[Paper Review] The Tamarkin--Tsygan calculus of an algebra a la Stasheff

Pedro Tamaroff|arXiv (Cornell University)|Jul 21, 2019

Advanced Topics in Algebra27 references4 citations

TL;DR

This paper provides an intrinsic, computable description of the Tamarkin–Tsygan calculus for associative algebras using non-commutative poly vector fields and differential forms on a quasi-free cofibrant replacement. It proves the operad $\mathsf{Calc}$ is inhomogeneous Koszul, enabling a $\mathsf{Calc}_{\infty}$-model that computes the calculus structure via explicit formulas for the cup product, Gerstenhaber bracket, and differential on forms, with applications to monomial algebras yielding explicit Hochschild cohomology and homology computations.

ABSTRACT

We show how to compute the Tamarkin-Tsygan calculus of an associative algebra by providing, for a given cofibrant replacement of it, a `small' $\mathsf{Calc}_\infty$-model of its calculus, which we make somewhat explicit at the level of $\mathsf{Calc}$-algebras. To do this, we prove that the operad $\mathsf{Calc}$ is inhomogeneous Koszul; to our best knowledge, this result is new. We illustrate our technique by carrying out some computations for two monomial associative algebra using the cofibrant replacement obtained by the author in 1804.01435.

Motivation & Objective

• To provide a homotopy-invariant, computable description of the Tamarkin–Tsygan calculus for associative algebras using intrinsic structures on cofibrant resolutions.
• To establish that the colored operad $\mathsf{Calc}$ is inhomogeneous Koszul, a result previously unknown.
• To construct a $\mathsf{Calc}_{\infty}$-algebra structure on the complex of non-commutative poly vector fields and differential forms associated to a quasi-free model of an associative algebra.
• To enable explicit computations of Hochschild cohomology and homology structures, including the Gerstenhaber bracket and action on cohomology, via this model.
• To extend the intrinsic description of the calculus à la Stasheff to a homotopy-coherent framework that recovers the classical calculus on homology.

Proposed method

• Use of the quasi-free associative algebra $B = (TV, d)$ as a cofibrant replacement for an associative algebra $A$, allowing computation of the calculus via derived structures.
• Definition of non-commutative poly vector fields $\mathcal{X}^*(B) = \operatorname{cone}(\operatorname{Ad}: B \to \operatorname{Der}(B))$ and non-commutative differential forms $\Theta_*(B) = \operatorname{cone}(C: V \otimes B \to B)$.
• Construction of a $\mathsf{Calc}_{\infty}$-algebra structure on $(\mathcal{X}^*(B), \Theta_*(B))$ via the inhomogeneous Koszul duality theory for operads.
• Explicit formulas for the calculus operations: the cup product via a brace operation $\{X,Y;d\}$, the Lie bracket as the commutator of derivations, and the differential $d\omega = \sum (-1)^\varepsilon v_{i+1}\cdots v_n v_1\cdots v_{i-1} dv_i$.
• Application of the Kontsevich–Soibelman operad $\mathsf{KS}$ and formality of $C_*(\mathsf{Cyl})$ to relate the $\mathsf{Calc}_{\infty}$-structure to the homology of the Hochschild complex.
• Computation of Hochschild cohomology $\mathrm{HH}^*(A)$ and homology $\mathrm{HH}_*(A)$ for monomial algebras using the derived structure on $B$, including explicit bases for cohomology groups.

Experimental results

Research questions

• RQ1 Can the Tamarkin–Tsygan calculus of an associative algebra be described intrinsically using the bar construction and coderivations, à la Stasheff?
• RQ2 Is the colored operad $\mathsf{Calc}$, governing the calculus structure, inhomogeneous Koszul?
• RQ3 Can a $\mathsf{Calc}_{\infty}$-algebra structure be constructed on the complex of non-commutative poly vector fields and differential forms of a quasi-free model of an algebra?
• RQ4 What are the explicit formulas for the cup product, Gerstenhaber bracket, and differential in the calculus structure on such a model?
• RQ5 How can this framework be used to compute the Gerstenhaber algebra structure and Hochschild cohomology for concrete monomial associative algebras?

Key findings

• The operad $\mathsf{Calc}$ is proven to be inhomogeneous Koszul, a new result that enables the construction of a cofibrant $\mathsf{Calc}_{\infty}$-model with underlying symmetric sequence isomorphic to $T(\delta) \circ \mathsf{PreCalc}^{\text{!}}$.
• The Tamarkin–Tsygan calculus $\mathsf{Calc}_A$ of an associative algebra $A$ is computable via the pair $(\mathcal{X}^*(B), \Theta_*(B))$ for any quasi-free model $B \to A$.
• The cup product is given by a brace operation $\{X,Y;d\}$, the Lie bracket is the standard commutator of derivations, and the differential on forms is $d\omega = \sum (-1)^\varepsilon v_{i+1}\cdots v_n v_1\cdots v_{i-1} dv_i$.
• For the monomial algebra $A = \Bbbk\langle x,y,\Lambda,\Gamma \mid x^2=0, yx=0, \Lambda x=0, \Gamma y=0 \rangle$, $\mathrm{HH}^2(A)$ and $\mathrm{HH}^3(A)$ are each two-dimensional, with explicit bases given by classes of derivations $\Phi_0, \Phi_1$ and $\Upsilon_0^0, \Upsilon_0^1$.
• The Gerstenhaber bracket action is computed explicitly: for example, $[E_{s+2}, \Phi_t] = 3\Xi_{s+t+1} - 2\Theta_{s+t}$, and $[F_0, -] = [G_0, -] = 0$ on $\langle \Upsilon_s^t, \Phi_s \rangle$, while $[F_0, -] = [G_0, -] = 2$ on $\langle \Omega_s^t \rangle$.
• Cyclic homology $\mathrm{HC}_*(A)$ is concentrated in degree zero and is isomorphic to $\Bbbk[\bar{y}]$, while $\mathrm{HH}_*(A)$ is trivial in degrees $>1$ and $d: \mathrm{HH}_0(A) \to \mathrm{HH}_1(A)$ is an isomorphism.