Symmetries of algebras captured by actions of weak Hopf algebras

Abstract

In this paper, we present a generalization of well-established results regarding symmetries of 𝕜-algebras, where 𝕜 is a field. Traditionally, for a 𝕜-algebra A, the group 𝕜-algebra automorphisms of A captures the symmetries of A via group actions. Similarly, the Lie algebra of derivations of A captures the symmetries of A via Lie algebra actions. In this paper, given a category C whose objects possess 𝕜-linear monoidal categories of modules, we introduce an object SymC (A) that captures the symmetries of A via actions of objects in C. Our study encompasses various categories whose objects include groupoids, Lie algebroids, and more generally, cocommutative weak Hopf algebras. Notably, we demonstrate that for a positively graded non-connected 𝕜-algebra A, some of its symmetries are naturally captured within the weak Hopf framework. 2020 Mathematics Subject Classification. 18B40, 16T05, 18M05.