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The action of the Nakayama automorphism of a Frobenius algebra on Hochschild cohomology

Mariano Suárez-Álvarez

TL;DR

The paper proves that the Nakayama automorphism $\sigma$ of a Frobenius algebra $A$ acts trivially on Hochschild cohomology $\mathrm{HH}^*(A)$, while generally acting nontrivially on Hochschild homology. Using this, it constructs cohomological invariants of $A$ and its automorphisms/derivations via the Nakayama Jacobian $\mathrm{jac}_\sigma$ and the Nakayama divergence $\mathrm{div}_\sigma$, organized into 1-cocycles and 1-forms that yield classes in $\mathrm{H}^1$-cohomology groups. The authors develop explicit computations for key examples (Grassmann algebras, trivial extensions, and a 4-dimensional quantum complete intersection), showing both trivial and nontrivial Jacobians and divergences, and they prove a Liouville-type formula connecting the Jacobian of flows to divergences. These results provide new invariants and insights into deformation theory, crossed products, and the structure of Frobenius algebras, with potential extensions to twisted Calabi–Yau settings.

Abstract

We prove that the Nakayama automorphism of a Frobenius algebra acts trivially on the Hochschild cohomology of the algebra. As an application of this fact, we show how to construct certain invariants attached to such algebras, and to their automorphisms and derivations.

The action of the Nakayama automorphism of a Frobenius algebra on Hochschild cohomology

TL;DR

The paper proves that the Nakayama automorphism of a Frobenius algebra acts trivially on Hochschild cohomology , while generally acting nontrivially on Hochschild homology. Using this, it constructs cohomological invariants of and its automorphisms/derivations via the Nakayama Jacobian and the Nakayama divergence , organized into 1-cocycles and 1-forms that yield classes in -cohomology groups. The authors develop explicit computations for key examples (Grassmann algebras, trivial extensions, and a 4-dimensional quantum complete intersection), showing both trivial and nontrivial Jacobians and divergences, and they prove a Liouville-type formula connecting the Jacobian of flows to divergences. These results provide new invariants and insights into deformation theory, crossed products, and the structure of Frobenius algebras, with potential extensions to twisted Calabi–Yau settings.

Abstract

We prove that the Nakayama automorphism of a Frobenius algebra acts trivially on the Hochschild cohomology of the algebra. As an application of this fact, we show how to construct certain invariants attached to such algebras, and to their automorphisms and derivations.

Paper Structure

This paper contains 21 sections, 63 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Frobenius algebras and their Hochschild cohomology
  3. Frobenius algebras
  4. The action on central elements and derivations
  5. The action on Hochschild cohomology
  6. The action on HH2(A)
  7. The action on Hochschild cohomology
  8. The action on homology
  9. Invariants for Frobenius algebras
  10. Jacobians
  11. An example: the Grassmann algebra
  12. An example: trivial extensions
  13. An example: the quantum complete intersection of dimension four
  14. An example: separable algebras and group algebras
  15. Crossed products
  16. ...and 6 more sections

Key Result

Theorem A

The Nakayama automorphism $\sigma:A\to A$ of a Frobenius algebra $A$ acts trivially on the Hochschild cohomology $\mathop{\mathrm{HH}}\nolimits^*(A)$ of $A$.

Theorems & Definitions (111)

  • Theorem A
  • Proposition B
  • Proposition C
  • Proposition D
  • Proposition E
  • Proposition H
  • Proposition I
  • Proposition J
  • Proposition K
  • Lemma 1
  • ...and 101 more