We investigate maximal tori in the Hochschild cohomology Lie algebra HH1(A) of a finite dimensional algebra A, and their connection with the fundamental groups associated to presentations of A. We prove that every maximal torus in HH1(A) arises as the dual of some fundamental group of A, extending the work by Farkas, Green, and Marcos; de la Peña and Saorín; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of A is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras.
Maximal Tori in HH1 and the Fundamental Group
Rubio y Degrassi L.
2023
Abstract
We investigate maximal tori in the Hochschild cohomology Lie algebra HH1(A) of a finite dimensional algebra A, and their connection with the fundamental groups associated to presentations of A. We prove that every maximal torus in HH1(A) arises as the dual of some fundamental group of A, extending the work by Farkas, Green, and Marcos; de la Peña and Saorín; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of A is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras.
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