Convolution bialgebra of a Lie groupoid and transversal

For a Lie groupoid W over a smooth manifold M we construct the adjoint action of the etale Lie groupoid W# of germs of local bisections of W on the Lie algebroid g of W. With this action, we form the associated convolution C-c(infinity)(M)/R-bialgebra C-c(infinity)(W-#, g). We represent this C-c(infinity)(M)/R-bialgebra in the algebra of transversal distributions on W. This construction extends the Cartier-Gabriel decomposition of the Hopf algebra of distributions with finite support on a Lie group. (C) 2022 Elsevier B.V. All rights reserved.