ] = ifa1a2 a3cℓ1,m1,ℓ2,m2 ℓ3,m3Ta3,ℓ3,m3 +(−1)m1km2 ga1a2δℓ1,ℓ2δm1+m2 , (46)[ J3, Ta,ℓ,m ] = mTa,ℓ,m . For the spherical harmonics, it is well known that Yℓ1m1 Yℓ2m2 = ℓ1+ℓ2∑ ℓ=|ℓ1−ℓ2| √ (2ℓ1 + 1)(2ℓ2 + 1) 2ℓ+ 1 ( ℓ1 0 ℓ2 0 ∣∣∣ℓ0 )( ℓ1 m1 ℓ2 m2 ∣∣∣ ℓ m1+m2 ) Yℓm1+m2 , (47) which leads to the cIJ K coefficients in (46). Potential applications can e.g. be conceived in Supergravity [44, 45], using the space SL(2,R)/U(1) related to the non-compact group SL(2,R), the unitary representations of which are known [46], and correspond to the discrete series (either lower or upper bounded) and the continuous series (principal and supplementary). Discrete (respectively continuous) series are characterised by a discrete (respectively continuous) spectrum of the Casimir op- erator of sl(2,R), from which we conclude that the discrete series are normalisable, whereas the continuous series are not. They can be found, for instance, in Section 2 of Ref. [32], and can be related to Eqs. (5.26) and (5.27) of Ref. [41]; (the former corresponding to the discrete series, while the latter, within the continuous series, requires to distinguish between “bosons” (n even in [32], p. 197) and “fermions” (n odd in [32]), with the correspondence: 2µ = E0− 1 2 +s and 2λ = −E0− 1 2 +s (µ, λ from Ref. [41]) with E0 = 0 for bosons and E0 = 1 2 for fermions and with s ∈ iR (resp. s ∈ R) for the principal (respectively complementary) series. In the context of harmonic functions, the discussion of non-compact groups rapidly shows to be considerably intrincate [47]. Indeed, as the number s for the continuous series (see [47]) is either real or purely imaginary, the expressions in Section 4.3 of this reference [47] are meaningless for the continuous series, whereas for discrete series, convergence occurs for spin greater than 1/2, whereas here, s = 0, 1/2. Harmonic analysis of homogeneous spaces G/H , with G a non-compact Lie group and H a compact closed subgroup, of the same type as SL(2,R)/U(1), has been considered to some extent in [33], Chapter 15. 5.3 Coset space SO(4)/SO(3) The manifold SO(4)/SO(3) is well-known to be isomorphic to the three-sphere S3. Its interest in our context is that it gives rise to an equivalent realisation of the algebra (44), but with a different Hilbert basis. This construction can furthermore be extended to the coset spaces SO(n)/SO(n− 1) ∼= Sn−1 for values n > 4. The sphere S3 can be parameterised by x1 = sinψ sin θ cosϕ , x2 = sinψ sin θ sinϕ , x3 = sinψ cos θ , x4 = cosψ ,    0 ≤ ψ ≤ π , 0 ≤ θ ≤ π , 0 ≤ ϕ < 2π , (48) 28 and endowed with a scalar product defined by (f, g) = 1 2π2 π∫ 0 sin2 ψdψ π∫ 0 sin θdθ 2π∫ 0 dϕf(θ, ϕ, ψ) g(θ, ϕ, ψ) . Again, using the terminology of Appendix A, the sphere S 3 is parameterised by 0 ≤ ϕ ≤ 2π , −1 ≤ u1 = cos θ ≤ 1 , 0 ≤ u2 = 1 2 ψ − 1 2 cosψ sinψ ≤ π 2 . Representations of SO(4) are characterised by their Dynkin labels or by a Young tableau associated to a tensor with a certain symmetry. Among tensors, only traceless nth-order symmetric tensors admit a scalar representation with respect to the embedding SO(3) ⊂ SO(4). Let us denote by Dn 2 ,n 2 , n ∈ N the representation corresponding to the set of traceless nth-order tensors, which is of dimension (n + 1)2. If Di (n)j are the corresponding matrix elements, from the result of Section 3.2 and because Dn 2 ,n 2 contains the scalar representation in the decomposition through the embedding SO(3) ⊂ SO(4), for any Fourier expansion on S3 we have to consider the indices i = i0, where D i0 (n)j are in the scalar representation of SO(3). These matrix elements can be easily obtained. To that purpose, we introduce the gener- ators of the Lie algebra so(4) in the usual {N0, N±, N ′ 0, N ′ ±} basis: [ N0, N± ] = ±N±,[ N+, N− ] = 2N0, [ N ′ 0, N ′ ± ] = ±N ′ ±,[ N ′ +, N ′ − ] = 2N ′ 0, [ Na, N ′ b ] = 0 . From the expression of the generators of the Lie algebra defined on S3 and the expansion on the sphere, we obtain: N0 = − i 2 [ ∂ ∂ϕ + cotψ sin θ ∂ ∂θ − cos θ ∂ ∂ψ ] , N+ = 1 2 eiϕ [( i cot θ − cotψ sin θ ) ∂ ∂ϕ + ( 1 + i cotψ cos θ ) ∂ ∂θ + i sin θ ∂ ∂ψ ] , N− = 1 2 e−iϕ [( i cot θ + cotψ sin θ ) ∂ ∂ϕ + ( − 1 + i cotψ cos θ ) ∂ ∂θ + i sin θ ∂ ∂ψ ] , N ′ 0 = − i 2 [ ∂ ∂ϕ − cotψ sin θ ∂ ∂θ + cos θ ∂ ∂ψ ] , N ′ + = 1 2 eiϕ [( i cot θ + cotψ sin θ ) ∂ ∂ϕ + ( 1− i cotψ cos θ ) ∂ ∂θ − i sin θ ∂ ∂ψ ] , N ′ − = 1 2 eiϕ [( i cot θ − cotψ sin θ ) ∂ ∂ϕ − ( 1 + i cotψ cos θ ) ∂ ∂θ − i sin θ ∂ ∂ψ ] . 29 We can construct spherical harmonics, since we have Φ 1 2 1 2 ; 1 2 , 1 2 (θ, ϕ, ψ) = √ 2eiϕ sin θ sinψ = √ 2 ( x1 + ix2) , Φ 1 2 ,− 1 2 ; 1 2 , 1 2 (θ, ϕ, ψ) = √ 2(− cos θ sinψ + i cosψ) = √ 2(−x3 + ix4) , Φ 1 2 , 1 2 ; 1 2 ,− 1 2 (θ, ϕ, ψ) = − √ 2(cos θ sinψ + i cosψ) = − √ 2(x3 + ix4) , Φ 1 2 ,− 1 2 ; 1 2 ,− 1 2 (θ, ϕ, ψ) = − √ 2e−iϕ sin θ sinψ = − √ 2 ( x1 − ix2) , denoting this representation by D 1 2 , 1 2 , and whose highest weight is given by Φ 1 2 1 2 ; 1 2 , 1 2 . Accord- ing to this prescription, the highest weight of the representation Dn 2 ,n 2 is given by Φn 2 ,n 2 ;n 2 ,n 2 (θ, ϕ, ψ) = √ n+ 1einϕ sinn θ sinn ψ . The remaining vectors Φn 2 ,m1; n 2 ,m2 (θ, ϕ, ψ) are explicitly obtained by the action of the op- erators N−, N ′ −, where m1, m2 indicate the eigenvalues of N0, N ′ 0. Moreover, we have the conjugacy properties Φ n 2 ,m1; n 2 ,m2 (θ, ϕ, ψ) = (−1)m1+m2Φn 2 ,−m1; n 2 ,−m2 (θ, ϕ, ψ) . Since the first and third indices are redundant, we set Φn 2 ,m1; n 2 ,m2 → Φn,m1,m2 . The representation space Dn 2 ,n 2 can also be obtained in another way. Introduce yi = rxi (see equation (48)) in spherical coordinates, as well as the Laplacian ∇2 in this system of coordinates (to be defined below, in equation (54)) and the space of polynomials of degree n, Rn[y1, y2, y3, y4], so that we have Dn 2 ,n 2 = { P (y1, y2, y3, y4)|r=1 where P ∈ Rn[y1, y2, y3, y4] s.t. ∇2P (y1, y2, y3, y4) = 0 } .(49) In equation (49), the Laplacian constraint simply projects on traceless polynomials. Then the Hilbert basis on S3 relevant for our purpose is defined by B = { Φn;m1,m2 , n ∈ N,−n 2 ≤ m1, m2 ≤ n 2 } . (50) We further have the orthonormality relations ( Φn,m1,m2 ,Φn′,m′ 1 ,m′ 2 ) = δnn′δm1 m′ 1 δm2 m′ 2 . In the notations above for Φn,m1,m2 , we have taken the following conventions: n labels the representation space, here Dn 2 ,n 2 , and m1 (resp. m2) labels the eigenvalue of N0 (resp. N ′ 0). From the results of Section 3.3, we would expect to need four labels to classify all repre- sentations of so(4). However, as only certain types of representations appear in the Fourier expansion, it will turn out that three labels are sufficient. In order to define the Lie alge- bra ĝ ( SO(4)/SO(3) ) , we introduce Ta,n,m1,m2 = TaΦn,m1,m2 (θ, ϕ, ψ) the Hermitian operator 30 N0, N ′ 0. The associated central charges k0 and k′0 are more involved and are obtained using the two-forms dual to N0, N ′ 0 given by γ0 = − i 2 k0 [ dθ ∧ dψ − cotψ sin θdϕ ∧ dψ − cos θdϕ ∧ dθ ] sin2 ψ sin θ , γ′0 = − i 2 k′0 [ dθ ∧ dψ + cotψ sin θdϕ ∧ dψ + cos θdϕ ∧ dθ ] sin2 ψ sin θ . The Lie brackets take the form (see (17)) [ Ta,n,m1,m2 , Ta′,n′,m′ 1 ,m′ 2 ] = ifaa′ a′′cn,m1,m2,n′,m′ 1 ,m′ 2 n′′,m′′ 1 ,m′′ 2Ta′′,n′′,m′′ 1 ,m′′ 2 , +(−1)m1+m2ga′a′δnn′δm1+m′ 1 δm1+m′ 2 (k0m ′ 1 + k′0m ′ 2) , (51)[ N0, Ta,n,m1,m2 ] = m1Ta,n,m1,m2 ,[ N ′ 0, Ta,n,m1,m2 ] = m2Ta,n,m1,m2 . Since the two Hilbert bases (42) and (50) are admissible bases for the space L2(S3), the alge- bra (51) is isomorphic to the algebra (44). The former is presented using the representation theory of S3 ∼= SU(2), whilst the latter is described in terms of the representation theory of S 3 = SO(4)/SU(3). The coefficients cIJ K in (51) can be obtained from the relation Φn,m1,m2 (θ, ϕ, ψ)Φn′,m′ 1 ,m′(θ, ϕ, ψ) = n+n′∑ N=|n−n′| λ(N, n, n′) ( n m1,m2 n′ m′ 1 ,m′ 2 ∣∣∣ N m1+m′ 1 ,m2+m′ 2 ) ×ΦN,m1+m′ 1 ,m2+m′ 2 (θ, ϕ, ψ), (52) where ( n m1,m2 n′ m′ 1 ,m′ 2 ∣∣∣ N m1+m′ 1 ,m2+m′ 2 ) are the Clebsch-Gordan coefficients of the decomposition Dn 2 ,n 2 ⊗Dn′ 2 ,n ′ 2 = n+n′⊕ N=|n−n′| DN 2 ,N 2 , and λ(N, n, n′) are coefficients which can be computed recursively. There is a third presentation of the algebra given in (51). The two Casimir operators of the so(4) algebra are given by Q = N2 0 + 1 2 (N+N− +N−N+) , Q′ = N ′ 0 2 + 1 2 (N ′ +N ′ − +N ′ −N ′ +) , or by C = 2(Q +Q′) , C ′ = 2(Q−Q′) . For the representation Dn 2 ,n 2 , we have C = n(n+ 2) , C ′ = 0 . (53) 31 However, the Laplacian in spherical coordinates (r, φ, θ, ψ) takes the form ∇2 = 1√ g ∂i( √ ggij∂j) = ∂2 ∂r2 + 3 r ∂ ∂r − C r2 , (54) with the metric given by ds2 = dr2 + r2 ( dψ2 + sin2 ψ dθ2 + sin2 ψ sin2 θ dφ2 ) . (55) Hence, the condition (53) for functions of the sphere S3 in the representation Dn 2 ,n 2 is equiv- alent to impose the constraint ∇2Y (θ, φ, ψ) = −n(n + 2) r2 Y (θ, φ, ψ) . (56) This definition is obviously equivalent to equation (49), as ∇2 ( rnY (θ, φ, ψ) ) = 0 . If we set Y (θ, φ, ψ) ≡ Ynℓm(θ, φ, ψ) = Hnℓ(ψ)Yℓm(θ, φ) , n ∈ N , 0 ≤ ℓ ≤ n , −ℓ ≤ m ≤ ℓ , then the condition (56) is equivalent to the second-order linear homogeneous ordinary dif- ferential equation d2Hnℓ(ψ) dψ2 + 2 cotψ dHnℓ(ψ) dψ + ( n(n+ 2)− ℓ(ℓ+ 1) sin2 ψ ) Hnℓ(ψ) = 0 . (57) Indeed, we can express the functions Hnℓ(ψ) as follows Hnℓ(ψ) = Nnℓ sin ℓ ψ C1+ℓ n−ℓ(cosψ) , where C1+ℓ n−ℓ are the Gegenbauer polynomials defined by (see e.g. [48, 49]) 1 (1 + x2 − 2x cosψ)α = ∞∑ n=0 Cα n (cosψ)x n , for |x| < 1 , and the normalisation coefficients are given by Nnℓ = (−1)ℓ(2ℓ)! √ (n+ 1)(n− ℓ)! (n+ ℓ+ 1)! . The harmonic functions Ynℓm are characterised by the eigenvalues of the Casimir operators in the embedding chain SO(4) ⊃ SO(3) ⊃ SO(2), with n being associated to SO(4), ℓ to SO(3) and m to SO(2). This is another illustration of the missing label problem studied in Section 3.3. The functions Ynℓm in (56) can be expressed in terms of the functions Φn,m1,m2 , by means of the decomposition Dn 2 ,n 2 = n⊕ k=0 Dn , 32 valid for subduced representations in the embedding SO(3) ⊂ SO(4). These functions can be considered as an alternative Hilbert basis on the 3-sphere, with (Ynℓm, Yn′ℓ′m′) = δnn′δℓℓ′δ m m′ , and constitute the standard basis used in the harmonic analysis on S3. It is possible to define the Lie algebra ĝ(SO(4)/SO(3)) by means of the generators Ta,n,ℓ,m = TaYnℓm, N0 + N ′ 0, N0 − N ′ 0 and k0 + k′0, k0 − k′0. However, the functions Ynℓm are not simultaneous eigenfunctions of the operators N0 and N ′ 0, but of their sum N0 +N ′ 0. This follows at once from the identities Y11±1 = Φ1, 1 2 ,± 1 2 , Y110 = 1√ 2 ( Φ1,− 1 2 , 1 2 + Φ1, 1 2 ,− 1 2 ) , Y100 = i√ 2 ( Φ1,− 1 2 , 1 2 − Φ1, 1 2 ,− 1 2 ) . Thus, in this new basis, the commutation relations are more involved. In particular, the fact that the Y s are not eigenfunctions of N0 − N ′ 0 in the second line of (51) implies that the coefficient of k0 − k′0 together with the commutator with N0 − N ′ 0 have to be consequently modified. The construction described in this section can be further extended to the coset spaces SO(n)/SO(n−1) ∼= Sn−1. As a matter of fact, the only representations of SO(n) that contain a scalar representation with respect to the embedding SO(n− 1) ⊂ SO(n) are the traceless symmetric nth-order tensors. These representations are obtained from the symmetric powers of the fundamental representation [1, 0n−1] after extracting the trace, and correspond to representations with Dynkin labels [n, 0, · · · , 0], in the notation of [50]. Similarly to what happened with the results of Section 5.2, the construction above could be potentially of interest for applications to either Supergravity or Einstein-Maxwell scalar theories in 4 + 1 dimensions, with a non-compact Riemannian version SO(3, 1)/SO(3) of our manifold SO(4)/SO(3), that can be seen as the special case for n = 3 of the se- quence SO(n, 1)/SO(n) of symmetric real manifolds in supergravity theories, which yield non-symmetric homogeneous special Kähler (respectively special quaternionic Kähler) man- ifolds [44, 45]. Unitary representations of SL(2,C) were originally obtained by Gel’fand [51], while the unitary representations of SO(1, n) were studied in [52]. 5.4 Coset space SU(3)/SU(2) In order to construct the generalised Kac-Moody algebra associated to the coset space SU(3)/SU(2), we must first conveniently parameterise the manifold SU(3). To this extent, we proceed in four steps: 1. If w1, w2, w3 are three orthonormal vectors of C3, i.e., satisfying w† iwj = δij , the matrix M1 = ( w1 w2 w3 ) subjected to the additional constraint detM1 = 1, is such that M † 1M1 = 1, and therefore belongs to SU(3). We choose the matrix M1 as M1 =   cos θeiω1 sin θ cos ξeiω2 sin θ sin ξe−i(ω1+ω2) − sin θeiω1 cos θ cos ξeiω2 cos θ sin ξe−i(ω1+ω2) 0 − sin ξeiω2 cos ξe−i(ω1+ω2)   , with 0 ≤ ω1, ω2 < 2π, 0 ≤ ξ, θ ≤ π 2 . 33 2. We introduce the diagonal matrix U1 =   eiλ 0 0 0 eiλ 0 0 0 e−2iλ   , 0 ≤ λ < 2π which belongs to U(1) ⊂ SU(3). 3. We further consider the matrix U2 =   1 0 0 0 α β 0 −β α   , with { α = cos ρeiψ1 , β = sin ρeiψ2 , 0 ≤ ρ ≤ π 2 , 0 ≤ ψ1, ψ2 < 2π . This implies that U2 ∈ SU(2) ⊂ SU(3). 4. Finally, we define the matrix U ≡ U2U1M1 ∈ SU(3) , which parameterises a point on the manifold SU(3). If we define λ+ ω1 = ϕ3, λ+ ω2 = ϕ1 and λ− ω1 − ω2 = ϕ2, then we have the following identities in the first row of the matrix U , U11 = cos θeiϕ3 , U12 = sin θ cos ξeiϕ1 , U13 = sin θ sin ξeiϕ2 . Similarly, we have U21 = −α sin θeiϕ3 , U22 = α cos θ cos ξeiϕ1 − β sin ξe−i(ϕ2+ϕ3) , U23 = α cos θ sin ξeiϕ2 + β cos ξe−i(ϕ1+ϕ3) , for the second row of U , and for the third row, U31 = β sin θeiϕ3 , U32 = −β cos θ cos ξeiϕ1 − α sin ξe−i(ϕ2+ϕ3) , U33 = −β cos θ sin ξeiϕ2 + α cos ξe−i(ϕ1+ϕ3) . The three rows {Ui1, Ui2, Ui3}, i = 1, 2, 3 span the fundamental three-dimensional represen- tation [1, 0] of SU(3), whereas {U i1, U i2, U i3}, i = 1, 2, 3 span the anti-fundamental three- dimensional representation [0, 1] of SU(3). It is well known that any representation of SU(3) 34 can be obtained from appropriate tensor products of the fundamental and anti-fundamental representations, from which we conclude that the Hilbert basis for the manifold SU(3) can be deduced from the Uij and U ij functions. From the Peter-Weyl theorem, for a given representation D of dimension d, we obtain, with the functions Uij and U ij , d copies of the representation D. If we denote by Dn,m the representation of highest weight nµ1 + mµ2 (see below for the notations), then it is straightforward to obtain the correct number of copies of the representations Dn,0 and D0,n, say (n + 2)!/(2!n!). For general representations Dn,m, however, it is more involved. For instance, we have eight copies of the adjoint representation D1,1, rather than nine, as it would seem at a first glance. Indeed, the highest weight of the adjoint representations are U 11U12, U 11U22, U11U32, U21U12, U21U22, U21U32, U 31U12, U 31U22, U 31U32, but because of the relation U11U12 + U21U12 + U 31U32 = 0, there are actually eight independent copies of the adjoint representation. Thus, having constructed all representations with the correct multiplicity, an appropriate Kac-Moody algebra ĝ(SU(3)) can be defined. It is worth noticing that the embedding of Kostant’s principal subalgebra su(2)P , iso- morphic to so(3), into the Lie algebra su(3), is actually maximal, constituting the unique symmetric embedding of Kostant’s su(2)P into any simple, compact Lie algebra [53]. However, we are mainly interested in the Fourier expansion on the coset space SU(3)/SU(2) ∼= S 5 . This isomorphism is related to the possible description of the corresponding tangent space as the 5-dimensional irreducible representation space of either so(5) ≃ usp(4) or su(2). Since this representation of su(2) is self-conjugate, this implies, by Theorem 1.5 of [54], the maximal embedding su(2) ⊂ usp(4), so that the subduced representation remains irreducible. It should be mentioned in this respect that the action of SU(2) on its 5-dimensional irreducible representation is an example of a θ-group [55, 56, 57], a remarkable class of linear groups of transformations related to symmetric, (pseudo-)Riemannian coset spaces that has shown to be of current interest in several physical applications. According to the results of Section 3.2, the only functions that must be considered are    z3 = U11 = cos θeiϕ3 , z2 = U13 = sin θ sin ξeiϕ2 , z1 = U12 = sin θ cos ξeiϕ1 , 0 ≤ θ ≤ π 2 , 0 ≤ ξ ≤ π 2 , 0 ≤ ϕi < 2π . They parameterise the five-sphere as a consequence of |z1|2+|z2|2+|z3|2 = 1. In the language of Appendix A, the parameterisation of S5 is given by [58] 0 ≤ ϕ1, ϕ2, ϕ3 < 2π , 0 ≤ u1 = 1 2 sin2 ξ ≤ 1 2 , 0 ≤ u2 = 1 4 sin4 θ ≤ 1 4 . The scalar product on the five-sphere is taken as (f, g) = 1 π3 π 2∫ 0 sin3 θ cos θdθ π 2∫ 0 sin ξ cos ξdξ 2π∫ 0 dϕ1 2π∫ 0 dϕ2 2π∫ 0 dϕ3 ×f(θ, ξ, ϕ1, ϕ2, ϕ3) g(θ, ξ, ϕ1, ϕ2, ϕ3) , 35 while the generators of the Lie algebra su(3) are E+ 1 = 1 2 ei(ϕ1−ϕ2) ( ∂ ∂ξ − i tan ξ ∂ ∂ϕ1 − i cot ξ ∂ ∂ϕ2 ) , E+ 2 = 1 2 ei(ϕ2−ϕ3) ( − sin ξ ∂ ∂θ − cot θ cos ξ ∂ ∂ξ − i cot θ sin ξ ∂ ∂ϕ2 − i tan θ sin ξ ∂ ∂ϕ3 ) , E+ 3 = 1 2 ei(ϕ1−ϕ3) ( − cos ξ ∂ ∂θ + cot θ sin ξ ∂ ∂ξ − i cot θ cos ξ ∂ ∂ϕ1 − i tan θ cos ξ ∂ ∂ϕ3 ) , E− 1 = 1 2 ei(−ϕ1+ϕ2) ( − ∂ ∂ξ − i tan ξ ∂ ∂ϕ1 − i cot ξ ∂ ∂ϕ2 ) , (58) E− 2 = 1 2 ei(−ϕ2+ϕ3) ( sin ξ ∂ ∂θ + cot θ cos ξ ∂ ∂ξ − i cot θ sin ξ ∂ ∂ϕ2 − i tan θ sin ξ ∂ ∂ϕ3 ) , E− 3 = 1 2 ei(−ϕ1+ϕ3) ( cos ξ ∂ ∂θ − cot θ sin ξ ∂ ∂ξ − i cot θ cos ξ ∂ ∂ϕ1 − i tan θ cos ξ ∂ ∂ϕ3 ) , h1 = −i ( ∂ ∂ϕ1 − ∂ ∂ϕ2 ) , h2 = −i ( ∂ ∂ϕ2 − ∂ ∂ϕ3 ) . Here (E± i , hi), i = 1, 2, are the generators associated with the two simple roots αi, i = 1, 2 of the complexification of su(3), and E± 3 = ±[E± 1 , E ± 2 ]. We denote the fundamental weights by µi with i = 1, 2. A direct computation with equation (58) shows that the functions ψnn1,n2,n3 (θ, ξ, ϕ) = √ (n + 2)! 2n1!n2!n3! sinn1+n2 θ cosn3 θ cosn1 ξ sinn2 ξ ei(n1ϕ1+n2ϕ2+n3ϕ3) , n1 + n2 + n3 = n span the representation with highest weight ∣∣nµ1 〉 , while the functions ψ n1,n2,n3 n span the representation with highest weight ∣∣nµ2 〉 . These highest weights are explicitly given by 〈 θ, ξ, ϕ1, ϕ2, ϕ3 ∣∣nµ1 〉 = ψnn,0,0(θ, ξ, ϕ1, ϕ2, ϕ3) = √ (n+ 2)! 2n! sinn θ cosn ξeinϕ1 , 〈 θ, ξ, ϕ1, ϕ2, ϕ3 ∣∣nµ2 〉 = ψ 0,0,n n (θ, ξ, ϕ1, ϕ2, ϕ3) = √ (n+ 2)! 2n! cos θne−inϕ3 . Furthermore, (ψnn1,n2,n3 , ψmm1,m2,m3 ) = δmn δ n1 m1 δn2 m2 δn3 m3 , (ψ n1,n2,n3 n , ψmm1,m2,m3 ) = 0 . The representations of highest weight ∣∣nµ1 +mµ2 〉 , denoted by Dn,m, are obtained from the highest weight ψn,mn,0,0;0,0,m = √ (n+m+ 2)! 2n!m! sinn θ cosn ξ cosm θei(nϕ1−mϕ2) , 36 which enables us to obtain the full representation from the operators E− i , where i = 1, 2. The functions ψn,mn1,n2,n3;m1,m2,m3 that span the representation Dn,m (which can alternatively be obtained by combining ψnn1,n2,n3 with ψ m1,m2,m3 m and substracting the trace). We now identify the minimal set of indices to characterise any function. From Section 3.3, each representation is characterised by two numbers. We denote by Dn,m the representation of highest weight nµ1 + mµ2. Then three internal labels are needed to distinguished elements inside each representation space. The inner states are distinguished by the eigenvalues of h1, h2 as well as the value of the Casimir operator of the su(2)-subalgebra generated by E± 1 , h1 that we take equal to Q = 1 4 h21 + 1 2 (E+ 1 E − 1 + E− 1 E + 1 ) . Then for n,m ∈ N, the representation space reduces to Dn,m = { ψn,m,n1,n2,ℓ, n1, n2 s.t. n1µ1 + n2µ2 is a weight, 0 ≤ ℓ ≤ 1 2 (n +m) } and the Hilbert basis of S5 adapted to our construction is defined by B = { ψn,m,n1,n2,ℓ, n,m ∈ N, n1, n2, s.t. n1µ1 + n2µ2 is a weight, 0 ≤ ℓ ≤ 1 2 (n +m) } . (59) In the notation above, n, m correspond to the representation Dn,m, n1, n2 are the eigenvalues of h1, h2 and ℓ is the eigenvalue of the additional internal label Q. In order to obtain the conjugacy relation, we observe that we have the following normalisation for the fundamental and anti-fundamental representations D1,0 = {√ 3z1, √ 3z2, √ 3z3 } , D1,0 = D1,0 = {√ 3z̄3,− √ 3z̄2, √ 3z̄3 } , (60) and thus ψ n,m,n1,n2,ℓ (θ, ξ, ϕ1, ϕ2, ϕ3) = (−1) 1 3 (n−m−n1+n2)ψm,n,−n1,−n2,ℓ(θ, ξ, ϕ1, ϕ2, ϕ3) . Then we define the Lie algebra ĝ ( SU(3)/SU(2) ) by introducing Ta,n,m,n1,n2,ℓ = Taψn,m,n1,n2,ℓ. We identify the maximal set of commuting operators by observing that relations (58) can be extended to define a differential realisation of the Lie algebra so(6) ⊃ su(3). Within this differential realisation, all the representations Dn,m are rearranged into representations of so(6) corresponding to symmetric traceless tensors. The two constructions based on S5 = SU(3)/SU(2) or on S5 = SO(6)/SO(5) lead to isomorphic algebras in straight analogy with Sections 5.1 and 5.3. We will not use the representations of so(6) to build the generalised Kac-Moody algebra. The Hermitian operators are taken to be h1, h2 and h = −i ( ∂ ∂ϕ1 + ∂ ∂ϕ2 + ∂ ∂ϕ3 ) , (61) the latter being associated to the Cartan subalgebra of so(6), and we have hψn,m,n1,n2,ℓ = (n−m)ψn,m,n1,n2,ℓ . 37 We finally introduce the associated central charge k1, k2, k (see (15) and (16)). The Lie brackets take the form (see (17)) [ Ta,n,m,n1,n2,ℓ, Ta′,n′,m′,n′ 1 ,n′ 2 ,ℓ′ ] = ifaa′ a′′cn,m,n1,n2,ℓ,n′,m′,n′ 1 ,n′ 2 ,ℓ′ n′′,m′′,n′′ 1 ,n′′ 2 ,ℓ′′Ta′′,n′′,m′′,n′′ 1 ,n′′ 2 ,ℓ′′ +(−1) 1 3 (n−m−n1+n2)(k1n2 + k2n ′ 1 + k(n′ −m′))× gaa′δnm′δmn′δn1+n′ 2 δn2+n′ 1 δℓℓ′ ,[ h1, Ta,n,m,n1,n2,ℓ ] = n1Ta,n,m,n1,n2,ℓ , (62)[ h2, Ta,n,m,n1,n2,ℓ ] = n2Ta,n,m,n1,n2,ℓ ,[ h, Ta,n,m,n1,n2,ℓ ] = (n−m)Ta,n,m,n1,n2,ℓ , Observe that, because of the identity Dn,m = Dm,n, the structure constants involve the normalisation term δmn′δnm′δn1+n′ 2 δn2+n′ 1 , and not δmm′δnn′δn1+n′ 1 δn2+n′ 2 as may be naively expected. Such a subtlety is only encountered when the Lie algebra g admits complex (i.e., neither real nor pseudo-real) representations, and is not present for the remaining examples studied in this article. The cIJ K coefficients are obtained from ψn,m,n1,n2,ℓψn′,m′,n′ 1 ,n′ 2 ,ℓ′ = ∑ λ(N,M, n, n′, m,m′, ℓ, ℓ′, L) ( n n1 m n2 ℓ n′ n′ 1 m′ n′ 2 ℓ′ ∣∣∣ N M n1+n′ 1 n2+n′ 2 L ) ×ψN,M,n1+n′ 1 ,n2+n′ 2 ,L. Here ( n n1 m n2 ℓ n′ n′ 1 m′ n′ 2 ℓ′ ∣∣∣ N M n1+n′ 1 n2+n′ 2 L ) are the Clebsch-Gordan coefficients of the decompo- sition Dn,m ⊗Dn′,m′ = ⊕ N,M DN,M . As before, the coefficients λ(N,M, n, n′, m,m′, ℓ, ℓ′, L) can be computed recursively. This construction can be extended naturally, along the same lines, to the generic coset space SU(n + 1)/SU(n) ∼= S2n+1. The only representations that contain the scalar repre- sentations with respect to the embedding SU(n − 1) ⊂ SU(n) are Dn,0,··· ,0,m (in the no- tations of [41]) and correspond to traceless tensor products of the fundamental and the anti-fundamental representations. It should be observed that the generalised Kac-Moody algebra that we obtain from SU(n− 1) ⊂ SU(n) is isomorphic to the construction from the coset space SO(2n+ 2)/SO(2n+ 1) ∼= S 2n+1. 5.5 Coset space G2/SU(3) Prior to the construction of parameterisations of the manifolds G2 and G2/SU(3), we briefly recall some fundamental properties of the exceptional Lie algebra g2 (see e.g. [59]). The Cartan matrix of g2 is given by ( 2 −1 −3 2 ) 38 and its simple roots and fundamental weights are given respectively by α1 = ∣∣2,−3 〉 , α2 = ∣∣− 1, 2 〉 , µ1 = ∣∣1, 0 〉 , µ2 = ∣∣0, 1 〉 . The representation of highest weight µ2 is seven-dimensional and real, explicitly: 〈 z ∣∣1,−2 〉 = z[1,−2] , 〈 z ∣∣− 1, 1 〉 = z[−1,1] , 〈 z ∣∣0, 1 〉 = z[0,1] ,〈 z ∣∣− 1, 2 〉 = z[−1,2] , 〈 z ∣∣1,−1 〉 = z[1,−1] , 〈 z ∣∣0,−1 〉 = z[0,−1] ,〈 z ∣∣0, 0 〉 = x0 . Because the representation is real, x0 is a real number, while z[1,−2], z[−1,1], z[0,1] are complex and conjugate to z[−1,2], z[1,−1], z[0,−1], respectively. A differential realisation of the algebra deduced from this representation is given by (see Chapter 10 in [60]) Eα1[2,−3] = z[1,−2]∂[−1,1] − z[1,−1]∂[−1,2] , E−α1[−2,3] = z[−1,1]∂[1,−2] − z[−1,2]∂[1,−1] , Eα2[−1,2] = z[0,1]∂[1,−1] − z[−1,1]∂[0,−1] + √ 2 ( x0∂[1,−2] − z[−1,2]∂0 ) , E−α2[1,−2] = z[0,−1]∂[−1,1] − z[1,−1]∂[0,1] + √ 2 ( x0∂[−1,2] − z[1,−2]∂0 ) , Eα1+α2[1,−1] = z[1,−2]∂[0,−1] − z[0,1]∂[−1,2] + √ 2 ( x0∂[−1,1] − z[1,−1]∂0 ) , E−α1−α2[−1,1] = z[−1,2]∂[0,1] − z[0,−1]∂[1,−2] + √ 2 ( x0∂[1,−1] − z[−1,1]∂0 ) , Eα1+2α2[0,1] = z[1,−1]∂[1,−2] − z[−1,2]∂[−1,1] + √ 2 ( x0∂[0,−1] − z[0,1]∂0 ) , (63) E−α1−2α2[0,1] = z[−1,1]∂[−1,2] − z[1,−2]∂[1,−1] + √ 2 ( x0∂[0,1] − z[0,−1]∂0 ) , Eα1+3α2[−1,3] = z[0,1]∂[1,−2] − z[−1,2]∂[0,−1] , E−α1−3α2[1,−3] = z[1,−2]∂[0,1] − z[0,−1]∂[−1,2] , E2α1+3α2[1,0] = z[0,1]∂[−1,1] − z[1,−1]∂[0,−1] , E−2α1−3α2[1,0] = z[−1,1]∂[0,1] − z[0,−1]∂[1,−1] , h1 = z[1,−2]∂[1,−2] − z[−1,1]∂[−1,1] − z[−1,2]∂[−1,2] + z[1,−1]∂[1,−1] , h2 = −2z[1,−2]∂[1,−2] + z[−1,1]∂[−1,1] + z[0,1]∂[0,1] +2z[−1,2]∂[−1,2] − z[1,−1]∂[1,−1] − z[0,−1]∂[0,−1] . Due to the embedding g2 ⊂ so(7), the quadratic form q(z) = x20 + 2z[1,−2]z[−1,2] + 2z[−1,1]z[1,−1] + 2z[0,1]z[0,−1] , is preserved i.e., for any element ξ in the realisation (63) of g2, we have ξ(q(z)) = 0. As a consequence, the representations with highest weight nµ2 can be described in terms of 39 nth-order polynomials in the variables z. In order to factor out those terms proportional to q(z), we establish that D0,n = {P ∈ Rn[x0, z[1,−2], z[−1,1], z[0,1], z[−1,2], z[1,−1], z[0,−1]] s.t. ∇2(P ) = 0} , (64) where ∇2 = ∂20 + 2∂[1,−2]∂[−1,2] + 2∂[−1,1∂[1,−1] + 2∂[0,1]∂[0,−1] , and Rn[x0, z[1,−2], z[−1,1], z[0,1], z[−1,2], z[1,−1], z[0,−1]] denotes the space of n th-order polynomials. For the embedding su(3) ⊂ g2, the branching rule for the adjoint representation (with highest weight µ1) of g2 is given by 14 ˜ = 8 ˜ ⊕ 6 ˜ . The adjoint representation is always real, whereas the two complex conjugate fundamental and anti-fundamental representations regroup into a six-dimensional real representation: 3 ˜ ⊕ 3 ˜ = 6 ˜ . (65) The embedding su(3) ⊂ g2 can be explicitly described as follows. Let β1 = 2α1 + 3α2 , β2 = −α1 , be the two simple roots of su(3), so that the generators of the subalgebra su(3) ⊂ g2 are expressed as Eβ1 = E2α1+3α2 , E−β1 = E−2α1−3α2 , Eβ2 = E−α1 , E−β2 = Eα1 , Eβ1+β2 = Eα1+3α2 , E−β1−β2 = E−α1−3α2 H1 = 2h1 + h2 , H2 = −h1 . By introducing also the fundamental weight, µ1 = 2 3 β1 + 1 3 β2 = α1 + 2α2 , µ2 = 1 3 β1 + 2 3 β2 = α2 , the generators of the coset g2/su(3) read as 3˜ 3˜ Eµ1 = Eα1+2α2 , Eµ2 = Eα2 , Eµ1−β1 = E−α1−α2 , Eµ2−β2 = Eα1+α2 , Eµ1−β1−β2 = E−α2 , Eµ2−β2−β1 = E−α1−2α2 . The next step in the construction is to derive a matrix representation. This can be easily done by means of the vectors Z = (z[−1,2], z[−1,1], z[0,1], z[1,−2], z[1,−1], z[0,−1], x0), ∂Z = (∂[1,−2], ∂[1,−1], ∂[0,−1], ∂[−1,2], ∂[−1,1], ∂[0,1], ∂0) t , 40 such that, to any (first-order) differential operator D of g2, we can associate the matrix M defined by D = ZM∂Z . The matrices Mγ associated to the roots γ of g2, as well as those h1, h2, corresponding to the Cartan subalgebra, can be constructed in a straightforward manner via this prescription. For the subalgebra su(3), these matrices reduce to Ti =   λi 0 0 0 −λi 0 0 0 0   , i = 1, . . . , 8 , where λi are the 3× 3 Gell-Mann matrices. For the coset g2/su(3) we have xiUi =   0 0 0 0 x5 + ix6 0 0 0 −x5 − ix6 0 0 0 0 −x3 − ix4 x1 + ix2 0 −x5 + ix6 −x3 + ix4 0 0 x5 − ix6 0 x1 − ix2 0 0 x3 − ix4 −x1 + ix2 0 0 0√ 2(x1 + ix2) √ 2(x3 + ix4) − √ 2(x5 + ix6) − √ 2(x1 − ix2) − √ 2(x3 − ix4) x3 + ix4 √ 2(x1 − ix2) −x1 − ix2 √ 2(x3 − ix4) 0 − √ 2(x5 − ix6) 0 − √ 2(x1 + ix2) 0 − √ 2(x3 + ix4) 0 √ 2(x5 + ix6)√ 2(x5 − ix6) 0   , with the matrices Ui defined by U1 = Mα2 −M−α2 , U2 = i(Mα2 +M−α2 ) , U3 = Mα1+α2 −M−α1−α2 , U4 = i(Mα1+α2 +M−α1−α2 ) , U5 = Mα1+2α2 −M−α1−2α2 , U6 = i(Mα1+2α2 +M−α1−2α2 ) . It should be observed that these matrices are not well adapted, because the representation is real. With equation (65), we consider the real basis X =   x[−1,2] = 1√ 2 (z[−1,2] + z[1,−2]) x[−1,1] = 1√ 2 (z[−1,1] + z[1,−1]) x[0,1] = 1√ 2 (z[0,1] + z[0,−1]) y[−1,2] = − i√ 2 (z[−1,2] − z[1,−2]) y[−1,1] = − i√ 2 (z[−1,1] − z[1,−1]) y[0,1] = − i√ 2 (z[0,1] − z[0,−1]) x0   . (66) 41 Over this basis, the generators of the su(3)-subalgebra take the form Si =   1 2 (λi − λi) i 2 (λi + λi) 0 − i 2 (λi + λi) 1 2 (λi − λi) 0 0 0 0   , while for the coset g2/su(3) the generators Vi are given by ixjVj =   0 − x6 − x4 0 − x5 − x3 2x2 x6 0 x2 − x5 0 x1 2x4 x4 − x2 0 − x3 − x1 0 −2x6 0 x5 x3 0 − x6 − x4 2x1 x5 0 x1 x6 0 − x2 −2x3 x3 − x1 0 x4 x2 0 2x5 −2x2 −2x4 2x6 −2x1 2x3 −2x5 0   . Given an appropriate real basis, we can construct a parameterisation of the manifold G2. From the coset space structure, we rewrite a matrix for G2 in the form G =M3M2 where M3 is a matrix that parameterises SU(3), and M2 a matrix that parameterises the factor space G2/SU(3). The matrix M3 can be directly obtained from Section 5.4: M3 =   1 2 (U + U) i 2 (U − U) 0 − i 2 (U − U) 1 2 (U + U) 0 0 0 1   . (67) The matrix M2 is constructed as follows. Take U1(θ) = eiθV1 and consider the specific point on G2/SU(3) given by U1(π). Next, introduce the matrix P = RR1R2R3R4R5, with the R−matrices being appropriate rotations: R angle −ϕ in the plane (x[−1,2], x[−1,1]), R1 angle −θ1 in the plane (x[−1,1], x[0,1]), R2 angle −θ2 in the plane (x[0,1], y[−1,,2]), R3 angle −θ3 in the plane (y[−1,,2], y[−1,1]), R4 angle −θ4 in the plane (y[−1,1], y[0,1]), and R5 angle −1/2θ5 in the plane (y[0,1], x0) with the notations of (66) and define M2 by M2 = P−1U1(π)P . The parameterisation of the manifold is thus given by G =M3P −1U1(π)P . (68) At a first glance, the matrix M2 seems not to be very illuminating. Fortunately, however, as follows from Section 3.2, only the last column of the matrix M2 will be relevant for the 42 harmonic analysis on G2/SU(3). It reduces to a very simple expression: M2,71 = sin θ1 sin θ2 sin θ3 sin θ4 sin θ5 sinϕ ≡ − i 2 (z1 − z̄1) , M2,72 = sin θ1 sin θ2 sin θ3 sin θ4 sin θ5 cosϕ ≡ 1 2 (z1 + z̄1) , M2,73 = cos θ1 sin θ2 sin θ3 sin θ4 sin θ5 ≡ 1 2 (z2 + z̄2) , M2,74 = cos θ2 sin θ3 sin θ4 sin θ5 ≡ − i 2 (z2 − z̄2) , (69) M2,75 = cos θ3 sin θ4 sin θ5 ≡ 1 2 (z3 + z̄3) , M2,76 = cos θ4 sin θ5 ≡ − i 2 (z3 − z̄3) , M2,77 = cos θ5 ≡ x0 , which is a parameterisation of the sphere S6 ∼= G2/SU(3) with 0 ≤ ϕ < 2π, 0 ≤ θi ≤ π, i = 1, · · · , 5, that corresponds to the usual spherical coordinates. Comparing with the approach of Appendix A, the sphere S 6 is parameterised by 0 ≤ ϕ ≤ 2π , −1 ≤ u1 = cos θ1 ≤ 1 , 0 ≤ u2 = 1 2 (θ2 − cos θ2 sin θ2) ≤ π 2 −2 3 ≤ u3 = −1 3 (sin2 θ3 cos θ3 + 2 cos θ3) ≤ 2 3 , 0 ≤ u4 = 1 8 ((3θ4 − 3 sin θ4 cos θ4 − 2 sin3 θ4 cos θ4) ≤ 3 8 π. − 8 15 ≤ u5 = − 8 15 cos θ5 − 4 15 sin θ25 cos θ5 − 1 5 sin θ45 cos θ5 ≤ 8 15 . If we define the scalar product on S6 by (f, g) = 15 16π3 π∫ 0 dθ5 sin 5 θ5 π∫ 0 dθ4 sin 4 θ4 π∫ 0 dθ3 sin 3 θ3 π∫ 0 dθ2 sin 2 θ2 π∫ 0 dθ1 sin θ1 2π∫ 0 dϕ f(θ1, θ2, θ3, θ4, θ5, ϕ) g(θ1, θ2, θ3, θ4, θ5, ϕ), without loss of generality (e.g. after having conjugated the matrix M2 by an appropriate permutation) we can introduce the harmonic functions Φ1;1,−2 = √ 7 4 ( M2,72 + iM2,71 ) = − √ 7 2 z[−1,2] = √ 7 2 eiϕ sin θ1 sin θ2 sin θ3 sin θ4 sin θ5 , Φ1;−1,2 = √ 7 4 ( M2,72 − iM2,71 ) = √ 7 2 z[1,−2] 43 = √ 7 2 e−iϕ sin θ1 sin θ2 sin θ3 sin θ4 sin θ5 , Φ1;−1,1 = √ 7 4 ( M2,73 + iM2,74 ) = − √ 7 2 z[−1,1] = √ 7 2 sin θ3 sin θ4 sin θ5 ( cos θ1 sin θ2 + i cos θ2 ) , Φ1;1,−1 = √ 7 4 ( M2,73 − iM2,74 ) = − √ 7 2 z[1,−1] (70) = √ 7 2 sin θ3 sin θ4 sin θ5 ( cos θ1 sin θ2 − i cos θ2 ) , Φ1;0,1 = √ 7 4 ( M2,75 + iM2,76 ) = √ 7 2 z[0,1] = √ 7 2 sin θ5 ( cos θ3 sin θ4 + i cos θ4 ) , Φ1;0,−1 = √ 7 4 ( M2,75 − iM2,76 ) = − √ 7 2 z[0,−1] = √ 7 2 sin θ5 ( cos θ3 sin θ4 − i cos θ4 ) , Φ1;0,0 = √ 7M2,77 = √ 7x0 = √ 7 cos θ5, which are orthonormal with respect to the scalar product on S6. The precise signs are obtained from (63). These functions parameterise the representation D0,1. The highest weight of the representation D0,n (see equation (64)) is therefore given by Φαn,0,n = √ 1 60 1 4n (2n+ 5)! n!(n + 2)! zn[0,1] = √ 1 60 1 4n (2n+ 5)! n!(n + 2)! sin θn5 ( cos θ3 sin θ4 + i cos θ4 )n , and D0,n is constructed by the action of the operators given in equation (63). Only at the very end, we substitute equation (70) into the nth-order polynomials of D0,n, given in equation (64), to obtain the corresponding harmonic functions. In contrast to the previous cases, for G2 we have a degeneracy problem. According to Proposition 3.2, we need 6 internal labels to separate states within an irreducible repre- sentation of G2, the Casimir operators of G2 being used to characterise the representation. Considering the reduction chain G2 ⊃ SU(3) ⊃ SU(2) ⊃ U(1), (71) provides us with five internal labels, namely the Casimir operators of SU(3) and SU(2), as well as the generators of the Cartan subalgebra. It is thus necessary to consider an additional label. This operator can be constructed by the method of elementary multiplets (see [61]) observing that the adjoint representation of G2 decomposes as the direct sum of an octet T and two conjugate triplets V, V of SU(3). The simplest labelling problem resulting from 44 this method is a cubic operator TV V in the generators of G2, such that in each monomial one generator belongs to the octet and each of the triples, respectively. This operator is Hermitian and commutes with the elements of SU(3) [62]. It is worthy to be observed that the sphere can be obtained in two different ways, either as the coset space S6 = G2/SU(3), or alternatively as S6 = SO(7)/SO(6). On the other hand, as the subduced representations Dn,0,0 of SO(7) are isomorphic to the representation representation D0,n of G2 (see e.g. [63]), it follows that the harmonic functions on G2/SU(3) are the same as the harmonic functions on SO(7)/SO(6), except that the former are labeled with the quantum numbers of G2, whereas the latter are labeled by the quantum numbers of SO(7). This, in particular, implies that we have the Lie algebra isomorphism ĝ ( G2/SU(3) ) ∼= ĝ ( SO(7)/SO(6) ) . This enables us to construct harmonic functions on the sphere S6 using two alternative ways, either using the representation theory of g2 or the representation theory of so(7). In the second case, we can extend the differential realisation of g2 given in (63) to a differential realisation of so(7). Moreover all representations D0,n turn out to be representations of so(7) corresponding to symmetric traceless tensors. The differential realisation of so(7) is given on page 407 in [41]. It is not useful to reproduce the expression for all the generators of so(7), but only for the Cartan subalgebra: h1 = −i ∂ ∂ϕ , h2 = −i ( cot θ2 sin θ1 ∂ ∂θ1 − cos θ1 ∂ ∂θ2 ) , (72) h3 = −i ( cot θ4 sin θ3 ∂ ∂θ3 − cos θ3 ∂ ∂θ4 ) . According to Proposition 3.2, we need nine internal labels. Actually, as we merely con- sider symmetric traceless tensors in fact, only 6 labels are required [64]. The reduction chain 5 SO(7) ⊃ SO(5) ⊃ SO(3) , provides three additional operators, namely the two Casimir operators of SO(5) and the Casimir operator of SO(3). Thus with the generators of the Cartan subalgebra we have identified six labels. We introduce the vector representation (with the notations of (69)) D1,0,0 = { Ψ1,1,0,0 = √ 7 2 z1,Ψ1,−1,0,0 = − √ 7 2 z̄1,Ψ1,0,1,0 = √ 7 2 z2,Ψ1,0,−1,0 = √ 7 2 z̄2, Ψ1,0,0,1 = √ 7 2 z3,Ψ1,0,0,−1 = − √ 7 2 z3,Ψ1,0,0,0 = √ 7x0 } , (73) 5As in Section 5.3, the set of Gegenbauer polynomials obtained from the reduction chain SO(7) ⊃ SO(6) ⊃ SO(5) ⊃ SO(4) ⊃ SO(3) ⊃ SO(2) of S6 does not constitute an adapted set of harmonic functions in our case. This will further hold for all n−spheres. 45 obtained explicitly (signs included) from the differential realisation of so(7). The first label is associated to the vector representation D1,0,0, whilst the last three indices correspond to the eigenvalues of the Cartan subalgebra. In a similar manner, with the highest weight vector of the representation Dn,0,0, n ∈ N being given by Ψn,n,0,0 = √ 1 60 1 4n (2n+ 5)! n!(n + 2)! zn1 = √ 1 60 1 4n (2n+ 5)! n!(n + 2)! einϕ sinn θ1 sin n θ2 sin n θ3 sin n θ4 sin n θ5 , the representation Dn,0,0 can be easily obtained. The labels introduced previously enables us to determine an adapted Hilbert basis: B = { Ψn,m1,m2,m3,ℓ1,ℓ2,ℓ3 , n ∈ N } . (74) In this notation, the first index corresponds to the representation Dn,0,0, the three last indices to the eigenvalues of the Cartan subalgebra of so(7) and the remaining indices to the additional internal labels. From (73) we have the conjugacy relation Ψ̄n,m1,m2,m3,ℓ1,ℓ2,ℓ3 = (−1)n1+n2Ψn,−m1,−m2,−m3,ℓ1,ℓ2,ℓ3 . The generators of ĝ(SO(7)/SO(6)) are then given by Ta,n,m1,m2,m3,ℓ1,ℓ2,ℓ3 = TaΨn,m1,m2,m3,ℓ1,ℓ2,ℓ3, the Hermitian operators (72). The corresponding 5−forms are thus γ1 = −ik1 dθ1 ∧ dθ2 ∧ dθ3 ∧ dθ4 ∧ dθ5 sin 5 θ5 sin 4 θ4 sin 3 θ3 sin 2 θ2 sin θ1 , γ2 = −ik2 ( − cot θ2 sin θ1dϕ ∧ dθ2 ∧ dθ3 ∧ dθ4 ∧ dθ5 − cos θ1dϕ ∧ dθ1 ∧ dθ3 ∧ dθ4 ∧ dθ5 ) × sin5 θ5 sin 4 θ4 sin 3 θ3 sin 2 θ2 sin θ1 γ3 = −ik3 ( − cot θ4 sin θ3dϕ ∧ dθ1 ∧ dθ2 ∧ dθ4 ∧ dθ5 − cos θ3dϕ ∧ dθ1 ∧ dθ2 ∧ dθ3 ∧ dθ5 ) × sin5 θ5 sin 4 θ4 sin 3 θ3 sin 2 θ2 sin θ1 and the associated central charges are noted k1, k2, k3. The Lie brackets take the form (see (17) [ Ta,n,m1,m2,m3,ℓ1,ℓ2,ℓ3, Ta′,n′,m′ 1 ,m′ 2 ,m′ 3 ,ℓ′ 1 ,ℓ′ 2 ,ℓ′ 3 ] = ifaa′ a′′ca,n,m1,m2,m3,ℓ1,ℓ2,ℓ3,a′,n′,m′ 1 ,m′ 2 ,m′ 3 ,ℓ′ 1 ,ℓ′ 2 ,ℓ′ 3 a′′,n′′,m′′ 1 ,m′′ 2 ,m′′ 3 ,ℓ′′ 1 ,ℓ′′ 2 ,ℓ′′ 3Ta′′,n′′,m′′ 1 ,m′′ 2 ,m′′ 3 ,ℓ′′ 1 ,ℓ′′ 2 ,ℓ′′ 3 +(−1)m1+n2(k1n ′ 1 + k2n ′ 2 + k3n ′ 3)gaa′δn,n′δℓ1,ℓ′1δℓ2,ℓ′2δℓ3,ℓ′3δm1+m′ 1 δm2+m′ 2 δm3+m′ 3 ,[ h1, Ta,n,m,n1,n2,ℓ ] = n1Ta,n,m,n1,n2,ℓ , (75)[ h2, Ta,n,m,n1,n2,ℓ ] = n2Ta,n,m,n1,n2,ℓ ,[ h3, Ta,n,m,n1,n2,ℓ ] = n3Ta,n,m,n1,n2,ℓ , The cIJ K coefficients can be obtained either from the Clebsch-Gordan coefficients in the decomposition D0,n ⊗D0,n′ = ⊕ 0,N D0,N , 46 associated to G2, or using the decomposition Dn,0,0 ⊗Dn′,0,0 = ⊕ N,0,0 DN,0,0 , corresponding to SO(7). A question that arises naturally in this context is whether for the non-compact real form G2(2) of the exceptional algebra G2, a consistent construction can be obtained for G2(2)/SL(3,R), which can be seen as a non-compact pseudo-Riemannian version of the non- symmetric coset G2/SU(3). This case is of great physical relevance due to its relation with the super-Ehlers embedding of minimal supergravity without matter coupling in 4+1 space- time dimensions (see [65]). 6 Concluding remarks We have considered a notion of generalised Kac-Moody algebras based on the set of smooth maps from an n−dimensional compact manifold M (associated to a compact Lie group Gc) to a real or complex Lie group G, and studied the conditions that ensure that such generalisations admit central extensions, that have been denoted ĝ(M). From this point of view, it turns out that the harmonic analysis on the manifoldM as well as the representation theory of Gc constitute a crucial ingredient to properly express the commutators in the generalised Kac-Moody algebra. We also observed that the non-centrally extended algebras g(M) can be obtained natu- rally from a (4+n)-dimensional Kaluza-Klein theory compactified on the compact manifold M, from which it is easily deduced that to any unitary representation of g there corresponds a uniquely determined unitary representation of g(M). The converse of this assertion also holds. This correspondence suggests to try an extrapolation of the condition obtained for M = Tr to the general case ĝ(M), for the case of non-vanishing central charges. This may provide an alternative tool to inspect highest weight unitary representations. In Section 2 we have seen that considering the set Diff(M) of vector fields on M, it is possible to define an algebra with a semidirect product structure Diff(M)⋉g(M), in analogy with the commutator structure described by equation (6). At this point, one may wonder whether the centrally extended algebra ĝ(M) is compatible with Diff(M). In this context, it turns out that the compatibility condition can be expressed in terms of the two-cocycles associated to the central extensions, leading to the constraint ω(L ·X, Y ) + ω(X,L · Y ) = 0 ∀X, Y ∈ g(M), ∀L ∈ Diff(M) , where L · X denotes the natural action of Diff(M) on g(M). Compatibility in the latter sense was discussed in [11, 30, 66]. On a different footing, and in the context of bosonic membranes, central extensions of Diff(S1 × S 1) and of Diff(S2) have been studied by several authors (see e.g. [67, 68, 69] and references therein). A question that remains currently unanswered is whether the symmetric nature of the coset manifolds has any consequences for the structural properties of the generalised Kac–Moody algebras. Albeit it seems that the answer is in the negative, as can be suspected from the examples presented, a definitive 47 answer requires a more detailed analysis, as well as a careful comparison with other examples, possibly in higher ranks. We hope to provide more evidence in this respect in future work. Several additional possibilities emerge from the generic approach described in this paper, such as the problem whether this notion of generalised Kac-Moody algebra can be applied and leads to useful insights in the description of extended objects, such as, for example, those aris- ing in the framework ofM−theory (M2− orM5−branes) [70]. The extension of these results to the non-compact case via the formalism provided by the Plancherel formula is certainly a problem worthy to be considered in detail, not only because of its geometrical significance, but also due to its current physical applications. In this context, it could be suggested that (super)membrane solutions of extended theories of (super)gravity in higher dimensions might be related to the various central extensions of the generalised Kac-Moody algebras introduced in the manuscript, likeM2− andM5−branes are central extensions of the N = 1, D = 10+1 M-theory superalgebra. An eventual extension of the generalised Kac-Moody algebras to non-compact Lie groups or non-compact (and possibly pseudo-Riemannian) coset manifolds (e.g. non-Euclidean tori) could then be associated to exotic versions of the M−theory, such as the M∗-theory of the M ′-theory [71]. However, as commented in Section 5.2, the exten- sion of this work to manifolds involving non-compact groups present subtleties that require additional techniques to surmount the difficulties posed by the non-compacity, the details of which have not yet been solved in fully satisfactory manner, but that warrant further investigation. As a final observation, also of physical interest, we point out that the motivation of the algebras g(M) and ĝ(M) in terms of current algebras is an aspect that deserves to be analysed more in detail, considering for instance specific fields, as it may lead to some concrete realisations of ĝ(M). Work in this direction is currently in progress. A Some identities Let M be a an n = p+q−dimensional compact real manifold of volume V with parametrisa- tion yA = (ϕi, ur) = (ϕ1, · · · , ϕp, u1, · · · , uq). Recall that there are two types of parameters. Angles ϕ1, · · · , ϕp such that functions on M are periodic in all ϕ−directions, as well as parameters, u1, · · · , up that do not correspond to angles and such that the functions on M are not periodic in all the u−directions. (For instance, for the sphere S2, the two parameters are the angle 0 ≤ ϕ < 2π and the parameter −1 ≤ u = cos θ ≤ 1.) From the integration measure, we can write ∫ M dµ(M) = 1 V ∫ M dpϕ dqu = 1 , and let B = { ρI(ϕ, u) , I ∈ I } , where I is a countable set (see Section 3.3), be a orthonormal Hilbert basis of L2(M). Assume further that all functions are bounded. Since B is a complete orthonormal basis we have ∫ M dµ(M) ρI(ϕ, u) ρJ(ϕ, u) = δIJ 48 and ρI(ϕ, u)ρ I(ϕ′, u′) = δp(ϕ− ϕ′)δq(u− u′) (76) (the sum over repeated indices is implicit). Since the functions are bounded and B is a complete Hilbert basis we have on the one hand ρI(ϕ, u) = ηIJρJ(ϕ, u) , ρI(ϕ, u) = ηIJρ J(ϕ, u) , with ηIJηJK = δIK , and on the other hand ρI(ϕ, u)ρJ(ϕ, u) = cIJ KρK(ϕ, u) , (77) ρI(ϕ, u)ρJ(ϕ, u) = cIJKρ K(ϕ, u) , where cIJK = cIJK = ηILηJMηKNcLM N . We now assume that M is either Gc of Gc/H , where Gc is a compact Lie group and H ⊂ Gc, so that the coefficients cIJ K can be expressed by means of Clebsch-Gordan coefficients. By using the standard Hilbert basis {∣∣I 〉 , I ∈ I } corresponding to all unitary representation of Gc, with the notations of Section 3.3, and setting ρI(ϕ, u) = 〈 ϕ, u ∣∣I 〉 , we can extend the usual techniques of quantum mechanics for the composition of spherical harmonics with Gc = SU(2) to other groups Gc. We thus obtain the relations { ρI(ϕ, u)ρJ(ϕ, u) = cIJ KρK(ϕ, u) , ρK(ϕ, u) = cIJKρI(ϕ, u)ρJ(ϕ, u) , and cIJ KcIJL = δKL , cIJ KcLMK = δLI δ M J . B Missing label operators As observed, it may be convenient to describe the representations of a semisimple Lie algebra g with respect to some distinguished (semisimple) subalgebra g′ that may correspond to an internal symmetry. The question that arises is whether in such a description the labels are sufficient to separate the degeneracies that may appear. This is known as the ‘internal labelling problem’ (see e.g. [72, 62]). The subalgebra g′ provides 1 2 (dim g′+ℓ′)) labels, where 49 it may happen that g′ and g have some Casimir operator in common. Therefore, subtracting the number ℓ0 of such common functions, we still need n0 = 1 2 (dim g− ℓ− dim g′ − ℓ′) + ℓ0 operators to separate the irreducible representations of g′ that appear with multiplicity greater than one in the decomposition of D. Such operators must necessarily commute with the generators of g′, and are commonly called ‘missing label operators’ or ‘subgroup scalars’. In order to prevent undesired interactions and to allow simultaneous diagonalisation, these operators are additionally required to commute with each other [61]. Among the various approaches, differential operators constitute a convenient procedure to determine internal labelling operators [34, 73, 74]: Given a Lie algebra g with generators {X1, . . . , Xn} and commutators 6 [Xi, Xj ] = fij kXk, the generators Xi are realised as differential operators in the space C∞ (g∗) by: X̂i = −fijkxk ∂ ∂xj , where {x1, . . . , xn} are the coordinates of a covector in a dual basis of {X1, . . . , Xn}. The invariants of g correspond to solutions of the system of partial differential equations: X̂iF = 0, 1 ≤ i ≤ n , with the number N (g) of independent solutions given by the formula N (g) := dim g− supx1,...,xnrank (A(g)) , where A(g) = ( fij kxk ) corresponds to the functional matrix associated with the commutator table of g over the given basis. For polynomial solutions, the standard symmetrisation map defined by Λ ( xi1 . . . xip ) = 1 p! ∑ σ∈Sp Xσ(i1) . . .Xσ(ip) with Sr the permutation group with p elements, allows to recover the Casimir operators in their usual form as elements belonging to the centre of the enveloping algebra U(g) [75]. If g′ ⊂ g is an embedding of Lie algebras, it induces branching rules of representations [63]. In particular, the adjoint representation of g decomposes as: ad(g) = ad(g′)⊕ R , where R is a (completely reducible) representation of g′ called the characteristic representa- tion.7 In order to compute the missing labels analytically, we can proceed as follows. Let {X1, . . . , Xm} be a basis of g′ and extend it to a basis B = {X1, . . . , Xm, Y1, . . . Yn−m} of g. The brackets adopt the form: [Xi, Xj] = fij kXk , [Xi, Yp] = gip qYq , [Yp, Yq] = Epq kXk + Fpq rYr , 6Pay attention that there is no i factor in the Lie brackets, which is more convenient to identify the missing label operators. 7Complete reducibility is actually ensured only if the subalgebra g′ is semisimple. 50 where i, j, k ∈ {1, . . . , m} and p, q, r ∈ {1, . . . , n−m}. Now we consider those differen- tial operators that are associated to generators of g′, i.e., the system of partial differential equations X̂i = −fij kxk ∂ ∂xj − gip qyq ∂ ∂yp , 1 ≤ i ≤ m , (78) where {x1, . . . , xm, y1, . . . , yn−m} are the coordinates in a dual basis of B. We observe that solutions F to the system (78) such that ∂F ∂yp = 0 for all 1 ≤ p ≤ n −m correspond to the Casimir invariants of the subalgebra, while a genuine missing label must explicitly depend on the variables {y1, . . . , yn−m}. Now the system (78) has exactly n− r′ independent solutions, where r′ denotes the rank of the m× n polynomial coefficient matrix. From these solutions, ℓ + ℓ′ − ℓ0 correspond to the Casimir operators of either g or g′, so that the number of available labelling operators is given by χ = n− r′ − ℓ− ℓ′ + ℓ0. It can be easily shown (see e.g. [73]) that m − r′ = ℓ0, which implies that χ = 2n0, showing that there are n0 more labels available than required. It should however be noted that among these 2n0 solutions, at most n0 correspond to operators that commute with each other [76]. Once a complete set of dim g+ ℓ 2 labelling operators has been found, they can be simulta- neously diagonalised, from which an orthonormal basis of states for the representation D is obtained. A practical recipe for the orthonormalisation can be found e.g. in [76]. Acknowledgements. The authors thank P. Baseilhac, G. Bossard, E. Dudas, N. Mo- hammedi, M. Slupinski and specially P. 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