Lie stacks and their enveloping algebras

Abstract

A Lie stack is an algebra morphisms:A→A⊗BwhereAandBare finite dimensional C-algebras withBbeing augmented local. We construct the enveloping algebraU(s) of a Lie stack and show that it is an irreducible Hopf algebra domain with a Poincaré-Birkhoff-Witt basis. We recover the enveloping algebrasU(g) of Lie algebras as special instances. We give conditions such thatU(s) is neither commutative nor cocommutative and we give such examples forBbeing any (non-commutative) augmented local algebra, and for beingAthe 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. © 1997 Academic Press.