# Noncommutative differential calculus

## Disambiguation

There are several notions of noncommutative differential calculus. See related pages regular differential operator in noncommutative geometry, bicovariant differential calculus, differential bimodule, differential monad, cyclic homology

This entry is about the version introduced by Boris Tsygan, Dmitri Tamarkin and Ryszard Nest, and studied also by Dolgushev, Kowalzig and others.

## Definition

Let $k$ be a unital commutative ground ring. A Gerstenhaber algebra $(V^\bullet,\cup)$ over $k$ is a $\mathbf{N}$-graded-commutative algebra with a graded Lie structure on $V$ satisfying the Leibniz rule (an analogue of a Poisson bracket in dg-world)

$\{ c, a\cup b\} = \{c, a\}\cup b + (-1)^{p q} a \cup \{ c, b\}$

A Gerstenhaber module? $(\Omega_\bullet,\cap)$ is a graded module over a Gerstenhaber algebra $(V^\bullet,\cup)$

$\cap : V\otimes \Omega\to\Omega,\,\,\,\,\,\,\, V^p\otimes\Omega_n\ni a\otimes x \mapsto a\cap x \in\Omega_{n-p}$

with a graded Lie algebra action of $V$,

$\mathcal{L} : V\otimes\Omega\to\Omega,\,\,\,\,\,\, V^{p+1}\otimes\Omega_n\ni a\otimes x \mapsto \mathcal{L}_a(x) \in\Omega_{n-p},$

satisfying the mixed Leibniz rule

$b \cap \mathcal{L}_a(x) = \{ b, a\}\cap x + (-1)^{p q} \mathcal{L}_a(b\cap x).$

A Gerstenhaber module $\Omega$ over a Gerstenhaber algebra $V$ is a Batalin-Vilkovisky module if it is equipped with a $k$-linear differential of degree +1,

$B : \Omega_\bullet\to\Omega_{\bullet+1},\,\,\,\,\,\,B^2 = 0,$

such that $\mathcal{L}_\alpha$ for $a\in V^p$ and $B$ satisfy the generalization of the Cartan's homotopy formula

$\mathcal{L}_a(x) = B(a\cap x) - (-1)^p a\cap B(x)$

A Tsygan-Tamarkin-Nest noncommutative (differential) calculus is a pair $(V, \Omega)$ of a Gerstenhaber algebra $V$ and a Batalin-Vilkovisky $V$-module $\Omega$.

• Dmitri Tamarkin, Boris Tsygan, Noncommutative differential calculus, homotopy BV algebras and formality conjectures, Metods of Functional Analysis and topology, 1, 2001 arxiv:math.KT/0002116v1

• R. Nest, B. Tsygan, On the cohomology ring of an algebra, Advances in geometry, Progr. Math. 172, Birkhauser 1999, pp. 337–370 arxiv:math.QA/9803132

• V. Dolgushev, D. Tamarkin, B. Tsygan, Noncommutative calculus and the Gauss-Manin connection, in: Higher structures in geometry and physics, 139–158, Progr. Math., 287, Birkhäuser/Springer, New York, 2011 , arXiv:0902.2202

• K. Behrend, B. Fantechi, Gerstenhaber and Batalin-Vilkovisky structures on Lagrangian intersections, pdf

Many examples of such noncommutative differential calculi are constructed using Hopf algebroids (Schauenburg’s version) in works

• Niels Kowalzig, Gerstenhaber and Batalin-Vilkovisky structures on modules over operads, arxiv:1312.1642; Batalin-Vilkovisky algebra structures on (Co)Tor and Poisson bialgebroids arXiv:1305.2992

• Niels Kowalzig, Ulrich Kraehmer, Batalin-Vilkovisky structures on Ext and Tor, arxiv:1203.4984