Abstract
AbstractIn this paper the class of Brauer graph algebras is proved to be closed under derived equivalence. For that we use the rank of the maximal torus of the identity component $$Out^0(A)$$ O u t 0 ( A ) of the group of outer automorphisms of a symmetric stably biserial algebra A.
Highlights
IntroductionBrauer graph algebras or equivalently symmetric special biserial algebras, originating from modular representation theory, are studied quite extensively
Brauer graph algebras or equivalently symmetric special biserial algebras, originating from modular representation theory, are studied quite extensively. They appear in classifications of various classes of algebras including blocks with cyclic or dihedral defect groups [14,15], blocks of Hecke algebras [9,10] and others
In [8], we revised the proof of the fact that the only algebras possibly stably equivalent to self-injective special biserial algebras are self-injective stably biserial
Summary
Brauer graph algebras or equivalently symmetric special biserial algebras, originating from modular representation theory, are studied quite extensively. In this paper we make a final step in the proof of the fact that Brauer graph algebras are closed under derived equivalence. 4, we revisit the known derived invariants for Brauer graph algebras [3,4,5,6] in arbitrary characteristic, providing simpler proofs of their invariance for the larger class of symmetric stably biserial algebras and correcting some inaccuracies in the existing literature. As a corollary of Theorems 1.1 and 1.2 and the fact that Brauer graph algebras can be derived equivalent only to symmetric stably biserial algebras [8] we obtain the following: Corollary 1.3 The class of Brauer graph algebras is closed under derived equivalence. Note that the list of invariants from Theorem 1.2 is crucial to the forthcoming joint work [26] of Opper and the second named author, where a complete classification of Brauer graph algebras up to derived equivalence will be provided
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
