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\title{The affine signed Brauer algebra}
\author[Zajj Daugherty]{Zajj Daugherty\\
\small
Joint with M.\ Balagovic, I.\ Entova-Aizenbud,  I.\ Halacheva, \\
J.\ Hennig, M.\ S.\ Im, 
G.\ Letzter, 
E.\ Norton,\\ 
V.\ Serganova, and  C.\ Stroppel \\~\\ arXiv:1801.04178\\
(See also arXiv:1610.08470)}
\date{April 22, 2018}





\begin{document}
\begin{frame}\maketitle\end{frame}



%%%%%%%
\begin{frame}{Lie superalgebras}
A \alert{Lie superalgebra} is a  $\ZZ_2$-graded vector space $\mathfrak{g} =\mathfrak{g}_0 \oplus \mathfrak{g}_1$ with a \alert{super} Lie bracket \only<1-4>{
$$[,]:\fg \otimes \fg \to \fg$$
satisfying \\ \smallskip
\centerline{$\displaystyle
[x,y] = - (-1)^{\bar x\bar y}[y,x]
$} \smallskip
and\\ \smallskip
\centerline{$\displaystyle
[x,[y,z]]] = [[x,y], z] + (-1)^{\bar x\bar y}[y ,[x,z]],
$} \smallskip
where $x, y, z$ are each homogeneous, and $\bar{x}$ means \alert{degree}.
}\only<5->{
$[,]:\fg \otimes \fg \to \fg$ satisfying a super symmetry and super Jacobi identity.
}
\pause \medskip

Three types: \alert{basic}, \alert{Cartan type}, and \alert{strange} (two families: \alert{periplectic} and queer).

\medskip

\pause 
Let $V = V_0 \oplus V_1 = \CC^{m|n}$ be a $\ZZ_2$-graded vector space over $\CC$. \\
\only<5->{For (homogeneous) $v \in V_i$, write $\bar{v}=i$ for its \alert{degree}.}




\medskip\pause
The \alert{general linear Lie superalgebra is} \\ \smallskip
\centerline{$\displaystyle
\fgl(m|n) = \End(V)\pause = \fg_0 \oplus \fg_1,
$} \smallskip
where 
\\ \smallskip
\only<5->{\centerline{$\displaystyle
\fg_0 = \left\{\left. \begin{pmatrix} A & 0 \\ 0 & D\end{pmatrix} ~\right|~ A \in \End(V_0), D \in \End(V_1)\right\},
$} 
\smallskip
\centerline{$\displaystyle
\fg_1 = \left\{\left. \begin{pmatrix} 0 & B \\ C & 0\end{pmatrix} ~\right|~ B \in \Hom(V_1,V_0), C \in  \Hom(V_0,V_1)\right\}.
$}} \smallskip


\pause 
\alert{Bracket:} $[x,y] = xy - (-1)^{\bar x\bar y}yx$.

\end{frame}

%%%%%%%
\begin{frame}{Lie superalgebras}
Let $\beta: V \otimes V \to \CC$ be an odd, nondegenerate, homogeneous, bilinear form satisfying \\ \smallskip
\centerline{$\displaystyle
\beta(v,w) = (-1)^{\bar{v}\bar{w}}\beta(w,v)\quad  \text{(\alert{supersymmetric})}.
$} \smallskip
\pause 

The \alert{periplectic} Lie superalgebra is \\ \smallskip
\centerline{$\displaystyle
\fp(V) = \fp(n)= \{x \in \End(V) ~|~ \beta(xv,w) + (-1)^{\bar{x}\bar{v}}\beta(v,xw)=0\}.
$}

\pause \medskip

Specifically, we have 
\\ \smallskip
\centerline{$\displaystyle
\fp(n) \cong \left\{\left. \begin{pmatrix} A & B \\ C & -A^t \end{pmatrix} \in \fgl(n|n) 
		~\right|~ B = B^t, C = -C^t \right\}.
$} \smallskip \pause 
Then, as vector spaces $\fp(n) = \fg_0 \oplus \fg_1 \oplus \fg_{-1} $, where  
\\ \smallskip
\centerline{$\displaystyle
\fg_0 = \left\{ \begin{pmatrix} A & 0 \\ 0 & -A^t \end{pmatrix}\right\} \cong \fgl(n)
$} \smallskip
\centerline{$\displaystyle
\fg_1  = \left\{ \begin{pmatrix} 0 & B \\ 0 & 0 \end{pmatrix}\right\}, \quad 
 \fg_{-1} = \left\{ \begin{pmatrix} 0 & 0 \\ C & 0 \end{pmatrix}\right\}.
$} \smallskip

\pause
\alert{Goal:} Study the representation theory of $\fp(n)$. \pause In particular,  study the category $\mathcal{F}_n$ of finite-dimensional integrable representations. Highest weight category!
\end{frame}


\begin{frame}{Translation functors}

\alert{Key ingredients for other cases:} a large center in  $\cU\fg$, and  translation functors given by tensoring with the natural representation followed by the projection onto a block.\pause \medskip


Namely, you study the action of $\cU\fg$ on 
\\ \smallskip
\centerline{$\displaystyle
M \otimes V \otimes V \otimes \cdots \otimes V = M \otimes V^{\otimes d},
$} \smallskip
where $V$ is $\fg$'s favorite module, and $M$ is another simple module, by constructing operators in $\End_{\fg}(M \otimes V^{\otimes d})$ that commute with the $\fg$-action. Many commuting operators are generated by taking coproducts of central elements. 

\pause 

\alert{Example:} If $\fg = \fso(V)$ or $\fsp(V)$, then the commuting operators generate the \alert{degenerate affine Brauer algebra}; when $\fg=\fsl(V)$, you get the graded Hecke algebra of type A. \pause \smallskip


\alert{Obstruction:} The center of $\cU\fp(V)$ is  trivial! \pause 
But we'll figure it out anyway\dots
\end{frame}


\begin{frame}{Example: $V \otimes V$}
The algebra $\End_{\fp(V)}(V\otimes V)$ is 3-dimensional with basis $1$, 
\\ \smallskip
\centerline{$\displaystyle
s: v\otimes w\mapsto (-1)^{p(v)p(w)} w\otimes v, \quad \text{and} \quad 
e={\beta^*}\circ\beta: v\otimes w \mapsto \beta(v,w)c,
$} \smallskip
where $c$ spans the (super) sign module.
\medskip\pause

Draw: {\def\SCALE{.65}
\\ \smallskip
\centerline{
$\phantom{\alert{marked Brauer}}$\hfill
$\displaystyle
s  = \TikZ{[diagram]
\foreach\x in {1, 2}{\node[v] (t\x) at (.75*\x,1){};\node[v] (b\x) at (.75*\x,0){};}
\draw (t1) to (b2) (t2) to (b1);
}
\quad \text{ and } \quad 
e = \TikZ{[diagram, xscale=1.5]
\foreach\x in {1, 2}{\node[v] (t\x) at (.75*\x,1){};\node[v] (b\x) at (.75*\x,0){};}
\only<1-3>{\draw (t1) to [bend right=50] (t2) (b2) to [bend right=50] (b1);}
\only<4->{\draw (t1) to [bend right=50] node[bT]{} (t2) (b2) to [bend right=50] node[D]{} (b1);}
}\ .
$ \qquad \uncover<4->{(\alert{marked Brauer})}}} 

\medskip\pause

Relation: $e\circ s=e=-s\circ e$. \pause \pause Also, $e^2 = 0$. \hfill(non-semisimple case)
\end{frame}



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



\def\SCALE{.75}


(Moon 2003, Kujawa-Tharp 2014) The \alert{marked Brauer algebra} $B_d(\delta, \epsilon)$, $\epsilon = \pm 1$, is the space spanned by \alert{marked Brauer diagrams}
$$d=\TikZ{[diagram]
\foreach\x in {1,2,...,5}{
\node[v] (t\x) at (\x,1.5){};\node[v] (b\x) at (\x,0){};}
\node[v] (t6) at (6,1.5){};\node[v] (b6) at (6,0){};
\draw (t1) to (b2) (t4) to (b6)
	(t2) to [bend right] node[bT]{} (t6) (t3) to [bend right] node[T]{} (t5)
	(b4) to [bend right=40] node[D]{} (b1) (b3) to [bend left] node[D]{} (b5);
}
\qquad 
\alert{\begin{array}{l}
\text{caps get one $\TikZ{[diagram]\node[D] at (0,0){};}$ each,}\\
\text{cups get one $\TikZ{[diagram]\node[T] at (0,0){};}$ or $\TikZ{[diagram]\node[bT] at (0,0){};}$ each, }\\
\text{no two markings at same height.}
\end{array}}
$$
 with equivalence up to isotopy except for the local relations 

\smallskip

\centerline{$\begin{matrix}\TikZ{[diagram]\draw (0,0) to [bend right=40] node[T]{} (1.5,0); \draw[thin, densely dotted] (0,-.5) rectangle (1.5,.1);}
=
\epsilon\ \TikZ{[diagram]\draw (0,0) to [bend right=40] node[bT]{} (1.5,0); \draw[thin, densely dotted] (0,-.5) rectangle (1.5,.1);}\\
\TikZ{[diagram]\draw[rounded corners] (0,0) to (.75,-.25) to [bend left=40] (0,-.25) to (.75,0); \node[T] at (.75/2, -.4) {}; \draw[thin, densely dotted] (0,-.6) rectangle (.75,.1);}
=
\epsilon\ \TikZ{[diagram]\draw (0,0) to [bend right=40] node[bT] {}(1,0); \draw[thin, densely dotted] (0,-.4) rectangle (1,.1);}
\end{matrix} 
\quad \text{and}\quad
\TikZ{[diagram]
\draw (0,.8) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$x$} (1,.8); \draw[thin, densely dotted] (0,.5) rectangle (1,1.1);
\draw (1.25,0) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$y$} (2.25,0);  \draw[thin, densely dotted] (1.25,.3) rectangle (2.25,-.3);}
=
\epsilon\ 
\TikZ{[diagram]
\draw (0,0) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$x$} (1,0); \draw[thin, densely dotted] (0,.3) rectangle (1,-.3);
\draw (1.25,.8) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$y$} (2.25,.8);  \draw[thin, densely dotted] (1.25,.5) rectangle (2.25,1.1);}
$}

\smallskip

for any \alert{adjacent} markings $\TikZ{[diagram] \node[draw, fill=white, circle, inner sep=1pt] at (0,0) {\scriptsize$x$};}$ and $\TikZ{[diagram] \node[draw, fill=white, circle, inner sep=1pt] at (0,0) {\scriptsize$y$};}$ (meaning no markings of height between these two). \pause Multiplication is given by vertical concatenation, with relations  
$\TikZ{[line width=.75pt, scale=.4] \draw (1,1) ellipse [x radius=.75cm,y radius=.5cm];}=\delta,$

$$\TikZ{[diagram]\draw (0,0) to [bend left] node[D]{} (.75,0) to [bend right] node[bT]{} (1.5,0); \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);} 
= \TikZ{[diagram]\draw (0,0) to (.75,0); \draw[thin, densely dotted] (0,-.3) rectangle (.75,.3);}
=\TikZ{[diagram]\draw (0,0) to [bend right] node[T]{} (.75,0) to [bend left] node[D]{} (1.5,0);  \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);}
\quad \text{ and } \quad 
\TikZ{[diagram]\draw (0,0) to [bend left] node[D]{} (.75,0) to [bend right] node[T]{} (1.5,0); \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);} 
= \epsilon\ \TikZ{[diagram]\draw (0,0) to (.75,0); \draw[thin, densely dotted] (0,-.3) rectangle (.75,.3);}
=\TikZ{[diagram]\draw (0,0) to [bend right] node[bT]{} (.75,0) to [bend left] node[D]{} (1.5,0);  \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);}\ .$$



\end{frame}



%%%%%%
\begin{frame}

(Moon 2003, Kujawa-Tharp 2014) The \alert{marked Brauer algebra} $B_d(\delta, \epsilon)$, $\epsilon = \pm 1$, is the space spanned by \alert{marked Brauer diagrams}\dots

\def\SCALE{.75}

$$\begin{matrix}\TikZ{[diagram]\draw (0,0) to [bend right=40] node[T]{} (1.5,0); \draw[thin, densely dotted] (0,-.5) rectangle (1.5,.1);}
=
\epsilon\ \TikZ{[diagram]\draw (0,0) to [bend right=40] node[bT]{} (1.5,0); \draw[thin, densely dotted] (0,-.5) rectangle (1.5,.1);}\\
\TikZ{[diagram]\draw[rounded corners] (0,0) to (.75,-.25) to [bend left=40] (0,-.25) to (.75,0); \node[T] at (.75/2, -.4) {}; \draw[thin, densely dotted] (0,-.6) rectangle (.75,.1);}
=
\epsilon\ \TikZ{[diagram]\draw (0,0) to [bend right=40] node[bT] {}(1,0); \draw[thin, densely dotted] (0,-.4) rectangle (1,.1);}
\end{matrix}  
\qquad
\TikZ{[diagram]
\draw (0,.8) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$x$} (1,.8); \draw[thin, densely dotted] (0,.5) rectangle (1,1.1);
\draw (1.25,0) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$y$} (2.25,0);  \draw[thin, densely dotted] (1.25,.3) rectangle (2.25,-.3);}
=
\epsilon\ 
\TikZ{[diagram]
\draw (0,0) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$x$} (1,0); \draw[thin, densely dotted] (0,.3) rectangle (1,-.3);
\draw (1.25,.8) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$y$} (2.25,.8);  \draw[thin, densely dotted] (1.25,.5) rectangle (2.25,1.1);}
$$
$$\TikZ{[line width=.75pt, scale=.4] \draw (1,1) ellipse [x radius=.75cm,y radius=.5cm];}=\delta$$
$$\TikZ{[diagram]\draw (0,0) to [bend left] node[D]{} (.75,0) to [bend right] node[bT]{} (1.5,0); \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);} 
= \TikZ{[diagram]\draw (0,0) to (.75,0); \draw[thin, densely dotted] (0,-.3) rectangle (.75,.3);}
=\TikZ{[diagram]\draw (0,0) to [bend right] node[T]{} (.75,0) to [bend left] node[D]{} (1.5,0);  \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);}
\qquad
\TikZ{[diagram]\draw (0,0) to [bend left] node[D]{} (.75,0) to [bend right] node[T]{} (1.5,0); \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);} 
= \epsilon\ \TikZ{[diagram]\draw (0,0) to (.75,0); \draw[thin, densely dotted] (0,-.3) rectangle (.75,.3);}
=\TikZ{[diagram]\draw (0,0) to [bend right] node[bT]{} (.75,0) to [bend left] node[D]{} (1.5,0);  \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);}$$






\bigskip

For example, \def\SCALE{.6}
\\\centerline{$\TikZ{[diagram]
\foreach\x in {1,2,...,6}{
\node[v] (t\x) at (\x,1.6){};\node[v] (b\x) at (\x,.1){};}
\draw (t1) to (b2) (t4) to (b6)
	(t2) to [bend right] node[bT, pos=.4]{} (t6) (t3) to [bend right] node[T]{} (t5)
	(b4) to [bend right=40] node[D]{} (b1) (b3) to [bend left] node[D, pos=.6, sloped]{} (b5);
\foreach\x in {1,2,...,6}{
\node[v] (bt\x) at (\x,-.1){};\node[v] (bb\x) at (\x,-1){};
\draw[thin] (bt\x) to (b\x);}
\draw (bt1) to (bb1) (bt2) to (bb2) (bt5) to (bb5) (bt6) to (bb6)
	(bt3) to [bend right] node[T]{} (bt4) (bb4) to [bend right] node[D]{} (bb3)
	;}\pause
= \epsilon\ 
\TikZ{[diagram]
\foreach\x in {1,2,...,6}{
\node[v] (t\x) at (\x,1.5){};\coordinate (b\x) at (\x,0); \node[v] (bb\x) at (\x,-.9){};}
\draw 
	(t2) to [bend right] node[bT, pos=.4]{} (t6) (t3) to [bend right] node[T]{} (t5) (bb4) to [bend right] node[D]{} (bb3);
\draw [rounded corners] (t1) to (b2) to (bb2)
	(t4) to (b6) to (bb6)
	 (bb1) to (b1) to [bend left=50] node[pos=.6] (a){} (b3)  to [bend right] node (b){} (b4) to [bend left] node (c){} (b5) to (bb5);
\draw[densely dotted, red] (3,-.5) rectangle (5,.5);
\node[D] at (a){}; \node[bT] at (b){}; \node[D] at (c){};
}\pause
=
\TikZ{[diagram]
\foreach\x in {1,2,...,6}{
\node[v] (t\x) at (\x,1.5){};\node[v]  (b\x) at (\x,0){};}
\draw 
	(t2) to [bend right] node[bT, pos=.4]{} (t6) (t3) to [bend right] node[T]{} (t5) 
	(b4) to [bend right] node[D]{} (b3) (b5) to [bend right] node[D]{} (b1)
	(t1) to (b2) (t4) to [bend left] (b6);
}
$}

\pause 
\alert{Note:}\\
\alert{(1)} $B_d(\delta, 1) = $ classical Brauer.\\
\alert{(2)} If $\epsilon = -1$, then multiplication is well-defined exactly when $\delta =0$.\\$\phantom{\quad}$ This case is called the \alert{signed Brauer algebra}. 

\end{frame}







\begin{frame}{Centralizer algebras}
\def\SCALE{.5}

The marked Brauer algebra $B_d(\delta, \epsilon)$ is generated by 
{\def\SCALE{.65}
\centerline{$s_i = \TikZ{[diagram]
\foreach\x in {1, 3, 4, 6}{
\node[v] (t\x) at (.75*\x,1){};\node[v] (b\x) at (.75*\x,0){};}
\draw (t1) to (b1) (t3) to (b4) (t4) to (b3) (t6) to (b6);
\node[above] at (t3) {\tiny\alert{$\phantom{+}i\ \phantom{+}$}};\node[above] at (t4) {\tiny\alert{$\ i\!+\!1$}};\node[below, white] at (b4) {\tiny{$\ i\!+\!1$}};
\node at (2*.75, .5){$\dots$}; \node at (5*.75, .5){$\dots$}; 
}
\quad \text{ and } \quad 
e_i = \TikZ{[diagram]
\foreach\x/\y in {1.5/1, 3/3, 4/4, 5.5/6}{\node[v] (t\y) at (\x,1){};\node[v] (b\y) at (\x,0){};}
\draw (t1) to (b1) (t3) to [bend right=40] node[bT]{} (t4) (b4) to [bend right] node[D]{} (b3) (t6) to (b6);
\node[above] at (t3) {\tiny\alert{$\phantom{+}i\ \phantom{+}$}};\node[above] at (t4) {\tiny\alert{$\ i\!+\!1$}};\node[below, white] at (b4) {\tiny{$\ i\!+\!1$}};
\node at (2+.25, .5){$\dots$}; \node at (5-.25, .5){$\dots$}; 
}\ ,
$}}
for $i=1, \dots, k-1$. 

\pause\medskip



Let $\fg = \fgl(V)^\beta$ ($\beta$-invariants) with $V = V_0 \oplus V_1$ (for $\beta$ non-deg, homog, super symmetric, bilinear). \pause If  \\ \smallskip
\centerline{$\displaystyle
\beta^* : \CC \to V \otimes V
 \quad \text{ and } \quad 
	\begin{array}{rl} s: V \otimes V &\to V \otimes V \\
				u \otimes v &\mapsto (-1)^{\bar{u}\bar{v}} v \otimes u\pause,\end{array}
$} \smallskip
then the map	\\ \smallskip
\centerline{$\displaystyle
e_i \mapsto 1^{\otimes i-1} \otimes \beta^* \beta \otimes 1^{d-i-1}, \quad 
	s_i \mapsto 1^{\otimes i-1} \otimes s \otimes 1^{d-i-1},
$} \smallskip
for $ i=1, \dots, d-1$,
 gives 
\\ \smallskip
\centerline{$\displaystyle
B_d(\delta, \epsilon) \to \End_{\fg}(V^{\otimes d})
$} \smallskip
 when $\delta = \dim V_0 - \dim V_1$ and $\epsilon = (-1)^{\bar{\beta}}$ [KT14].




\end{frame}







\begin{frame}{Classical: Jucys-Murphy elements and the Casimir}
\only<1-6>{For $i<j$, let 
{\def\SCALE{.65}
\centerline{$s_{i,j} = \TikZ{[diagram]
\foreach\x/\y in {1/1, 2/2, 2.5/3, 4.5/4, 5/5, 6/6}{\node[v] (t\y) at (\x,1){};\node[v] (b\y) at (\x,0){};}
\draw (t1) to (b1) (t2) to (b5) (t3) to (b3) (t4) to (b4)  (t5) to (b2) (t6) to (b6);
\node[above] at (t2) {\tiny\alert{$\phantom{j}i\ \phantom{j}$}};\node[above] at (t5) {\tiny\alert{$j$}};%\node[below, white] at (b4) {\tiny{$j$}};
\node at (1.5, .5){\tiny$\dots$}; \node at (5.5, .5){\tiny$\dots$};  \node at (3.5, .75){\tiny$\dots$}; 
}
\quad \text{ and } \quad 
e_{i,j} = \TikZ{[diagram]
\foreach\x/\y in {1/1, 2/2, 2.5/3, 4.5/4, 5/5, 6/6}{\node[v] (t\y) at (\x,1){};\node[v] (b\y) at (\x,0){};}
\draw (t1) to (b1)  (t3) to (b3) (t4) to (b4)(t6) to (b6)
		(t2) to [bend right=20] node[bT]{} (t5) (b5) to [bend right=20] node[D]{} (b2) ;
\node[above] at (t2) {\tiny\alert{$\phantom{j}i\ \phantom{j}$}};\node[above] at (t5) {\tiny\alert{$j$}};%\node[below, white] at (b4) {\tiny{$j$}};
\node at (1.5, .5){\tiny$\dots$}; \node at (5.5, .5){\tiny$\dots$};  \node at (3.5, .5){\tiny$\dots$}; 
}\ .
$}}}\pause 


The Brauer algebra $B_d(\delta) = B_d(\delta, 1)$ has Jucys-Murphy elements 
\\ \smallskip
\centerline{$\displaystyle
x_j =c+ \sum_{i=1}^{j-1} s_{i,j} - e_{i,j}, \quad c\in \CC,\ j=1, \dots, d,
$} \smallskip
that pairwise commute (Nazarov 1996).
\pause Define the degenerate affine version  by \\ \smallskip
\centerline{$\displaystyle
\VW_d(\delta) = \CC[y_1, \dots, y_d] \otimes B_d(\delta) \otimes \CC[y_1, \dots, y_d]/\text{(relations)}.
$} \smallskip
\pause 

\alert{Action on tensor space:} Fix $\fg = \fso(V)$ or $\fsp(V)$. Let
$\Omega \in \cU\fg \otimes \cU \fg$ be the 
split Casimir invariant, given by
$$
\Omega = 2\sum_{b \in \Lambda} b \otimes b^*,$$
where $\Lambda$ is a basis of $\fg$, and $\{b^*~|~b \in \Lambda\}$ is the dual basis w.r.t.\ $\beta$. \pause
Then $\Omega$ acts on $V \otimes V$ as as $s_1 - e_1$. \pause So the action of $x_j$ on $V^{\otimes d}$ is the same as that of 
$\sum_{i=1}^{j-1} \Omega_{i,j}$.
%
%\pause \alert{Goal:} Define degenerate affine marked Brauer and cyclotomic Brauer algebras, and classify their representation theory. 
\end{frame}


\begin{frame}{Classical: Action on $M \otimes V^{\otimes d}$}
Define the degenerate affine version by 
$$\VW_d(\delta) = \CC[y_1, \dots, y_d] \otimes B_d(\delta)\otimes\CC[y_1, \dots, y_d]/\text{(relations)},$$
where relations for the $y_i$'s are those satisfied between the $x_i$'s in $B_d(\delta)$.  Let $M$ be a $\fg$-simple module,  and let 
$$y_j \text{ act on } M \otimes V^{\otimes d} \text{ by } \sum_{i = 0}^{j-1} \Omega_{i,j}.$$
\pause  Then letting the finite part act on $V^{\otimes d}$ as before, and as the identity on $M$, we have a representation 
$$\VW_d(\delta) \to \End_\fg (M \otimes V^{\otimes d}).$$




\end{frame}

\begin{frame}{Periplectic J-M elements and action on tensor space}

But back to periplectic land\dots\\
The center is trivial, so there's no Casimir element in $\cU\fp(n)$, and there's no split Casimir in $\cU\fp(n) \otimes \cU\fp(n)$. 

\medskip\pause

\alert{Key observation:} With respect to the inner product\\ \smallskip
\centerline{$\displaystyle
\<x,y\> = \str(xy) = \tr\left(xy|_{V_0}\right) - \tr\left(xy|_{V_1}\right),
$} \smallskip
on $\fgl(n|n)$, we have 
\\ \smallskip
\centerline{$\displaystyle
\fp(n)^* = \fp(n)^\perp \qquad \text{so} \qquad \fgl(n|n) = \fp(n) \oplus \fp(n)^*.
$} \smallskip
\pause We define the \alert{split (fake) Casimir} element of \\ \smallskip
\centerline{$\displaystyle
\fp(n) \otimes \fp(n)^* \subset \fp(n) \otimes\mathfrak{gl}(n|n)
$} \smallskip by 
\\ \smallskip
\centerline{$\displaystyle
\Omega=2\sum b\otimes b^* ,
$} \smallskip
where the sum is over a basis of $\fp(n)$ and its dual in $\fp(n)^*$. \pause \medskip

Then for any $\fp(n)$-module $M$, $\Omega$ acts on $M \otimes V$, and that action commutes with the action of $\fp(n)$. [BDEHHILNSS]

%\smallskip\pause 
%
%\centerline{So \qquad $\sVW_d \to \End_\fp (M \otimes V^{\otimes d})$. \qquad
%}

\end{frame}

\begin{frame}
Draw Jucys-Murphy elements as 
 {\def\SCALE{.65}
\\ \smallskip
\centerline{
$y_i  = \TikZ{[diagram]
\foreach\x in {1, 3, 4, 5, 7}{\node[v] (t\x) at (.75*\x,1){};\node[v] (b\x) at (.75*\x,0){};
	\draw (t\x) to (b\x);}
	\node[B] at (.75*4, .5){};
	\foreach \x in {2, 6}{\node at (.75*\x,.5) {$\cdots$};}
	\node[above] at (t4) {\small $i$};
	}
$}} 
\pause 

Define the \alert{affine signed Brauer algebra} $\sVW_d$ as the algebra generated by $B_d(0, -1)$ and $y_1, \dots, y_d$, together with relations 
{\def\SCALE{.65}
\\ \smallskip
\centerline{
$\TikZ{[diagram]
\foreach\x in {1, 2}{\node[v] (t\x) at (.75*\x,1){};\node[v] (b\x) at (.75*\x,0){};
	\draw (t\x) to (b\x);}
	\node[B] at (.75*2, .5){};
	}
	=
	\TikZ{[diagram]
\foreach\x in {1, 2}{\node[v] (t\x) at (.75*\x,1){};\node[v] (b\x) at (.75*\x,0){};
	}
	\draw[rounded corners] (t2) to (.75*1,.5) to (b2);
	\draw[rounded corners] (t1) to (.75*2,.5) to (b1);
	\node[B] at (.75*1+.1, .5){};
	}
	+ \TikZ{[diagram]
\foreach\x in {1, 2}{\node[v] (t\x) at (.75*\x,1){};\node[v] (b\x) at (.75*\x,0){};}
\draw (t1) to (b2) (t2) to (b1);
}
	+ \TikZ{[diagram, xscale=1.5]
\foreach\x in {1, 2}{\node[v] (t\x) at (.75*\x,1){};\node[v] (b\x) at (.75*\x,0){};}
\draw (t1) to [bend right=50] node[bT]{} (t2) (b2) to [bend right=50] node[D]{} (b1);
}
$,  \qquad $
\begin{matrix}
\TikZ{[diagram, scale=1.5]
\foreach\x in {1, 2}{\node[v] (t\x) at (.75*\x,1){};}
\draw (t1) to [bend right=100] node[bT]{} 
	node[pos=.75, B]{}
	(t2);
}
=\TikZ{[diagram, scale=1.5]
\foreach\x in {1, 2}{\node[v] (t\x) at (.75*\x,1){};}
\draw (t1) to [bend right=100] node[bT]{} 
	node[pos=.25, B]{}
	(t2);
}-
\TikZ{[diagram, scale=1.5]
\foreach\x in {1, 2}{\node[v] (t\x) at (.75*\x,1){};}
\draw (t1) to [bend right=100] node[bT]{} 
%	node[pos=.25, B]{}
	(t2);
},\\\text{and}\\
\TikZ{[diagram, scale=1.5]
\foreach\x in {1, 2}{\node[v] (b\x) at (.75*\x,0){};}
\draw (b2) to [bend right=100] node[D]{} 
	node[pos=.25, B]{}
	(b1);
}
=
\TikZ{[diagram, scale=1.5]
\foreach\x in {1, 2}{\node[v] (b\x) at (.75*\x,0){};}
\draw (b2) to [bend right=100] node[D]{} 
	node[pos=.75, B]{}
	(b1);
}+
\TikZ{[diagram, scale=1.5]
\foreach\x in {1, 2}{\node[v] (b\x) at (.75*\x,0){};}
\draw (b2) to [bend right=100] node[D]{} 
%	node[pos=.75, B]{}
	(b1);
}.
\end{matrix}
$}} 

\pause

\alert{Some results: } ([BDEHHILNSS-2])
\begin{itemize}
\item Presentation of $\sVW_d$ and related algebras/categories.
\item Action on tensor space and translation functors.
\item Filtrations and specializations.
\item Basis and spanning sets.
\item Center.
\end{itemize}\pause
Many other result about the category of  finite-dimensional integrable $\fp(V)$-modules itself in [BDEHHILNSS-1] as well! 




\end{frame}


\begin{frame}
\vspace{-2pt}
\hbox{\hspace{-.51in}
\begin{minipage}{1in}\includegraphics[width=5.15in]{groupphoto-text}\end{minipage}}
\end{frame}




\end{document}
