Lie stacks and their enveloping algebras
The purpose of this clearly written article is to introduce the notion of ``Lie stack'' and to study the first general properties of this mathematical object. A Lie stack is an algebra morphism \({\mathfrak s}\colon A\to A\otimes B\) where \(A\) and \(B\) are finite dimensional (associative) \(\mathbb{C}\)-algebras with \(B\) augmented local. Let \(\mathfrak g\) be a finite dimensional Lie algebra with basis \(g_1,\dots,g_n\) and realize it as a Lie subalgebra of \(\text{Der }A\) for some finite dimensional associative algebra \(A\), e.g. by Ado's theorem. Let \(B=\mathbb{C}[y_1,\dots,y_n]/(y_iy_j)\); then \({\mathfrak s}\colon A\to A\otimes B\), \({\mathfrak s}(a)=a\otimes 1+\sum_i g_i(a)\otimes y_i\) is a Lie stack. So the notion of ``Lie stack'' extends in a way the notion of Lie algebra. To each Lie stack \(\mathfrak s\), the author attaches an irreducible pointed Hopf algebra \(U({\mathfrak s})\); if \(\mathfrak s\) comes from a Lie algebra \(\mathfrak g\) as above then \(U({\mathfrak s})\simeq U({\mathfrak g})\). Simple conditions on \(\mathfrak s\) are given which are equivalent to the commutativity, resp. cocommutativity, of \(U({\mathfrak s})\); as a consequence new examples of non-commutative non-cocommutative Hopf algebras are presented. Further, regularity (homological) properties of \(U({\mathfrak s})\) are investigated. It is shown that \(U({\mathfrak s})\) is a domain with a PBW basis; if \(\text{GKdim }U({\mathfrak s})\) is finite then \(U({\mathfrak s})\) is an Auslander regular noetherian domain satisfying the Cohen-Macaulay property. The author also explains one of his main motivations, namely to interpret Lie stacks as deformations of Lie algebras. For more information see [\textit{L. Le Bruyn}, J. Algebra 196, No. 2, 552-577 (1997; Zbl 0892.17013)].
scientific article; zbMATH DE number 1066078
scientific article