Castrillón López, MarcoMuñoz Masqué, Jaime2023-06-202023-06-2020010035-759610.1216/rmjm/1008959668https://hdl.handle.net/20.500.14352/58902Let p:E→M be a vector bundle of dimension n+m and (xλ,yi), λ=1,…,n, i=1,…,m, be fibre coordinates. A vertical vector field X on E is said to be algebraic [respectively, algebraic homogeneous of degree d] if its coordinate expression is of the type X=∑mi=1Pi∂/∂yi, where Pi are polynomials [respectively, homogeneous polynomials of degree d] in coordinates yi. A vertical distribution over E is said to be algebraic [respectively, homogeneous algebraic of degree d] if all local generators are homogeneous algebraic [respectively, homogeneous algebraic of the same degree d] vector fields. It is proved that a vertical distribution locally spanned by vector fields X1,…,Xr is homogeneous algebraic of degree d if and only if an r×r matrix A=(aij), aij∈C∞(E), exists which is equal to d−1 times the identity matrix along the zero section of E, and such that [χ,Xj]=∑ri=1aijXi, j=1,…,r, where χ is the Liouville vector field.engjournal articlehttp://projecteuclid.org/euclid.rmjm/1181070239http://projecteuclid.org/euclid.rmjmopen access514.7Adjoint bundlealgebraic morphism of vector bundlesalgebraic vector fieldinvolutive distributiongauge algebralinear representationLie group bundleLiouville's vector field.Geometría diferencial1204.04 Geometría Diferencial