Symmetries of Algebras Captured by Actions of Weak Hopf Algebras

2022/09/24 by Fabio Calderón, Hongdi Huang, Calderón, Fabio +5 · 1 citation
Mathematics · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Advanced Combinatorial Mathematics

paper · pdf · doi:10.1007/s10468-024-10295-5

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 C whose objects possess \Bbbk-linear monoidal categories of modules, we introduce an object Sym_C(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 \Bbbk-algebra A, some of its symmetries are naturally captured within the weak Hopf framework.

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