Lie stack deformations
A Lie stack is an algebra morphism \({\mathfrak s}: A \to A\otimes B\) where \(A\), \(B\) are finite dimensional \(\mathbb{C}\)-algebras with \(B\) augmented local [ cf. \textit{L. Le Bruyn}, Adv. Math. 130, 103-135 (1997), Commun. Algebra 25, 2211-2226 (1997)]. Given a Lie stack \({\mathfrak s}\), one can construct its enveloping algebra \(U({\mathfrak s})\), a Hopf algebra with trivial coradical. The author is concerned in this paper with a class of Lie stacks \({\mathfrak s}_{d}: A \to A\otimes B\), where \(A\) is the path algebra of a connected quiver \(Q\) without oriented cycles, divided out by the two-sided ideal generated by the arrows; \(B\) is an arbitrary augmented local algebra of dimension \(n +1\), say \(B = \mathbb C 1\oplus M\) with \(M = \mathbb{C} b_{1} \oplus \dots \oplus b_{n}\) is the maximal ideal; \(d: M \to \text{Der }A\) is an embedding of the abelian Lie algebra \({\mathfrak M}\) and \({\mathfrak s}_{d}\) is given by \({\mathfrak s}_{d}(a) = a\otimes 1 + \sum_{i= 1}^{n} d(b_{i}) (a) \otimes b_{i}\). These stacks are degenerations of stacks \({\mathfrak s}_{g}\) associated to a Lie algebra \(g \subseteq \text{Der } A\). The author studies the algebra structure of \(U({\mathfrak s}_{d})\) and gives conditions for \(U(s_{d})\) to be commutative or cocommutative. As a consequence of his investigations, the author exhibits noncocommutative comultiplications on certain enveloping algebras of Lie algebras.
scientific article; zbMATH DE number 1088598