Beijing Institute of Mathematical Sciences and Applications

A natural language for the calculus on a manifold or an algebraic variety is in terms of the corresponding algebras of functions, be it smooth or algebraic. The goal of non-commutative calculus is to extend the calculus to the case of associative algebras. While most of what is done below works in the case of $A_\infty$ algebras, we will, at least until the end, avoid the $\infty$-language.

Let $A$ be an algebra. Let $D$ be a derivation of $A$. When $A$ is commutative, say $A=C^\infty (M)$ for a smooth manifold $M$, $D$ is a vector field on $M$ and as such it acts on the de Rham complex $(\Omega^\bullet_M,d)$ by Lie derivative, as it does on any natural tensor construction applied to $A$. We will denote this action by $L_D$ .

One also defines the contraction $\iota_D$ by D and the classical calculus is summarised by the "change of variables" identities
\begin{equation}
[d, \iota_D]=L_D; \; \iota_D^2=0
\end{equation}Introducing a formal variable $u$ of degree $-2$, and working over power series $\mathbb{C}[[u]]$, one can combine these into a single identity
\begin{equation}
(ud+\iota_D)^2=uL_D\in End_{\mathbb{C}[[u]]}( \Omega^\bullet(M)[[u]])
\end{equation}Now let $A$ be an associative algebra and $D$ a derivation of $A$. The commonly used replacement for de Rham complex is the periodic cyclic complex where. In the first approximation, the differential graded space of differential forms get replaced by the Hochschild chain complex $(C_\bullet(A),b)$.


The counterpart of the classical Cartan formula above has the form
\begin{equation}
(b+uB+\mathcal{I}(D))^2=u(e^D-1)
\end{equation}where $I(D)$ is an exponential series $I(D)$. All the homogenous components $I_{D^n}$ of $\mathcal{I}(D)$ are defined over ${\mathbb Z}$. This is also a typical appearance of the Todd class $D^{-1} (e^D-1)$, which seems to appear whenever one replaces total anti-symmetrisation with cyclic anti-symmetrisation as in cyclic homology.

We will spend some time studying cyclic periodic complex and and some of its immediate applications, the first one being the Chern character from topological/algebraic K-theory to topological/negative cyclic homology.

We will also spend some time with the symplicial approach to cyclic homology, restricting ourselves to algebras over field of characteristic zero.

The classical calculus has an immediate extension from vector fields to multi-vector fields, generalising the standard formulas
\begin{equation}\label{L and iota func}
\iota_f(\omega)=f\omega; L_f(\omega)=df\wedge \omega
\end{equation}for a function $f$ and a form $\omega$. The noncommutative Cartan calculus extends to the full DGA (differential graded algebra) of Hochschild cochains, where the appropriate notion seems to be that of calculus. We will spend some time describing the appearing structures and give some explicit formulas. We will apply these methods to algebraic index theorems and their analytic counterparts.

The course will be based partially on the introduction to the non-commutative geometry chapter of the Encyclopedia of Mathematical Physics and the preliminary book version (see Boris Tsygan homepage).