Erdogan Esin | Searcharxiv

arXiv2013

Prolongations of Lie Algebra Representations

In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we define prolongations of representations of Lie algebras. We show that if a Lie algebra representation corresponds to a Lie group representation, then prolongation of Lie algebra representation corresponds to the prolonged Lie group representation.

math.DG↗

arXiv2013

On the Prolongations of Representations of Lie Groups

In this paper, we introduce a study of prolongations of representations of Lie groups. We obtain a faithful (one-to-one) representation of TG where G is a finite-dimensional Lie group and TG is the tangent bundle of G, by using (not necessarily faithful) representations of G. We show that tangent functions of Lie group actions correspond to prolonged representations. We prove that if two representations are equivalent, then their prolongations are equivalent too. We show that if U is an invariant subspace for a representation, then TU is an invariant subspace for the prolongation of the given representation and vice versa. We prove that if the prolongation of Φis an irreducible representation, then Φis also an irreducible representation. Finally we show that prolongations of direct sum of two representations are direct sum of their prolongations.

math.DG↗