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\begin{document}

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\section*{noncommutative differential calculus}

\hypertarget{noncommutative_differential_calculus}{}\section*{{Noncommutative differential calculus}}\label{noncommutative_differential_calculus}

\noindent\hyperlink{disambiguation}{Disambiguation}\dotfill \pageref*{disambiguation} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{disambiguation}{}\subsection*{{Disambiguation}}\label{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]]\ldots{}

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

\hypertarget{definition}{}\subsection*{{Definition}}\label{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)

\begin{displaymath}
\{ c, a\cup b\} = \{c, a\}\cup b + 
(-1)^{p q} a \cup \{ c, b\}
\end{displaymath}
A [[Gerstenhaber module]] $(\Omega_\bullet,\cap)$ is a graded module over a Gerstenhaber algebra $(V^\bullet,\cup)$

\begin{displaymath}
\cap : V\otimes \Omega\to\Omega,\,\,\,\,\,\,\,
V^p\otimes\Omega_n\ni a\otimes x \mapsto a\cap x
\in\Omega_{n-p}
\end{displaymath}
with a graded Lie algebra action of $V[1]$,

\begin{displaymath}
\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},
\end{displaymath}
satisfying the mixed Leibniz rule

\begin{displaymath}
b \cap \mathcal{L}_a(x) = \{ b, a\}\cap x +
(-1)^{p q} \mathcal{L}_a(b\cap x).
\end{displaymath}
A Gerstenhaber module $\Omega$ over a Gerstenhaber algebra $V$ is a \textbf{Batalin-Vilkovisky module} if it is equipped with a $k$-linear differential of degree +1,

\begin{displaymath}
B : \Omega_\bullet\to\Omega_{\bullet+1},\,\,\,\,\,\,B^2 = 0,
\end{displaymath}
such that $\mathcal{L}_\alpha$ for $a\in V^p$ and $B$ satisfy the generalization of the [[Cartan's homotopy formula]]

\begin{displaymath}
\mathcal{L}_a(x) = B(a\cap x) - (-1)^p a\cap B(x)
\end{displaymath}
A \textbf{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]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[differential calculus]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Dmitri Tamarkin, Boris Tsygan, \emph{Noncommutative differential calculus, homotopy BV algebras and formality conjectures}, Metods of Functional Analysis and topology, 1, 2001 \href{http://arxiv.org/abs/math/0002116v1}{arxiv:math.KT/0002116v1}


\item R. Nest, B. Tsygan, \emph{On the cohomology ring of an algebra}, Advances in geometry, Progr. Math. \textbf{172}, Birkhauser 1999, pp. 337--370 \href{http://arxiv.org/abs/math/9803132}{arxiv:math.QA/9803132}


\item V. Dolgushev, D. Tamarkin, B. Tsygan, \emph{Noncommutative calculus and the Gauss-Manin connection}, in: Higher structures in geometry and physics, 139--158, Progr. Math., 287, Birkh\"a{}user/Springer, New York, 2011 , \href{http://arxiv.org/abs/0902.2202}{arXiv:0902.2202}


\item K. Behrend, B. Fantechi, \emph{Gerstenhaber and Batalin-Vilkovisky structures on Lagrangian intersections}, \href{https://www.math.ubc.ca/~behrend/GeBaViLa.pdf}{pdf}



\end{itemize}
Many examples of such noncommutative differential calculi are constructed using [[Hopf algebroid]]s (Schauenburg's version) in works

\begin{itemize}%
\item Niels Kowalzig, \emph{Gerstenhaber and Batalin-Vilkovisky structures on modules over operads}, \href{http://arxiv.org/abs/1312.1642}{arxiv:1312.1642}; \emph{Batalin-Vilkovisky algebra structures on (Co)Tor and Poisson bialgebroids} \href{http://arxiv.org/abs/1305.2992}{arXiv:1305.2992}


\item Niels Kowalzig, Ulrich Kraehmer, \emph{Batalin-Vilkovisky structures on Ext and Tor}, \href{http://arxiv.org/abs/1203.4984}{arxiv:1203.4984}



\end{itemize}
[[!redirects noncommutative calculus]] [[!redirects noncommutative differential calculus]] [[!redirects noncommutative differential calculi]] [[!redirects Batalin-Vilkovisky module]] [[!redirects Batalin–Vilkovisky module]] [[!redirects Batalin--Vilkovisky module]]



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