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% This file is a solution template for:

% - Giving a talk on some subject.
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% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
%
% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
% modify this file as you see fit and even to delete this copyright
% notice.


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{
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  % or ...
  % or whatever (possibly just delete it)
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\newtheorem*{question}{Question}
\newtheorem*{mypicture1}{Setuation 1}
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\newtheorem*{keylemma}{Key Lemma}
\newtheorem*{cokeylemma}{co-Key Lemma}

%\newtheorem*{example}{Example}
%\newtheorem*{theorem}{Theorem}
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%\newtheorem{cor}[lemma]{Corollary}
%\newtheorem{exm}[lemma]{Example}
%\newtheorem{exc}[lemma]{Exercise}
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%\newtheorem{conc}[lemma]{Conclusion}



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% You can add any graphics to every slide by following command:
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% Uncomment this, if you want the table of contents before each subsection.
% However, to edit slides in TeXWord avoiding this feature is good idea.
% \AtBeginSubsection[]
% {
%   \begin{frame}<beamer>
%     \frametitle{Outline}
%     \tableofcontents[currentsection,currentsubsection]
%   \end{frame}
% }

% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command:
% \beamerdefaultoverlayspecification{<+->}

\begin{document}

% Title Data. We keep it after \begin{document}
% to enable editing text in BaKoMa TeX Word.

\title{A Quantum Analogue of Kostant's Theorem for the General
Linear Group}

% Use the \inst command to identify several affiliations.
\author{Avraham Aizenbud}
\institute{Department of Mathematics, Massachusetts Institute of Technology}
%Dima? I think ``Department of Mathematics'' is irrelevant.
%But if you want it to stay - separate it by \\ or by ,


%\date[TUG 2005]{Annual TUG Conference, 2005}

%\subject{Desktop Publishing} % Should be passed to PDF [YNI]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
\begin{frame}
%\maketitle
\titlepage
\begin{center}Joint with Oded Yacobi \end{center}
$ $\\
\begin{center}\url{http://math.mit.edu/~aizenr/}  \end{center}

\end{frame}
%\newpage

\begin{frame}
%\thispagestyle{empty}
    \frametitle{Kostant Theorem}
\begin{theorem}[Kostant 1963]
 Let $G$ be an algebraic reductive group (e.g. $GL_n$). 
\pause
Let $\g$ be the corresponding Lie algebra. \pause Then 
\center{$O(\g)$ is free over $O(\g)^G$}
\end{theorem}
\pause
\begin{theorem}[Kostant 1963]
 $U(\g)$ is free over its center.
\end{theorem}
\pause
\begin{theorem}[Richardson 1979]
 Suppose $G$ is semi-simple and simply connected. Then $O(G)$ is free over $O(G)^G$.
\end{theorem}
\end{frame}
%\newpage

\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{Bernstein-Lunts Proof}

\begin{lemma}[Bernstein-Lunts]
 Let $M$ be a filtrated module over a filtrated algebra $A$.
\pause
 Then
$$gr (M) \text{ is free over } gr (A) \quad \Longrightarrow \quad M \text{ is free over } A$$
\end{lemma}
\pause

\begin{itemize}
 \item $O(\g)=K[x_{ij}].$
\pause
 \item $deg(x_{ij})=\delta_{ij}.$
\pause
  \item $\g = \h \oplus \n^{+} \oplus \n^{-}.$
\pause
  \item $gr(O(\g))=O(\h) \otimes O(\n^{+} \oplus \n^{-}).$
\pause
 \item  $gr(O(\g)^G)=O(\h)^W.$
\end{itemize}




\end{frame}
%\newpage

\begin{frame}
\thispagestyle{empty}
%
    \frametitle{Quantum $GL_2$}

\begin{definition}[Quantum plane]
$O(K^2_q):= K \langle x,y \rangle /\langle [x,y]_q \rangle :=K\langle x,y \rangle /\langle xy-qyx \rangle $
\end{definition}
\pause
$O(Mat_2(K)_q)=K\langle a,b,c,d \rangle /\langle ? \rangle $\\
\pause
We want
\begin{itemize}
 \item $(a \otimes x + b \otimes y)(c \otimes x + d \otimes y)=q(a \otimes x + b \otimes y)(c \otimes x + d \otimes y)$
\pause
 \item $(a \otimes x + c \otimes y)(b \otimes x + d \otimes y)=q(a \otimes x + c \otimes y)(b \otimes x + d \otimes y)$
\end{itemize}
\pause
\begin{definition}
\begin{itemize}
 \item $O((Mat_2)_q) = K \langle a,b,c,d \rangle / $ $ \langle[a,b]_q,[c,d]_q, [a,c]_q , [b,d]_q ,[b,c],[a,d]-(q-q^{-1})bc \rangle $
\pause
 \item $det_q=ad-qbc \in Z(O(Mat_2(K)_q))$
\pause
 \item $O((GL_2)_q)=det_q^{-1} O((Mat_2)_q)$
\end{itemize}
\end{definition}
%\begin{equation*}
%\xymatrix{ {a} \ar[d]\ar@{~>}[dr]\ar[r]
%& {b} \ar[d] \ar@{-}[ld] \\
%{c} \ar[r] & {d} }
%\end{equation*}
\pause
\begin{equation*}
\xymatrix{ {a} \ar[d]\ar@{~>}[dr]\ar[r]
& {b} \ar[d]  \\
{c} \ar[r] & {d} }
\end{equation*}
\end{frame}
%\newpage
\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{Quantum $GL_n$}

\begin{definition}
\begin{itemize}
 \item $A:=O((Mat_n)_q)=K \langle x_{ij} \rangle / \langle [x_{ij},x_{il}]_q,[x_{kj},x_{kl}]_q,[x_{ij},x_{kj}]_q,[x_{il},x_{kl}]_q,$ $
 [x_{il},x_{kj}],[x_{ij},x_{kl}]-(q-q^-1)x_{il}x_{kj} \rangle $
\pause
 \item $det_q=\sum_{w \in S_{n}}(-q)^{l(w)}x_{{1}w({1})} \cdots x_{nw(n)}
 \in Z(A)$
%O(Mat_n(K)_q)
\pause
 \item $O((GL_n)_q)=det_q^{-1} A$
%$O((Mat_n)_q)$
\end{itemize}
\end{definition}
%\begin{equation*}
%\xymatrix{ {\bullet} & {\bullet} \ar@/_/[dd]\ar@{~>}[ddr]\ar[r]
%& {\bullet} \ar@{-}[ddl] \ar@/^/[dd] \\
%{\bullet} & {\bullet} & {\bullet} \\
%{\bullet} & {\bullet} \ar[r] & {\bullet} }
%\end{equation*}

\pause
\begin{equation*}
\xymatrix{ {\bullet} & {\bullet} \ar@/_/[dd]\ar@{~>}[ddr]\ar[r]
& {\bullet}  \ar@/^/[dd] \\
{\bullet} & {\bullet} & {\bullet} \\
{\bullet} & {\bullet} \ar[r] & {\bullet} }
\end{equation*}

\end{frame}

\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{Quantum $GL_n$}

\begin{definition}
\begin{itemize}
 \item $A:=O((Mat_n)_q)=K \langle x_{ij} \rangle / \langle [x_{ij},x_{il}]_q,[x_{kj},x_{kl}]_q,[x_{ij},x_{kj}]_q,[x_{il},x_{kl}]_q,$ $
 [x_{il},x_{kj}],[x_{ij},x_{kl}]-(q-q^-1)x_{il}x_{kj} \rangle $
 \item $det_q=\sum_{w \in S_{n}}(-q)^{l(w)}x_{{1}w({1})} \cdots x_{nw(n)}
 \in Z(A)$
%O(Mat_n(K)_q)
 \item $O((GL_n)_q)=det_q^{-1} A$
%$O((Mat_n)_q)$
\end{itemize}
\end{definition}
%\begin{equation*}
%\xymatrix{ {\bullet} & {\bullet} \ar@/_/[dd]\ar@{~>}[ddr]\ar[r]
%& {\bullet} \ar@{-}[ddl] \ar@/^/[dd] \\
%{\bullet} & {\bullet} & {\bullet} \\
%{\bullet} & {\bullet} \ar[r] & {\bullet} }
%\end{equation*}
\begin{equation*}
\xymatrix{ {\bullet}  \ar@{~>}@/_/[ddrr] \ar@{~>}[drr] \ar@{~>}[dr] \ar@/^/[rr] \ar[d] \ar@/_/[dd]\ar@{~>}[ddr]\ar[r] & {\bullet} \ar@{~>}[dr] \ar[d]  \ar@/^/[dd] \ar@{~>}[ddr]\ar[r] \ar[d]
& {\bullet}  \ar@/^/[dd] \\
{\bullet} \ar@{~>}[drr] \ar@{~>}[dr] \ar@/^/[rr] \ar[r] \ar[d] & {\bullet} \ar@{~>}[dr] \ar[r] \ar[d] & {\bullet} \ar[d] \\
{\bullet}  \ar@/_/[rr] \ar[r] & {\bullet} \ar[r] & {\bullet} }
\end{equation*}

\end{frame}
%\newpage

\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{Invariants}

\begin{itemize}
 \item $A$ is bi-algebra
%$O((Mat_n)_q)$
\pause
 \item $O((GL_n)_q)$ is Hopf algebra
\pause
 \item We have the adjoint coaction of $O((GL_n)_q)$ on $A$
%$O((Mat_n)_q)$
\end{itemize}
\pause

\begin{theorem}[Domokos-Lenagan 2003]
Suppose that $q$ is not root of unity. 
\pause
 $$I:= A^{O((GL_n)_q)}=K[\Delta_1 \dots \Delta_n]$$ where
%$O((Mat_n)_q)$
\pause
$$\Delta_{d}=\sum_{\overset{Ind\subset \{1 \dots n\}}{|Ind|=d}}det_{q}(\{x_{ij}\}_{i,j \in Ind}) \in A$$
%O((Mat_n)_q)
%$det_{q,I}=\sum_{w \in S_{I}}(-q)^{l(w)}x_{i_{1}w(i_{1})} \cdots
%x_{i_{d}w(i_{d})}$
\end{theorem}

\end{frame}
%\newpage

\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{Main Result}
\begin{theorem}[Aizenbud-Yacobi 2010]
Suppose that  $q$ is not root of unity. Then
$A$
%$O((Mat_n)_q)$ 
is free over $I$.
% $O((Mat_n)_q)^{O((GL_n)_q)}$
\end{theorem}
\pause
 
\begin{theorem}[Joseph-Letzter 1994]
Suppose that $q$ is generic and $G$ is semi-simple algebraic group. Then the locally finite part of
the quantum enveloping algebra $U_{q}(\mathfrak{g})$ is free over its center.
\end{theorem}
\pause

\begin{theorem}[Baumann 2000]
Suppose that $q$ is generic. And $G$ is semi-simple and simply connected algebraic group. Then
$O(G_q)$ is free $O(G_q)^{O(G_q)}$.
\end{theorem}

\end{frame}
%\newpage

\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{First Attempt}
Bernstein-Lunts approach:\\
\pause
recall $deg(x_{ij})=\delta_{ij}$\\
\pause
$n=2$ -- OK\\
\pause
$n=3$ -- relations not compatible with the filtration --
\pause

\begin{equation*}
\xymatrix{ {\bullet} \ar@{~>}[dr]  & {\bullet}  & {\bullet}   \\
{\bullet}  & {\bullet}  & {\bullet}\\
{\bullet}   & {\bullet}  & {\bullet} }
\end{equation*}
$deg([x_{11},x_{22}])\leq 1$, indeed $deg(x_{12}x_{21})=0$

\end{frame}

\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{First Attempt}
Bernstein-Lunts approach:\\
recall $deg(x_{ij})=\delta_{ij}$\\
$n=2$ -- OK\\
$n=3$ -- relations not compatible with the filtration --

\begin{equation*}
\xymatrix{ {\bullet}   & {\bullet} \ar@{~>}[dr] & {\bullet}   \\
{\bullet}  & {\bullet}  & {\bullet}\\
{\bullet}   & {\bullet}  & {\bullet} }
\end{equation*}
%\pause
$deg([x_{12},x_{23}])=0 $ but $deg(x_{22}x_{13})=1$

\end{frame}

\begin{frame}
%%\thispagestyle{empty}
%  
    \frametitle{Proof}
\begin{itemize}
 \item  $deg_{(k)}(x_{ij})= 1 \text{ if } |i-j| < k$ and $0$ otherwise
\pause
 \item  Enough to show: 
%$gr_{(n)}(O((Mat_n)_q))$  
$gr_{(n)}(A)$  
is free over  
%$gr_{(n)}(O((Mat_n)_q)^{O((GL_n)_q)})$
$gr_{(n)}(I)$
\pause
 \item 
 Enough to show: 
%$gr_{(n-1)}(gr_{(n)}(O((Mat_n)_q)))$
$gr_{(n-1)}(gr_{(n)}(A))$
  is free over  
%$gr_{(n-1)}(gr_{(n)}(O((Mat_n)_q)^{O((GL_n)_q)}))$
$gr_{(n-1)}(gr_{(n)}(I))$
\pause
\item  $\cdots$
\end{itemize}

\pause
\begin{equation*}
\xymatrix{ {\bullet} \ar@{~>}[ddr] \ar@{~>}[dr] \ar@{~>}@/^/[ddrr]
\ar@{~>}[drr] & {\bullet} \ar@{~>}[ddr] \ar@{~>}[dr]
& {\bullet} \\
{\bullet} \ar@{~>}[dr] \ar@{~>}[drr] & {\bullet} \ar@{~>}[dr] & {\bullet} \\
{\bullet} & {\bullet} & {\bullet} }
\end{equation*}
\end{frame}


\begin{frame}
%%\thispagestyle{empty}
%  
    \frametitle{Proof}
\begin{itemize}
 \item  $deg_{(k)}(x_{ij})= 1 \text{ if } |i-j| < k$ and $0$ otherwise
 \item  Enough to show: 
$gr_{(n)}(A)$  
is free over  
$gr_{(n)}(I)$
 \item 
 Enough to show: 
$gr_{(n-1)}(gr_{(n)}(A))$
  is free over  
$gr_{(n-1)}(gr_{(n)}(I))$
\item  $\cdots$
\end{itemize}

\begin{equation*}
\xymatrix{ 
{\bullet} \ar@{~>}[dr] & {\bullet} & {\bullet} \\
{\bullet} & {\bullet} \ar@{~>}[dr] & {\bullet} \\
{\bullet} & {\bullet} & {\bullet} }
\end{equation*}

\end{frame}

\begin{frame}
%%\thispagestyle{empty}
%  
    \frametitle{Proof}
\begin{itemize}
 \item  $deg_{(k)}(x_{ij})= 1 \text{ if } |i-j| < k$ and $0$ otherwise
 \item  Enough to show: 
$gr_{(n)}(A)$  
is free over  
$gr_{(n)}(I)$
 \item 
 Enough to show: 
$gr_{(n-1)}(gr_{(n)}(A))$
  is free over  
$gr_{(n-1)}(gr_{(n)}(I))$
\item  $\cdots$
\end{itemize}

\begin{equation*}
\xymatrix{ 
{\bullet}  & {\bullet} & {\bullet} \\
{\bullet} & {\bullet}  & {\bullet} \\
{\bullet} & {\bullet} & {\bullet} }
\end{equation*}

\end{frame}


\end{document}




