nLab 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 of “Cartan calculus” for associative algebras introduced by Boris Tsygan, Dmitri Tamarkin and Ryszard Nest, and studied also by Dolgushev, Kowalzig and others.

Some early ideas of extending Cartan calculus to noncommutative geometry are from Gelfand et al. and from Reinhart.

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 $V[1]$ 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 $V[1]$,

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

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 $\Omega$.

See also the case of Batalin-Vilkovisky algebra.

Related concepts

• differential calculus

References

• I. M. Gel’fand, Yu. L. Daletskii, B. L. Tsygan, On a variant of noncommutative differential geometry, Dokl. Akad. Nauk SSSR 308:6 (1989) 1293–1297; Engl. transl.: Dokl. Math. 40:2 (1990) 422–426 mathnet.ru

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

• D. Tamarkin, B. Tsygan, The ring of differential operators on forms in noncommutative calculus, Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math. 73, Amer. Math. Soc. 2005, pp. 105–131 doi MR2131013

• 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

• Pedro Tamaroff, The Tamarkin-Tsygan calculus of an algebra a la Stasheff, Homology, Homotopy and Applications 23:2 (2021) 257–282 arXiv:1907.08888 doi

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, Journal für die reine und angewandte Mathematik 697 (2014)

  doi arXiv:1203.4984