Convolution bialgebra of a Lie groupoid and transversal distributions

TL;DR

This paper constructs a convolution bialgebra associated with a Lie groupoid and its Lie algebroid, representing it in the algebra of transversal distributions, extending classical decomposition results.

Contribution

It introduces a new convolution bialgebra for Lie groupoids and represents it via transversal distributions, extending the Cartier-Gabriel decomposition.

Findings

Constructed the adjoint action of the etale Lie groupoid on the Lie algebroid.

Formed a convolution C_c(M)/R-bialgebra associated with the groupoid and algebroid.

Extended classical decomposition of distributions on Lie groups.

Abstract

For a Lie groupoid G over a smooth manifold M we construct the adjoint action of the etale Lie groupoid G# of germs of local bisections of G on the Lie algebroid g of G. With this action, we form the associated convolution C_c(M)/R-bialgebra C_c(G#,g). We represent this C_c(M)/R-bialgebra in the algebra of transversal distributions on G. This construction extends the Cartier-Gabriel decomposition of the Hopf algebra of distributions with finite support on a Lie group.