nLab noncommutative differential calculus

Noncommutative differential calculus

Noncommutative differential calculus


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.


Let kk be a unital commutative ground ring. A Gerstenhaber algebra (V ,)(V^\bullet,\cup) over kk is a N\mathbf{N}-graded-commutative algebra with a graded Lie structure on V[1]V[1] satisfying the Leibniz rule (an analogue of a Poisson bracket in dg-world)

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

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

:VΩΩ,V pΩ naxaxΩ np \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]V[1],

:V[1]ΩΩ,V p+1Ω nax a(x)Ω np, \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 a(x)={b,a}x+(1) pq a(bx). b \cap \mathcal{L}_a(x) = \{ b, a\}\cap x + (-1)^{p q} \mathcal{L}_a(b\cap x).

A Gerstenhaber module Ω\Omega over a Gerstenhaber algebra VV is a Batalin-Vilkovisky module if it is equipped with a kk-linear differential of degree +1,

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

such that α\mathcal{L}_\alpha for aV pa\in V^p and BB satisfy the generalization of the Cartan's homotopy formula

a(x)=B(ax)(1) paB(x) \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,Ω)(V, \Omega) of a Gerstenhaber algebra VV and a Batalin-Vilkovisky VV-module Ω\Omega.

See also the case of Batalin-Vilkovisky algebra.


  • 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

Last revised on May 26, 2023 at 07:41:14. See the history of this page for a list of all contributions to it.