%
% file: opava.tex
% purpose: ``A century of Noether's theorem and beyond'', Opava, Dec 2018
% created: pasha nov 30 2018
% modified:
% modification:
%
\documentclass{beamer}
\usepackage{mathrsfs}
\DeclareMathOperator{\dcobound}{d}
\DeclareMathOperator{\Homol}{H}
%\DeclareMathOperator{\HomtwoNilp}{Hom2Nilp}
\def\liebrack {\ensuremath{[\,\cdot\, , \cdot\,]}}
\setcounter{framenumber}{-1}
\setbeamertemplate{navigation symbols}{}
\title{Structure functions and Spencer cohomology \\
in zero and positive characteristics}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
``A Century of Noether's Theorem and Beyond'' \\ Opava \\ December 1, 2018
}
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\titlepage
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% from now on, display counters
\setbeamertemplate{headline}
{
\vskip2pt\hskip1pt\insertframenumber /\inserttotalframenumber
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Structure functions}
Let $G \subseteq GL(n)$ be a real of complex Lie group, $M$ an $n$-dimensional
real of complex manifold.
\medskip
A \emph{$G$-structure} on $M$ is a reduction of the principal $GL(n)$-bundle to
the principal $G$-bundle.
\medskip
\emph{Structure functions} are obstructions to integrability
(= local flattening) of $M$ endowed with a $G$-structure).
\medskip
Some (well known) particular cases:
\medskip
{\small
\begin{tabular}{|l|l|l|}
\hline
\textbf{$G$} & \textbf{name of a $G$-structure} &
\shortstack[l]{\textbf{name of a} \\ \textbf{structure function}} \\ \hline
$O(n)$ & Riemann metric & Riemann tensor \\ \hline
$O(n) \times \mathbb R^*$ & almost conformal structure & Weyl tensor \\ \hline
$GL(n,\mathbb C) \subset GL(2n,\mathbb R)$ & almost complex structure &
Nijenhuis tensor \\ \hline
\end{tabular}
}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Spencer cohomology}
Structure functions are interpreted in terms of the \emph{Spencer cohomology}
$H^*(\mathscr L_{-1}, \mathscr L)$ of a graded Lie algebra
$\mathscr L = \bigoplus_{n \ge -1} \mathscr L_n$.
\medskip
Major examples of $\mathscr L$: Lie algebras of Cartan type
$W_n$, $S_n$, $H_{2n}$.
\bigskip
\begin{block}{Theorem (Serre)}
The Spencer cohomology vanishes in degrees $>0$ for $W_n$ and $S_n$, and is
fully computed for $H_{2n}$.
\end{block}
\bigskip
Spencer cohomology is also responsible for \emph{filtered deformations} of
a graded Lie algebra $\mathscr L$, and therefore important for characteristic
$p>0$ analogs of Lie algebras of Cartan type.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{The algebras $\mathbb W(L,A)$}
Let $L$ be an abelian Lie algebra acting by derivations on an associative
commutative algebra $A$, such that $AL$ is a free submodule of $Der(A)$.
\medskip
The Lie algebra $\mathbb W(L,A)$ is defined as the vector space $AL \simeq L \otimes A$
with multiplication
$$
[x \otimes a, y \otimes b] = y \otimes ax(b) - x \otimes by(a) .
$$
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{The algebras $\mathbb W(L,A)$ (cont.)}
Particular cases of the construction from the previous slide are:
\begin{enumerate}
\item
$A = K[t_1, \dots, t_n]$,
$L = \langle \frac{\dcobound}{\dcobound t_1}, \dots, \frac{\dcobound}{\dcobound t_n} \rangle$:
$\mathbb W(L,A)$ = one-sided Jacobson--Witt algebra =
infinite-dimensional Lie algebra of the general Cartan type $W_n$ =
Lie algebra of polynomial vector fields on the plane $K^n$.
\item
$A = K[t_1, t_1^{-1}, \dots, t_n, t_n^{-1}]$,
$L = \langle \frac{\dcobound}{\dcobound t_1}, \dots, \frac{\dcobound}{\dcobound t_n} \rangle$:
$\mathbb W(L,A)$ = two-sided Jacobson--Witt algebra =
Lie algebra of polynomial vector fields on the $n$-dimensional sphere.
\item
$K$ is of characteristic $p>0$, $A = O(n; \overline{m})$,
the algebra of divided powers in $n$ variables with shearing parameters
$\overline{m} = (m_1, \dots, m_n)$,
$L = \langle \partial_1, \dots, \partial_n \rangle$:
$\mathbb W(L,A)$ = finite-dimensional Lie algebra of the general Cartan type
$W(n; \overline{m})$.
\end{enumerate}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{A unified approach to calculation of the Spencer cohomology:
case $W_n$}
\begin{block}{Theorem}
Let $L$ has a basis $D_1, \dots, D_n$ such that the algebra $A$ decomposes
as the tensor product of algebras $A_1 \otimes \dots \otimes A_n$, with $D_i$
acting on $A_i$. Then
$$
\Homol^k(L, A) \simeq \bigoplus_{1 \le i_1 < \dots < i_k \le n}
A_1^{D_1} \otimes \dots \otimes (A_{i_1})_{D_{i_1}} \otimes \dots \otimes
(A_{i_k})_{D_{i_k}} \otimes \dots \otimes A_n^{D_n} .
$$
\end{block}
\begin{block}{Sketch of the proof}
1) $H^k(L, \mathbb W(L,A)) \simeq L \otimes H^k(L,A)$. \\
2) Apply the K\"unneth formula.
\end{block}
\begin{block}{Corollaries}
Serre's vanishing result in $p=0$, and non-vanishing result in $p>0$.
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{A unified approach to calculation of the Spencer cohomology:
case $S_n$}
The Lie algebra $\mathbb S(L,A)$ is defined as the kernel of homomorphism
$$
div: \mathbb W(L,A) \to A, \quad x \otimes a \to x(a) .
$$
To compute the corresponding Spencer cohomology, apply the cohomology long exact
sequence associated with the short exact sequence of $L$-modules
$$
0 \to \mathbb S(L,A) \to \mathbb W(L,A) \overset{div}\to L(A) \to 0.
$$
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{A unified approach to calculation of the Spencer cohomology:
case $H_{2n}$}
Let $(D_1, \dots, D_n)$ and $(F_1, \dots, F_n)$ be two $n$-element sets of
pairwise commuting derivations of $A$. Then $A$ equipped with the bracket
\begin{equation*}
[a,b] = \sum_{i=1}^n \Big(D_i(a) F_i(b) - F_i(a) D_i(b)\Big)
\end{equation*}
is a generalization of all kinds of Hamiltonian Lie algebras.
\medskip
To compute the corresponding Spencer cohomology, apply considerations based on
the K\"unneth formula, similar to the case of $\mathbb W(L,A)$ (but more
cumbersome).
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% don't include this final frame into the total count;
% TeX twice for that!
\newcounter{finalframe}
\setcounter{finalframe}{\value{framenumber}}
\setbeamertemplate{headline}{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{center}
{\Huge
That's all. Thank you.
}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{framenumber}{\value{finalframe}}
\end{document}
% eof