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Homogeneous algebraic distributions

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Castrillón López, Marco and Muñoz Masqué, Jaime (2001) Homogeneous algebraic distributions. Rocky Mountain Journal of Mathematics, 31 (1). pp. 57-75. ISSN 0035-7596

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Official URL: http://projecteuclid.org/euclid.rmjm/1181070239




Abstract

Let 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.


Item Type:Article
Uncontrolled Keywords:Adjoint bundle; algebraic morphism of vector bundles; algebraic vector field; involutive distribution; gauge algebra; linear representation; Lie group bundle; Liouville's vector field.
Subjects:Sciences > Mathematics > Differential geometry
ID Code:24262
Deposited On:22 Jan 2014 18:26
Last Modified:12 Dec 2018 15:13

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