%
% file:         spb-2011-algsem.tex
% purpose:      talk at Spb. algebraic seminar at POMI, Apr 25 2011
%               "Cohomology of current algebras"
% created:      pasha apr 21-23 2011
% modified:     pasha may 2 2011
% modification: comments
%

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\title{Cohomology of current algebras}
\author{Pasha Zusmanovich}
\institute{Tallinn University of Technology}
\date{April 25, 2011}

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$L$ - Lie algebra

$A$ - associative commutative algebra

\bigskip

\textbf{Current Lie algebra} is a vector space
$L \otimes A$ under the bracket
$$
[x \otimes a, y \otimes b] = [x,y] \otimes ab
$$
where $x,y\in L$, $a,b\in A$.

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\textbf{What current Lie algebras are good for?}

\begin{block}{Kac-Moody algebras}
are central extensions of current Lie algebras
$\mathfrak g \otimes \mathbb C[t,t^{-1}]$:
$$
\mathfrak g \otimes \mathbb C[t,t^{-1}] + \mathbb C t\frac{d}{dt} + \mathbb C z
$$
$$
[x \otimes f, y \otimes g] = [x,y] \otimes fg + (x,y)Res \frac{df}{dt}g \> z
$$
where $x,y\in \mathfrak g$, $f,g\in \mathbb C[t,t^{-1}]$

\medskip

$\form$ is the Killing form on $\mathfrak g$.
\end{block}

\uncover<2->{
\begin{block}{Modular semisimple Lie algebras}
$$
S \otimes K[x_1, \dots, x_n]/(x_1^p, \dots, x_n^p) + 1 \otimes D
$$
\end{block}
}

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\begin{block}{Question}
\begin{itemize}
\item
How $H^*(L \otimes A, K)$ is expressed through invariants of $L$ and $A$?
\item
How $H^*(L \otimes A, M \otimes V)$ is expressed
through invariants of $(L,M)$ and $(A,V)$?
\end{itemize}
\end{block}

\uncover<2->{
\begin{block}{Applications}
\begin{itemize}
\item
degree 2 (deformations and central extensions): 
structure theory of modular Lie algebras, physics.
\item
degree 3: $2$-Lie algebras and ``physics'' again. 
\item
all degrees: combinatorial identities.
\end{itemize}
\end{block}
}

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\begin{block}{An elementary observation} 
Cocycles of the form 
$$
\Phi(x_1 \otimes a_1, \dots, x_n \otimes a_n) = 
\varphi(x_1, \dots, x_n) \otimes a_1 \cdots a_n \bullet v
$$
for some $v \in V$ give rise to
$$
H^*(L,M) \otimes V \subseteq H^*(L \otimes A, M \otimes V) .
$$
\end{block}

\uncover<2->{
\begin{block}{Another elementary observation} 
Cocycles of the form 
$$
\Phi(x_1 \otimes a_1, x_2 \otimes a_2) = [x_1,x_2] \otimes F(a_1,a_2)
$$
give rise to
$$
Har^2(A,V) \subseteq H^2(L \otimes A, L \otimes V) .
$$
\end{block}
}

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\begin{block}{A naive desire}
$$
H^*(L \otimes A, M \otimes V) \simeq 
\bigoplus_i \mathcal F_i(L,M) \otimes \mathcal G_i(A,V)
$$
for some functors $\mathcal F_i$ and $\mathcal G_i$.
\end{block}

\uncover<2->{
\bigskip
\textbf{It fails miserably in general.}
}
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\begin{block}{What is known about (co)homology of current Lie algebras?}

\begin{itemize}
\item
$L$ is classical simple, $A$ close to polynomial: 
Feigin, Garland \& Lepowsky, Hanlon.
\item
$L$ is algebra of infinite matrices, $A$ arbitrary: additive K-theory
Loday \& Quillen, Feigin \& Tsygan.
\item
$L$, $A$ (almost) arbitrary, (co)homology of low degree.
\end{itemize}

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\textbf{Feigin, 1970-1990s:} 

\begin{block}{An example}
$H_*(\mathfrak g \otimes \mathbb C[t], \mathbb C) \simeq 
H_*(\mathfrak g, \mathbb C)$

Tool: comparison of spectral sequences arising from the triangular
decomposition 
$\mathfrak g \otimes \mathbb C[t] = 
\widehat{\mathfrak n}_- \oplus \mathfrak h \oplus \widehat{\mathfrak n}_+$
and using Kostant-like results about $H_*(\widehat{\mathfrak n}_+, \mathbb C)$.
\end{block}

\uncover<2->{
\begin{block}{Another example}
Partial results about $H^*_{continuos}(\mathfrak g^{M})$.

Tool: a map from the Weil complex 
$\bigwedge^*(\mathfrak g) \otimes S^*(\mathfrak g)$ to a certain bicomplex
$$\begin{CD}
\bullet @>\text{de Rham complex of $M$}>> \bullet @>>> \bullet @>>> \dots \\
@V{C^*_{continuos}(\mathfrak g^{M})}VV    @VVV         @VVV    \\
\bullet @>>> \bullet @>>>   \bullet @>>> \dots   \\
@VVV         @VVV           @VVV                 \\
\bullet @>>> \bullet @>>>   \bullet @>>> \dots   \\
\end{CD}$$
 
\end{block}
}

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\textbf{Garland \& Lepowsky, 1976}: 
$H^*(\mathfrak g \otimes t\mathbb C[t],\mathbb C)$

\textbf{Hanlon, 1986}: \hskip 67pt
$H^*(\mathfrak g \otimes \mathbb C[t]/(t^n),\mathbb C)$

Tool: eigenvectors of the Laplacian (Gelfand--Fuchs style).

\uncover<2->{
\bigskip\bigskip\bigskip
\textbf{Tsygan, 1983 and Loday \& Quillen, 1984}:
$$
H_*(gl(A),K) \simeq \bigwedge(HC_*(A))
$$

Lesson: cyclic (co)homology is involved!
}

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\begin{block}{How cyclic cohomology appears in current Lie algebras cohomology?}

\textit{First degree cyclic cohomology}:
$HC^1(A) = 
\set{\alpha: A \times A \to K}{\alpha(ab,c) + \alpha(ca,b) + \alpha(bc,a) = 0}$

\medskip
 
Let $\varphi \otimes \alpha \in Z^2(L\otimes A, K)$, $\varphi:L \times L \to K$,
$\alpha:A \times A \to K$:
\begin{align*}
     &\varphi([x,y],z) \otimes \alpha (ab,c) \\
+ \> &\varphi([z,x],y) \otimes \alpha (ca,b) \\
+ \> &\varphi([y,z],x) \otimes \alpha (bc,a) = 0
\end{align*}
for any $x,y,z\in L$, $a,b,c\in A$.

Cyclically permute $x,y,z$ and sum up the 3 equalities obtained:
\begin{multline*}
\Big( \varphi([x,y],z) + \varphi([z,x],y) + \varphi([y,z],x) \Big)
\\ \otimes
\Big( \alpha(ab,c) + \alpha(bc,a) + \alpha(ca,b) \Big) = 0.
\end{multline*}

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\begin{block}{Low degree cohomology}

Nice formulae:

\begin{itemize}
\item 
$
H^1(L \otimes A, M \otimes V) \simeq 
H^1(L,M) \otimes V + Hom_L(L,M) \otimes Der(A,V)
$
(Zusmanovich, 2005)

\item
$H^2(L \otimes A, K) \simeq H^2(L,K) \otimes A^* + B(L) \otimes HC^1(A)$ \\
(Haddi, 1992 and Zusmanovich, 1994) \\
(both assuming $[L,L] = L$)

\item
$H^2(\mathfrak g \otimes A, \mathfrak g \otimes A) \simeq Har^2(A,A)$
(Cathelineau, 1987)

\item
If $W_1(n)$ is the modular Zassenhaus algebra, then
\begin{multline*}
H^2(W_1(n) \otimes A, W_1(n) \otimes A) \\ \simeq 
H^2(W_1(n),W_1(n)) \otimes A + Der(A) + Der(A) + Har^2(A,A)
\end{multline*}
(Zusmanovich, 2003)

\item
$H^3(\mathfrak g \otimes A, K) \simeq HC^2(A)$ or $HD^2(A)$ 
(Cathelineau, 1987)

\end{itemize}

\end{block}
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\begin{block}{Low degree cohomology}

% this is copied from low.tex
... and ugly formulae: 

some part (not all!) of $H^2(L\otimes A, M\otimes V)$ is isomorphic to
{\tiny
\begin{multline*}
  H^2(L,M) \otimes V 
+ H_M^2(L) \otimes \frac{Hom(A,V)}{V \oplus Der(A,V)}
+ \mathscr H(L,M) \otimes Der(A,V)
+ \mathscr B(L,M) \otimes \frac{Har^2(A,V)}{\mathscr P_+(A,V)}
\\+ C^2(L,M)^L \otimes \mathscr P_+(A,V)
+ \mathscr X(L,M) \otimes \frac{\mathscr A(A,V)}{\mathscr P_+(A,V)}
+ \mathscr T(L,M) \otimes \frac{D(A,V)}{Der(A,V)}
\\+ Poor_-(L,M)\otimes \frac{S^2(A,V)}{Hom(A,V) + D(A,V) + Har^2(A,V) + \mathscr A(A,V)}
\end{multline*}
where:

$d^{[\>]} \varphi (x,y,z) = \varphi([x,y],z) + \curvearrowright$;
%d^\bullet \varphi(x,y,z) &= x\bullet\varphi(y,z) + \curvearrowright .

$\wp \alpha(a,b,c) = \alpha(ab,c) + \curvearrowright$;

$D\alpha(a,b,c) = a\bullet\alpha(b,c) + \curvearrowright$;

$\mathscr B(L,M) = 
\set{\varphi\in C^2(L,M)}{\varphi([x,y],z) + z\bullet\varphi(x,y) = 0; 
                          d^{[\>]}\varphi(x,y,z) = 0}$;

$Q^2(L,M) = \set{d\psi}{\psi\in Hom(L,M); x\bullet\psi(y) = y\bullet\psi(x)}$;

$H_M^2(L) = (Z^2(L,M^L) + Q^2(L,M))/Q^2(L, M)$;

%$\mathscr K(L,M) = \set{\varphi\in C^2(L,M)}
%$                       {d^{[\>]}\varphi(x,y,z) = 2x\bullet\varphi(y,z)}$;

$\mathscr J(L,M) = 
\set{\varphi\in C^2(L,M)}
{\varphi(x,y) = 
\psi([x,y]) - \frac 12 x\bullet\psi(y) + \frac 12 y\bullet\psi(x) \text{ for } \psi\in Hom(L,M)}$;

$\mathscr H(L, M) = (\mathscr K(L,M) + \mathscr J(L,M))/\mathscr J(L,M)$.

$\mathscr X(L,M) = \set{\varphi\in C^2(L,M)}
{2\varphi([x,y],z) = z\bullet\varphi(x,y); \varphi([x,y],z) = \varphi([z,x],y)}$;

$\mathscr T(L,M) = \set{\varphi\in C^2(L,M)}
{3\varphi([x,y],z) = 2z\bullet\varphi(x,y); \varphi([x,y],z) = \varphi([z,x],y)}$;

$Poor_-(L,M) = \set{\varphi\in C^2(L,M^L)}{\varphi([L,L],L) = 0}$;

%$Poor_+(L,M) = \set{\varphi\in S^2(L,M^L)}{\varphi([L,L],L) = 0}$;

%$Sym^2 (L,M) = \set{\varphi\in S^2(L,M)}{x\bullet\varphi(y,z) = y\bullet\varphi(x,z)}$;

%$SB^2 (L,M) = \set{\varphi\in S^2(L,M)}
%{\varphi(x,y) = x\bullet\psi(y) + y\bullet\psi(x) \text{ for } \psi \in Hom(L,M)}$;

%$SH^2(L,M) = (Sym^2(L,M) + SB^2(L,M))/SB^2(L,M)$.

%$$
%z \circ \varphi(x, y) = z\bullet\varphi(x,y) + \varphi([x,z],y) + \varphi(x,[y,z]).
%$$
%$\mathscr S^2(L,M) = \set{\varphi\in S^2(L,M)^L}
%{\varphi([x,y],z) + \curvearrowright = 0}$.

$D(A,V) = \set{\beta\in Hom(A,V)}
{\beta(abc) = a\bullet\beta(bc) - bc\bullet\beta(a) + \curvearrowright}$;

%$HC^1(A,V) = \set{\alpha\in C^2(A,V)}{\wp\alpha = 0}$.

%$\mathscr C^2(A,V) = \set{\alpha\in C^2(A,V)}
%{\alpha(ac,b) - \alpha(bc,a) + a\bullet\alpha(b,c) - b\bullet\alpha(a,c)
% + 2c\bullet\alpha(a,b) = 0}$.

%$\mathscr P_-(A,V) = \set{\alpha\in C^2(A,V)}
%{\alpha(ab,c) = a\bullet\alpha(b,c) + b\bullet\alpha(a,c)}$;

$\mathscr P_+(A,V) = \set{\alpha\in S^2(A,V)}
{\alpha(ab,c) = a\bullet\alpha(b,c) + b\bullet\alpha(a,c)}$;

$\mathscr A(A,V) = \set{\alpha\in S^2(A,V)}{2D\alpha = \wp\alpha}$.
}

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\begin{block}{How to (methodically) ``compute'' cohomology of general current Lie algebras?}

Cauchy formula:
$$
\bigwedge\nolimits^n(L\otimes A) \simeq \bigoplus_{\lambda \vdash n} Y_\lambda(L) \otimes Y_{\lambda^\sim}(A)
$$
$Y_\lambda$ - \textit{Schur functor} associated with the Young diagram $\lambda$. \\
Examples:

\Yboxdim5pt
$Y_{\yng(1,1,1)} = \frac{1}{3!} \sum_{\sigma\in S_3} (-1)^\sigma \sigma$  \\
$Y_{\yng(3)}     = \frac{1}{3!} \sum_{\sigma\in S_3}             \sigma$  \\
$Y_{\yng(2,1)}   = \frac{1}{3} (e + (12) - (13) - (123))$

\bigskip

$\lambda^\sim$ - obtained from $\lambda$ by interchanging rows and columns

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\begin{block}{How Young symmetrizers interact with the differential?}

{\small
each Young diagram $\lambda$ represents 
$Hom(Y_\lambda(L),M) \otimes Hom(Y_\lambda(A),V)$
}
% all arrows
\begin{diagram}[width=2.3em,height=2.5em]
&&&&&& \yng(1) &&&&&&&                                       \\ %1
&&&&& \ldTo && \rdTo &&&&&                                      \\ %2 
&&&& \yng(1,1) &&&& \yng(2) &&&&                                \\ %3
&&& \ldTo && \rdTo\rdTo(5,2) & & \ldTo(6,2)\ldTo && \rdTo &&&                          \\ %4
&& \yng(1,1,1) &&&& \yng(2,1) &&&& \yng(3) &&                    \\ %5
& \ldTo && \rdTo\rdTo(4,2)\rdTo(6,2)\rdTo(10,2) && \ldTo(6,2)\ldTo & \dTo & \rdTo\rdTo(6,2) && \ldTo(10,2)\ldTo(6,2)\ldTo(4,2)\ldTo && \rdTo &  \\ %6
\yng(1,1,1,1) &&&& \yng(2,1,1) && \yng(2,2) && \yng(3,1) &&&& \yng(4)  \\ %7
\dots         &&&& \dots       && \dots     && \dots     &&&& \dots
\end{diagram}

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\begin{block}{How Young symmetrizers interact with the differential?}

{\small
each Young diagram $\lambda$ represents 
$Hom(Y_\lambda(L),M) \otimes Hom(Y_\lambda(A),V)$
}
% non-vanishing arrows (taken from low.tex)
\begin{diagram}[width=2.3em,height=2.5em]
&&&&&& \yng(1) &&&&&&&                                       \\ %1
&&&&& \ldTo && \rdTo &&&&&                                      \\ %2 
&&&& \yng(1,1) &&&& \yng(2) &&&&                                \\ %3
&&& \ldTo && \rdTo & & \ldTo(6,2)\ldTo && \rdTo &&&                          \\ %4
&& \yng(1,1,1) &&&& \yng(2,1) &&&& \yng(3) &&                    \\ %5
& \ldTo && \rdTo\rdTo(4,2) && \ldTo(6,2)\ldTo & \dTo & \rdTo && \ldTo(10,2)\ldTo(6,2)\ldTo(4,2)\ldTo && \rdTo &  \\ %6
\yng(1,1,1,1) &&&& \yng(2,1,1) && \yng(2,2) && \yng(3,1) &&&& \yng(4)  \\ %7
\dots         &&&& \dots       && \dots     && \dots     &&&& \dots
\end{diagram}

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\begin{block}{Which arrows do not vanish?}
\begin{itemize}
\item going from ``right'' to ``left''
\item the source Young diagram included into the target Young diagram
\item the target Young diagram is of the shape:
\begin{align*}
                   & \yng(2,2,1)   \\
                   & \dots         \\
                   & \yng(1)
\end{align*}
\end{itemize}
\end{block}
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\begin{block}{A filtration and a spectral sequence}

$F^kC^*$ = ``closure'' under non-vanishing arrows of 
\begin{align*}
                   & \yng(1,1)   \\
k+1 \>             & \dots       \\
                   & \yng(1)
\end{align*}
\end{block}

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{\Huge That's all. Thank you.}
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\vskip30pt
Slides at \texttt{http://justpasha.org/math/spb-2011-algsem.pdf}
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