Symmetries of Algebras Captured by Actions of Weak Hopf Algebras

In this paper, we present a generalization of well-established results regarding symmetries of \(\Bbbk \)-algebras, where \(\Bbbk \) is a field. Traditionally, for a \(\Bbbk \)-algebra A, the group of \(\Bbbk \)-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 \(\mathcal {C}\) whose objects possess \(\Bbbk \)-linear monoidal categories of modules, we introduce an objec \(\operatorname {Sym}_{\mathcal {C}}(A)\) that captures the symmetries of A via actions of objects in \(\mathcal {C}\). 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 \(\Bbbk \)-algebra A, some of its symmetries are naturally captured within the weak Hopf framework.