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\title{Structure functions and Spencer cohomology \\
in zero and positive characteristics}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
Seminar \\ ``Cohomology in algebra, geometry, physics and statistics''
\\
Prague
\\
May 29, 2019
}
\begin{document}
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\titlepage
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\begin{frame}{Structure functions}
Let $G \subseteq GL(n)$ be a real or complex Lie group, $M$ an $n$-dimensional
real or 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}
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\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}
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\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}
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\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}
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\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
\begin{multline*}
\shoveleft{\Homol^k(L, A) \simeq L \otimes \hfill} \\
\Big(\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}\Big) .
\end{multline*}
(at $i_1, \dots, i_k$ are coinvariants, at the other places, invariants).
\end{block}
\begin{block}{Sketch of the proof}
1) $\Homol^k(L, \mathbb W(L,A)) \simeq L \otimes H^k(L,A)$. \\
2) Apply the K\"unneth formula.
\end{block}
\end{frame}
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\begin{frame}{A unified approach to calculation of the Spencer cohomology:
case $W_n$ (cont.)}
\begin{block}{Corollaries}
Serre's vanishing result in $p=0$, and non-vanishing result in $p>0$.
In particular,
$$
\dim \Homol^k (W(n; \overline{m})_{-1}, W(n; \overline{m})) = n \binom nk .
$$
\end{block}
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\begin{frame}{Digression: Nijenhuis tensors}
\begin{block}{Theorem}
The space of structure functions of a real $2n$-dimensional manifold endowed
with a $GL(n,\mathbb C)$-structure is $2n^2 (n-1)$-dimensional.
\end{block}
\begin{block}{Proof}
For any associative commutative unital algebra $A$,
{\small
\begin{align*}
\Homol^2((W_n)_{-1} &\otimes A, W_n \otimes A) \simeq \\
&\Big(B^{2,-1}((W_n)_{-1}, W_n) \otimes \frac{S^2(A,A)}{A \oplus Der(A)}\Big) \\
\oplus &\Big(S^2((W_n)_{-1}, (W_n)_{-1}) \otimes
\frac{C^2(A,A)}{\set{\alpha \in C^2(A,A)}{\alpha(a,b) = a\beta(b) - b\beta(a)}} \Big).
\end{align*}
}
Substitute $A = \mathbb C_{\mathbb R}$.
\end{block}
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\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
$$
0 \to \mathbb S(L,A) \to \mathbb W(L,A) \overset{div}\to L(A) \to 0.
$$
\begin{block}{Corollary}
$\Homol^k((S_n)_{-1}, S_n) = 0$ for $k>0$.
\end{block}
\end{frame}
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\begin{frame}{The algebras $\mathbb P(A, \overline D, \overline F)$}
Let $\overline{D} = (D_1, \dots, D_n)$ and $\overline F = (F_1, \dots, F_n)$ 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*}
denoted by $\mathbb P(A, \overline D, \overline F)$, is a generalization of all
kinds of Hamiltonian Lie algebras.
\end{frame}
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\begin{frame}{A unified approach to calculation of the Spencer cohomology:
case $H_{2n}$}
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). Under similar assumptions,
\begin{multline*}
\Homol^k\Big(\mathbb P(A, \overline D, \overline F)_{-1}, \mathbb P(A, \overline D, \overline F)\Big) \\
\simeq
\bigoplus_{k_1 + \dots + k_n = k}
\Homol^{k_1}(A_1, \overline D_1, \overline F_1)
\otimes \dots \otimes
\Homol^{k_n}(A_n, \overline D_1, \overline F_1) .
\end{multline*}
For example, the number of different summands in the ``classical'' case, where
each set of derivations $\overline D_i$, $\overline F_i$ consists of one
element, and each $\mathbb P(A_i, \overline D_i, \overline F_i)_{-1}$ is
$2$-dimensional, the number of different summands in this formula is
$$
\sum_{n_0 + n_1 + n_2 = n, \> n_1 + 2n_2 = k} \frac{n!}{n_0! n_1! n_2!} ,
$$
where $n_0, n_1, n_2$ is the number of occurrences of $0$th, $1$st, and $2$nd
cohomology respectively.
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{\Huge
That's all. Thank you.
}
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