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\title{Novikov structures on Kac-Moody and modular semisimple Lie algebras}
\author{Pasha Zusmanovich}
\institute{Tallinn University of Technology}
\date{May 30, 2011}
\begin{document}
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\begin{frame}
\titlepage
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\begin{frame}{What is a left-symmetric structure?}
Left-symmetric (aka Vinberg, pre-Lie, chronological) identity:
$$
x(yz) - (xy)z = y(xz) - (yx)z
$$
Equivalently:
\begin{itemize}
\item $(x,y,z) = (y,x,z)$, where $(x,y,z) = (xy)z - x(yz)$ is the associator.
\item $[L_x,L_y] = L_{[x,y]}$, where $L_x(a) = xa$.
\end{itemize}
\uncover<2->{
Left-symmetricity $\Rightarrow$ Lie-admissibility:
$$
[x,y] = xy - yx
$$
satisfies the Jacobi identity.
\begin{block}{Question}
Describe left-symmetric structures on a given Lie algebra.
\end{block}
\begin{block}{Origin}
Theory of affine manifolds (Auslander, Milnor).
\end{block}
}
\end{frame}
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\begin{frame}{What is a Novikov structure?}
Left-symmetric +
$$
(xy)z = (xz)y
$$
Equivalently:
$$
[R_x,R_y] = 0
$$
where $R_x(a) = ax$.
\begin{block}{Question}
Describe Novikov structures on a given Lie algebra.
\end{block}
\begin{block}{Origin (of Novikov algebras)}
\begin{itemize}
\item Integrability of dynamical systems (Gelfand \& Dorfman).
\item Poisson brackets of hydrodynamic type (Balinsky \& Novikov).
\end{itemize}
\end{block}
\end{frame}
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\begin{frame}{What is known about Novikov structures?}
($p=0$, finite-dimensional, unless stated otherwise)
\begin{itemize}
\item
Helmstetter (1979): a Lie algebra admitting a left-symmetric structure
is not perfect ($[L,L] \ne L$).
\item
Osborn (1992) and Xu (1996): Novikov structures on the Zassenhaus algebra $W_1(n)$
($p>0$).
\item
Osborn \& Zelmanov (1995):
an inverse problem: Lie structures on some infinite-dimensional
Novikov Witt-type algebras.
\item
Xu (2000): same inverse problem; Novikov structures on the infinite-dimensional
Witt algebra.
\item
Bai \& Meng (2001): Novikov structures on $4$-dimensional nilpotent Lie algebras.
\item
Burde (2006): a Lie algebra admitting a Novikov structure is solvable.
\item
Burde, Dekimpe \& Vercammen (2008): Novikov structures on Lie algebras
of (strictly) upper triangular $n \times n$ matrices exist only for small $n$.
\end{itemize}
\end{frame}
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\begin{frame}{Quadratic vs. linear problems}
Suppose a Lie-admissible algebra satisfies a number of identities:
\begin{equation*}
\sum_{\sigma\in S_3}
\Big(\alpha_\sigma^i (x_{\sigma(1)} x_{\sigma(2)}) x_{\sigma(3)} +
\beta_\sigma^i x_{\sigma(1)} (x_{\sigma(2)} x_{\sigma(3)}) \Big) = 0, \quad i=1, \dots, n
\end{equation*}
Rewrite these identities in terms of $\liebrack$ and $\circ$, where
$$
xy = \frac 12 [x,y] + \frac 12 x \circ y \qquad
\text{(so $x \circ y = xy + yx$)}:
$$
\begin{align*}
& (\alpha_e^i + \alpha_{(12)}^i + \beta_e^i + \beta_{(12)}^i) (x_1 \circ x_2) \circ x_3
\\ +
& (\alpha_{(23)}^i + \alpha_{(132)}^i + \beta_{(23)}^i + \beta_{(132)}^i)
(x_3 \circ x_1) \circ x_2
\\ +
& (\alpha_{(13)}^i + \alpha_{(123)}^i + \beta_{(13)}^i + \beta_{(123)}^i)
(x_2 \circ x_3) \circ x_1
\\ +
&\text{(terms linear with respect to $\circ$)} = 0.
\end{align*}
\begin{block}{Quadratic problem}
For a given Lie algebra $(L, \liebrack)$, find symmetric maps $L \circ L \to L$,
satisfying these identities.
\end{block}
\end{frame}
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\begin{frame}{Quadratic vs. linear problems (continuation)}
If
$$
rk
{\tiny
\begin{pmatrix}
\alpha_e^1 + \alpha_{(12)}^1 + \beta_e^1 + \beta_{(12)}^1
&
\alpha_{(23)}^1 + \alpha_{(132)}^1 + \beta_{(23)}^1 + \beta_{(132)}^1
&
\alpha_{(13)}^1 + \alpha_{(123)}^1 + \beta_{(13)}^1 + \beta_{(123)}^1
\\ \\
\dots & \dots & \dots
\\ \\
\alpha_e^n + \alpha_{(12)}^n + \beta_e^n + \beta_{(12)}^n
&
\alpha_{(23)}^n + \alpha_{(132)}^n + \beta_{(23)}^n + \beta_{(132)}^n
&
\alpha_{(13)}^n + \alpha_{(123)}^n + \beta_{(13)}^n + \beta_{(123)}^n
\end{pmatrix}
}
< n
$$
then this quadratic problem has a \textbf{linear} consequence.
\medskip
\uncover<2->{
{\bf
Left-symmetric structures - quadratic
\smallskip
Novikov (and LR, and probably some others?) structures - linear!
}
}
\end{frame}
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\begin{frame}{2-sided Alia algebras}
\hskip -20pt (Dzhumadil'daev, 2009)
\begin{align*}
[x,y]z + [z,x]y + [y,z]x &= 0 \\
z[x,y] + y[z,x] + x[y,z] &= 0
\end{align*}
$$
\begin{array}{l}
\text{commutative} \\
\text{Lie} \\
\text{Novikov} \\
\text{LR}
\end{array}
\Rightarrow \text{2-sided Alia} \Rightarrow \text{Lie-admissible}
$$
\begin{block}{Question}
Describe 2-sided Alia structures on a given Lie algebra.
\end{block}
\uncover<2->{
\begin{block}{Equivalent question}
Describe commutative $2$-cocycles on a given Lie algebra.
\end{block}
}
\end{frame}
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\begin{frame}{What are commutative $2$-cocycles?}
Each 2-sided Alia structure on a Lie algebra $L$ is given by
$$
xy = [x,y] + \varphi(x,y)
$$
where $\varphi: L \times L \to K$ is a \textbf{commutative $2$-cocycle} on $L$, i.e.:
\begin{enumerate}
\item $\varphi$ is symmetric
\item $\varphi([x,y],z) + \varphi([z,x],y) + \varphi([y,z],x) = 0$
\end{enumerate}
Cocycles with $\varphi([L,L],L) = 0$ are called \textbf{trivial}.
\medskip
The space of all $K$-valued commutative $2$-cocycles on $L$ is denoted as $Z^2_{comm}(L)$.
\end{frame}
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\begin{frame}{Commutative $2$-cocycles on current Lie algebras}
A \textbf{current Lie algebra} is a Lie algebra of the form $L \otimes A$,
where $L$ is a Lie algebra, $A$ is an associative commutative algebra, with the Lie
bracket
$$
[x \otimes a, y \otimes b] = [x,y] \otimes ab
$$
for $x,y\in L$, $a,b\in A$.
\begin{block}{Fact}
Under some technical assumptions,
\begin{align*}
&Z^2_{comm} (L\otimes A) \\
&\simeq Z^2_{comm}(L) \otimes A^* \\
&\oplus
\set{\varphi: L \times L \to K}{\varphi \text{ is skew; } \varphi([x,y],z) = \varphi([z,x],y)}
\otimes HC^1(A) \\
&\oplus \text{(trivial cocycles)}
\end{align*}
\end{block}
\end{frame}
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\begin{frame}{No Novikov structures on Kac-Moody}
Kac--Moody:
$$
\mathfrak g \otimes \mathbb C[t,t^{-1}] + \mathbb C t\frac{d}{dt} + \mathbb C z
$$
\begin{block}{Corollary}
All commutative $2$-cocycles on affine Kac-Moody algebras are trivial.
\end{block}
\begin{block}{Fact}
A Lie algebra which is not $2$-step solvable and all whose commutative $2$-cocycles
are trivial, do not admit a Novikov structure.
\end{block}
\begin{block}{Theorem}
Affine Kac--Moody algebras do not admit Novikov structures.
\end{block}
\end{frame}
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\begin{frame}{Novikov structures on modular semisimples}
Modular semisimple Lie algebras:
$$
S \otimes K[t_1, \dots, t_n]/(t_1^p, ..., t_n^p) + 1 \otimes D
$$
\begin{block}{Another corollary}
$Z^2_{comm}$ of such algebras is isomorphic to $Z^2_{comm}(D)$.
\end{block}
\begin{block}{Another theorem}
A modular semisimple Lie algebra admits a Novikov structure if and only if
$W_1(n)$ is ``involved''.
\end{block}
\uncover<2->{
Simples:
\begin{itemize}
\item Burde (1994): left-symmetric structures on classical, examples for Cartan type.
\item
Dzhumadil'daev \& Zusmanovich (2010):
nonzero commutative $2$-cocycles exist only on $sl(2)$ and $W_1(n)$.
\end{itemize}
Semisimples: by the corollary above.
}
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\begin{center}
{\Huge That's all. Thank you.}
\bigskip
Slides at \texttt{http://justpasha.org/math/oostende.pdf}
\end{center}
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