%
% file:    iceland-2009.tex
% purpose: talk at Iceland, Sep 7 2009, 
%          "Low-dimensional cohomology of current Lie algebras"
% created: pasha sep 2-7 2009
%

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\title{Low-dimensional cohomology of current Lie algebras}
\author{Pasha Zusmanovich}
\date{September 7, 2009}

\begin{document}

\setcounter{framenumber}{-1}


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\begin{frame}
\titlepage
\end{frame}
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\begin{frame}{What is a Lie algebra?}

anticommutativity:
$[x,y] = - [y,x]$

Jacobi identity:
$[[x,y],z] + [[z,x],y] + [[y,z],x] = 0$

\bigskip

Example: algebra of all $n \times n$ matrices under $[X,Y] = XY - YX$

\medskip

Another (boring) example: \textit{abelian} Lie algebra: $[x,y] = 0$

\medskip

Another (interesting) example: $Der(A)$ for any algebra $A$

\smallskip

$D: A \to A$ is a \textit{derivation of $A$} if             \\            
$D(ab) = D(a)b + aD(b)$ for any $a,b\in A$

\smallskip

$[D_1, D_2] = D_1 \circ D_2 - D_2 \circ D_1$ 


\uncover<2->{
\begin{block}{What is a current Lie algebra?}

$L$ - Lie algebra
\quad
$A$ - associative commutative algebra

$L \otimes A$

$[x \otimes a, y \otimes b] = [x,y] \otimes ab$
\quad for $x,y\in L$, $a,b\in A$.
\end{block}
}

\end{frame}
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\begin{frame}{What current Lie algebras are good for?}

\begin{block}{Kac-Moody algebras}

$\mathfrak g \otimes \mathbb C[t,t^{-1}] + \mathbb C t\frac{d}{dt} + \mathbb C z$

\medskip

$[x \otimes f, y \otimes g] = [x,y] \otimes fg + (x,y)Res \frac{df}{dt}g \> z$ \\
for $x,y\in \mathfrak g$, $f,g\in \mathbb C[t,t^{-1}]$

\medskip

$\form$ - \textit{symmetric invariant bilinear form} on $\mathfrak g$ 
$\quad ([x,y],z) = (x,[z,y])$ \\
example: $(X,Y) = Tr(XY)$ on the matrix algebra

\medskip

It is a \textit{central extension} of the current Lie algebra 
$\mathfrak g \otimes \mathbb C[t,t^{-1}]$:

$0 \to Z \to \text{(central extension)} \to L \to 0$ 
\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}
}

\end{frame}
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\begin{frame}{What current Lie algebras are good for? (cont.)}

\begin{block}{Physics}

\textbf{Gauge theory}: how point particles transform 
as they move along paths in spacetime:
$$
\begin{xy}
{\ar@/^/ (-10,0)*+{\bullet};(10,0)*+{\bullet}}
\end{xy}
$$

spacetime = smooth manifold $M$, for example, a cylinder $S^1 \times \mathbb R$ \\
transformation = element of a smooth (Lie) group $G$ acting on $M$.

\smallskip

\textbf{String theory}: \\
string = (bunch of) loops \\
gauge group = loop group \{smooth functions $S^1 \to G$\}.

\smallskip

\textit{Loop algebra}: 
\{smooth functions $S^1 \to L$\} = $L \otimes \mathbb R[t,t^{-1}]$ 

\smallskip

\textbf{Quantum mechanics}:

Phases are unobservable, so one considers representations of groups
``up to a phase'', i.e. \textit{projective representations} =
representations of a central extension.

Central extension of a loop algebra = Kac-Moody algebra.
\end{block}

\end{frame}
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\begin{frame}{What is a cohomology of Lie algebras?}

$C^0(L,M) \overset{d}\to C^1(L,M) \overset{d}\to C^2(L,M) \overset{d}\to C^3(L,M) 
\overset{d}\to \dots$ 

\smallskip 

$C^n(L,M) = \{\text{skew-symmetric multilinear maps}: L \times \dots \times L \to M\}$

\smallskip

$d: C^n(L,M) \to C^{n+1}(L,M)$
\begin{align*}
d\varphi&(x_1, \dots, x_{n+1}) \\ 
= 
&\sum_{1 \le i < j \le n+1} (-1)^{i+j-1} 
\varphi([x_i,x_j],x_1, \dots, \widehat{x_i}, \dots, \widehat{x_j}, \dots, x_{n+1}) \\ 
+
&\quad\>\sum_{i=1}^{n+1} \quad\>\> (-1)^i x_i \bullet \varphi(x_1, \dots, \widehat{x_i}, \dots, x_n)
\end{align*}

\end{frame}
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\begin{frame}{What cohomology of Lie algebras is good for?}
\begin{block}{Low-dimensional interpretations}

\begin{itemize}
\item Abelian extensions

\uncover<2->{
$L$ - Lie algebra, $M$ - $L$-module

$0 \to M \to {?} \to L \to 0 \quad$ described by $H^2(L,M)$
}

\item Derivations

\uncover<3->{
derivations = 1-cocycles  \\
$D([x,y]) - [x,D(y)] + [y,D(x)] = 0$

inner derivations = 1-coboundaries \\
$D(x) = [x,a]$

outer derivations = $H^1(L,L)$
}

\item Deformations

\uncover<4->{
$L \text{ over } K \leadsto \mathcal L \text{ over } K((t))$

$\{x,y\} = [x,y] + \varphi_1(x,y)t + \varphi_2(x,y)t^2 + \dots$

``infinitesimal'' deformations = $H^2(L,L)$

obstructions to ``integrability'' = $H^3(L,L)$ 

(Gerstenhaber, 1960s)
}

\end{itemize}

\end{block}
\end{frame}
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\begin{frame}{What cohomology of Lie algebras is good for? (cont.)}
\begin{block}{Combinatorial identities}

\smallskip

$\mathcal C: 
C_0 \overset{d}\to C_1 \overset{d}\to C_2 \overset{d}\to C_3 \overset{d}\to \dots$

\textit{Euler--Poincar\'e characteristic} 
$\chi(\mathcal C) = \sum\limits_{n\ge 0} (-1)^n \dim C_n$

Euler--Poincar\'e principle: $\chi(\mathcal C) = \chi(H^*(\mathcal C))$.

\uncover<2->{
$$
\prod_{n=1}^\infty (1 - t^n) = 1 + 
\sum_{n=1}^\infty (-1)^n (t^{\frac{3n^2 - n}{2}} + t^{\frac{3n^2 + n}{2}})
$$
Euler, 1740s -- conjectured \\
Garland \& Lepowsky, 1975-1976 -- proved using cohomology of some subalgebras
of $\mathfrak g \otimes \mathbb C[t,t^{-1}]$.
}

\end{block}
\end{frame}
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\begin{frame}{What cohomology of Lie algebras is good for? (cont.)}
\begin{block}{$2$-Lie algebras}

''Categorified'' Lie algebras = Lie-like structures on a category in the category of 
vector spaces (i.e. objects and morphisms are vector spaces)
\\
Used in ``higher gauge theory'': how strings transform as they move along surfaces in
spacetime:
$$
\begin{xy}
(-10,0)*+{\bullet}="left"; (10,0)*+{\bullet}="right";
{\ar@/^1.25pc/ "left";"right"};
{\ar@/_1.25pc/ "left";"right"};
{\ar@{=>} (0,3)*{};(0,-3)*{}} ;
\end{xy}
$$

Classified in terms of $H^3$ of ordinary Lie algebras. \\
(Baez \& Co., 2002-2009)

\end{block}
\end{frame}
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\begin{frame}{What is known about (co)homology of some particular current Lie algebras?}

$H^*$ of 
$\mathfrak g \otimes \mathbb C[t,t^{-1}]$,
$\mathfrak g \otimes \mathbb C[t]$,
$\mathfrak g \otimes \mathbb C[t]/(t^n)$, etc.

(Feigin, Garland, Hanlon, Lepowsky, and others, 1975--1996)

\bigskip\bigskip\bigskip

\uncover<2->{
$H_*(gl(A)) \simeq \bigwedge(HC_*(A))$ and additive K-theory 
(Tsygan, 1983 and Loday--Quillen, 1984)

\smallskip

\textit{Cyclic (co)homology}:
$HC^1(A) = 
\set{\alpha: A \times A \to K}{\alpha(ab,c) + \alpha(ca,b) + \alpha(bc,a) = 0}$
}

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

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 $x,y,z\in L$, $a,b,c\in A$.

\bigskip

\uncover<2->{
But what for the general cocycle $\sum_{i\in I} \varphi_i \otimes \alpha_i \in Z^n(L\otimes A)$?
}

\end{frame}
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\begin{frame}{How to compute cohomology of ``general'' current Lie algebras?}

Task: to ``compute'' $H^n(L\otimes A, M\otimes V)$.
\begin{multline*}
Z^n(L\otimes A, M\otimes V) \subset Hom((L \otimes A)^{\otimes n}, M \otimes V)
\\ \simeq Hom(L^{\otimes n}, M) \otimes Hom(M^{\otimes n}, V)
\end{multline*}
Various symmetrizations of the cocycle equation $d\Phi = 0$ 
lead to conditions $(S \otimes T) \Phi = 0$, where  \\
$S\in Hom(Hom(L^{\otimes n}, M))$,
$T\in Hom(Hom(A^{\otimes n}, V))$.

\uncover<2->{
For example:
fully symmetrize the cocycle equation
$d\Phi(x_1 \otimes a_1, x_2 \otimes a_2, x_3 \otimes a_3) = 0$ with respect to $x_i$'s, 
where $\Phi = \sum_{i\in I} \varphi_i \otimes \alpha_i$,
$\varphi_i: L \times L \to M$, $\alpha_i: A \times A \to V$:
\begin{align*}
\sum\nolimits_{i\in I} \Big( 
( &x_1 \bullet \varphi_i(x_2,x_3) + x_1 \otimes \varphi_i(x_3,x_2) \\ 
+ &x_2 \bullet \varphi_i(x_1,x_3) + x_2 \otimes \varphi_i(x_3,x_1) \\ 
+ &x_3 \bullet \varphi_i(x_1,x_2) + x_3 \otimes \varphi_i(x_2,x_1) ) \\ 
& \otimes
(
- a_1 \bullet \alpha_i (a_2, a_3)
+ a_2 \bullet \alpha_i (a_1, a_3)
- a_3 \bullet \alpha_i (a_1, a_2)
)
\Big) = 0
\end{align*}
}

\end{frame}
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\begin{frame}{What is known about (co)homology of ``general'' current Lie algebras?}

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) \\
(both assuming $[L,L] = L$)

\item
If $\mathfrak g$ is a simple Lie algebra ($p=0$), then \\ 
$H^2(\mathfrak g \otimes A, \mathfrak g \otimes A) \simeq Har^2(A,A)$
(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{frame}
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\begin{frame}{Application to structure theory of modular Lie algebras}

A bit more about $H^2(W_1(n) \otimes A, W_1(n) \otimes A)$...

\medskip

$W_1(n) = \langle e_{-1}, e_0, e_1, \dots, e_{p^n-2} \rangle$, 
\begin{equation*}
[e_i, e_j] = \Big(\binom {i + j + 1}j - \binom {i + j + 1}i \Big) e_{i+j} .
\end{equation*}

Description of modular semisimple Lie algebras with a solvable maximal subalgebra
(Weisfeiler, 1984): filtered deformations of 
$W_1(n) \otimes K[x_1, \dots, x_n]/(x_1^p, \dots, x_n^p) + 1 \otimes D$.

\medskip

Filtered deformation: (graded algebra) $\leadsto$ (filtered algebra)

filtered algebra: 
$\mathcal L = \mathcal L_{-1} \supset \mathcal L_0 \supset \mathcal L_1 \supset \dots$

associated graded algebra: $L = \bigoplus_{i\ge -1} \mathcal L_i/\mathcal L_{i+1}$

\end{frame}
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\begin{frame}{What is known about (co)homology of ``general'' current Lie algebras? 
(cont.)}

% this is copied from low.tex
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)}$;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{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

\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{A spectral sequence...}

% 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}
\small 
each Young diagram $\lambda$ represents $Hom(Y_\lambda(L),M) \otimes Hom(Y_\lambda(A),V)$
\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{A spectral sequence...}

% 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}
\small
each Young diagram $\lambda$ represents $Hom(Y_\lambda(L),M) \otimes Hom(Y_\lambda(A),V)$
\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{A spectral sequence... (abelian $L$)}

% case of abelian $L$
\begin{diagram}[width=2.3em,height=2.5em]
&&&&&& \yng(1) &&&&&&&                                       \\ %1
&&&&& \ldTo && \rdTo &&&&&                                      \\ %2 
&&&& \yng(1,1) &&&& \yng(2) &&&&                                \\ %3
&&& \ldTo && \rdTo & & \ldTo && \rdTo &&&                          \\ %4
&& \yng(1,1,1) &&&& \yng(2,1) &&&& \yng(3) &&                    \\ %5
& \ldTo && \rdTo && \ldTo & \dTo & \rdTo && \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}
\small
each Young diagram $\lambda$ represents $Hom(Y_\lambda(L),M) \otimes Hom(Y_\lambda(A),V)$
\end{frame}
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\begin{frame}{A spectral sequence (cont.)}

Which arrows do not vanish?

\begin{itemize}
\item going from ``right'' to ``left''
\item the source Young diagram included into the target Young diagram
(the only case when $L$ is abelian)
\item the target Young diagram is of the shape:
\begin{align*}
                   & \yng(2,2,1)   \\
                   & \dots         \\
                   & \yng(1)
\end{align*}
\end{itemize}

\uncover<2->{
Filtration: $F^kC^*$ = ``closure'' under non-vanishing arrows of 
\begin{align*}
                   & \yng(1,1)   \\
k+1 \>             & \dots       \\
                   & \yng(1)
\end{align*}
}

\end{frame}
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% TeX twice for that!
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\begin{frame}
\begin{center}
{\Huge That's all. Thank you.}
\end{center}

\vskip30pt
Slides at \texttt{http://justpasha.org/math/iceland-2009.pdf} .
\end{frame}
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\setcounter{framenumber}{\value{finalframe}}

\end{document}
% end of iceland-2009.tex