Symmetries of Algebras Captured by Actions of Weak Hopf Algebras

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