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.
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)
A Gerstenhaber module? $(\Omega_\bullet,\cap)$ is a graded module over a Gerstenhaber algebra $(V^\bullet,\cup)$
with a graded Lie algebra action of $V[1]$,
satisfying the mixed Leibniz rule
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,
such that $\mathcal{L}_\alpha$ for $a\in V^p$ and $B$ satisfy the generalization of the Cartan's homotopy formula
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.
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
Last revised on October 14, 2014 at 21:58:11. See the history of this page for a list of all contributions to it.