Lie stacks and their enveloping algebras

A Lie stack is an algebra morphism s: A --> A x B where A and B are finite dimensional C-algebras with B being augmented local. We construct the enveloping algebra U(s) of a Lie stack and show that it is an irreducible Hopf algebra domain with a Poincare-Birkhoff-Witt basis. We recover the enveloping algebras U(g) of Lie algebras as special instances. We give conditions such that U(s) is neither commutative nor cocommutative and we give such examples for B being any (non-commutative) augmented local algebra, and for being A the path algebra of a suitable bipartite quiver. By studying orbit closes in the variety of Lie stacks of fixed dimension one obtains in this way deformations of enveloping algebras of Lie algebras. (C) 1997 Academic Press.