Convolution bialgebra of a Lie groupoid and transversal distributions
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 Cc(M)/R-bialgebra Cc(G#,g). We represent this Cc(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.
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