The Coxeter transformation as an automorphism of the Tamarkin--Tsygan calculus

Abstract

Let $A$ be a finite-dimensional algebra over a field $\kk$. We show that the Auslander--Reiten bimodule $\ar_A:=\D A[-1]$ is central in the derived Picard group of $A$ and that, when $\gldim A<\infty$, it induces through the derived-invariance functor $\HB$ of \cite{Armenta19,ArmentaKeller17,ArmentaKeller19} a canonical automorphism $σ_A$ of the Tamarkin--Tsygan calculus of $A$; the pair $(\HB(A),σ_A)$ is invariant under derived equivalence. We then compute both components of $σ_A$. On the Hochschild homology of an elementary algebra, which is concentrated in degree zero, the matrix of $σ_A$ in the basis of idempotent traces is $-C_A^{-1}C_A^{\mathrm{T}}$, so its characteristic polynomial is the Coxeter polynomial; the enriched calculus strictly refines both the calculus and the Coxeter polynomial, as the path algebras of quivers of types $\mathbb{A}_4$ and $\mathbb{D}_4$ show, although it is not a complete derived invariant, as the smallest cospectral pair of trees shows. On Hochschild cohomology we prove that $σ_A$ is the identity: the left and right actions of $\HH^\bullet(A)$ on the bimodule $\D A$ coincide for every finite-dimensional $A$. This yields a short conceptual proof that the Nakayama automorphism of a Frobenius algebra acts trivially on Hochschild cohomology, recovering a recent theorem of Suárez-Álvarez. Finally we extend the construction to smooth and proper differential graded algebras, hence to perfect derived categories of smooth projective varieties; the enrichment degenerates precisely on Calabi--Yau categories, and on $\PP^n$ it is governed by the Coxeter polynomial $(x+(-1)^n)^{n+1}$ of the Beilinson algebra. Happel's trace formula and de la Peña's cyclotomicity theorem for fractionally Calabi--Yau algebras become statements internal to the enriched calculus.