Symmetries of algebras captured by actions of weak Hopf algebras

TL;DR

This paper generalizes classical symmetry concepts of algebras by introducing a framework that captures symmetries via actions of objects in monoidal categories, including weak Hopf algebras, broadening the scope of algebraic symmetry analysis.

Contribution

It introduces the object Sym_{}(A) that captures algebra symmetries through category actions, extending classical group and Lie algebra symmetries to weak Hopf algebra actions.

Findings

Symmetries of -algebras are captured by weak Hopf algebra actions.

The framework applies to categories including groupoids and Lie algebroids.

Positively graded non-connected algebras exhibit symmetries within the weak Hopf framework.

Abstract

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 $\Bbbk$-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 object $\operatorname{Sym}_{\mathcal{C}}(A)$ 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.