\include{preamble}

\begin{document}

Speaker: Yoshiyuki Kimura

Title: Graded quiver varieties and quantum cluster algebras

(joint work with Fan Qin)

February 22, 2012

\section{Introduction.}

Set $\Sigma = (\vec{x}, \vec{\rm ex}, Q)$ where $\vec{x}$ is a set of variables, $\vec{\rm ex}$ is a subset of the variables (the exchangeable variables), and $Q$ is a quiver without loops or 2-cycles. We assume that $Q_0 = \vec{x}$. The frozen variables are $\vec{\rm fr} = \vec{x} \setminus \vec{\rm ex}$.

\subsection{Mutation.}

Given $x \in \vec{\rm ex}$, we first add new arrows $[\beta\alpha]$ for each subquiver $\bullet \xrightarrow{\alpha} x \xrightarrow{\beta} \bullet$. Then we reverse all arrows incident with $x$ and remove all ``maximal'' 2-cycles. This new quiver is denoted $\mu_x(Q)$. It is easy to check that $\mu_x^2 = 1$ for all $x \in Q_0$.

Let $\cF_\Sigma = \bQ(x \mid x \in \vec{x})$. Define $x^*$ by the exchange formula
\[
x^* x = \prod_{\substack{h \in Q\\ {\rm out}(h) = x}} y^{\#(x\to y)} + \prod_{\substack{h \in Q \\ {\rm in}(h) = x}} y^{\#(y\to x)}.
\]

Define $\mu_x(\Sigma) = (\vec{x'}, \vec{\rm ex'}, \mu_x(Q))$ where $\vec{x'} = \vec{x} \setminus \{x\} \cup \{x^*\}$ and $\vec{\rm ex'} = \vec{\rm ex} \setminus \{x\} \cup \{x^*\}$. Given $(x_1, \dots, x_\ell)$ exchangeable, we can define $x_1 \in \vec{\rm ex}, x_2 \in \mu_{x_1}(\vec{\rm ex}), \dots, x_\ell \in \mu_{x_{\ell-1}}( \cdots \mu_{x_1}(\vec{\rm ex}))$. Set 
\begin{align*}
{\rm Mut}(\Sigma) &= \{\mu_{x_\ell} \cdots \mu_{x_1}(\Sigma) \mid (x_1, \dots, x_\ell) \text{ is exchangeable}\}\\
\cX(\Sigma) &= \bigcup_{\vec{x} \in {\rm Mut}(\Sigma)} \vec{x} \subset \cF_{\Sigma}\\
\sA(\Sigma) &= \langle \cX(\Sigma) \rangle_{\rm algebra} \subset \cF_\Sigma\\
\cM(\Sigma) &= \bigcup_{\vec{x} \in {\rm Mut}(\Sigma)} \{ \prod_{x \in \vec{x}} x^{a_x} \mid a_x \in \bZ_{\ge 0}\}.
\end{align*}

\subsection{Laurent phenomena.}

\begin{theorem}[Fomin--Zelevinsky 2001]
\[
\cA(\Sigma) \subset \bigcap_{(\vec{x}, \vec{\rm ex}, Q) \in {\rm Mut}(\Sigma)} \bZ[x \mid x \in \vec{\rm fr}][x^{\pm 1} \mid x \in \vec{\rm ex}]
\]
\end{theorem}

\begin{conjecture}
\[
\cA(\Sigma) \subset \bigcap_{(\vec{x}, \vec{\rm ex}, Q) \in {\rm Mut}(\Sigma)} \bZ_{\ge 0}[x \mid x \in \vec{\rm fr}][x^{\pm 1} \mid x \in \vec{\rm ex}]
\]
\end{conjecture}

\begin{theorem}[Fomin--Zelevinsky, Caldero--Keller]
\begin{compactenum}[\rm 1.]
\item If $\# \cX(\Sigma) < \infty$, then $Q_{\vec{\rm ex}}$ is a Dynkin quiver.
\item If $\# \cX(\Sigma) < \infty$, then $M(\Sigma)$ is a free basis of $\sA(\Sigma)$ over $\bZ$.
\end{compactenum}
\end{theorem}

\begin{conjecture}[``Strong'' positivity conjecture]
There exists a basis $\cB(\Sigma)$ of $\sA(\Sigma)$ with the following properties:
\begin{compactenum}[\rm 1.]
\item $\cM(\Sigma) \subset \cB(\Sigma)$
\item $\cB(\Sigma)$ has positive structure constants.
\end{compactenum}
\end{conjecture}

\section{Graded quiver varieties.}

This will give us a connection between two kinds of categorification of cluster algebras: additive and monoidal.

Let $(I,\Omega)$ be an acyclic quiver. Let $\hat{I} = I \times \frac{1}{2} \bZ = \hat{I}_0 \cup \hat{I}_1$ where $\hat{I}_0 = I \times \bZ$ and $\hat{I}_1 = I \times (\frac{1}{2} + \bZ)$.

Let $V,W$ be $\hat{I}$-graded vector spaces over $\bC$. Set 
\begin{align*}
E_\Omega(V,W)^{[n]} &= \bigoplus_{\substack{h \in \Omega\\ a \in \hat{I}}} \hom(V_{{\rm out}(h)}(a), W_{{\rm in}(h)}(a+n))\\
L(V,W)^{[n]} &= \bigoplus_{(i,a) \in \hat{I}} \hom(V_i(a), W_i(a+n))a
\end{align*}
Now let $V$ and $W$ be $\hat{I}_1$- and $\hat{I}_0$-graded, respectively and define
\[
M_\Omega(V,W) = E_\Omega(V,V)^{[0]} \oplus E_{\Omega^*}(V,V)^{[-1]} \oplus L(V,W)^{[-1/2]} \oplus L(W,V)^{[-1/2]}
\]

We define the ``moment map''
\begin{align*}
\mu \colon M_\Omega(V,W) &\to L(V,V)^{[-1]}\\
(B,\alpha, \beta) &\mapsto [B,B] + \alpha \beta
\end{align*}

We say that $(B,\alpha, \beta)$ is {\bf stable} if for all $V' \subset V$ such that $BV' \subset V'$ and $V' \subset \ker \beta$, we have that $V' = 0$. The stable locus is denoted by a superscript $s$. Define $M(V,W) = \mu^{-1}(0)^s / G_V$ and $M_0(V,W) = \mu^{-1}(0) /\!\!/ G_V$ where $G_V = \prod_{(i,a) \in \hat{I}} \GL(V_i(a))$. We have a projective morphism $\pi \colon M(V,W) \to M_0(V,W)$ and $M(V,W)$ is smooth and connected. There is a closed embedding $M_0(V,W) \subset M_0(V',W)$ if $\dim  V'_i(a) \ge \dim V_i(a)$ for all $(i,a) \in \hat{I}$. We set $M_0(W) = \bigcup_v M_0(V,W)$.

Let $\cP_W = \{L \mid L \text{ is a direct summand of } \pi_! \ul{\bC}_{M(V,W)}[d] \text{ for some } V \text{ and } d\}$ and $Q_W = \langle \cP_W \rangle$ (generated by direct summands as a subset of $D^b_c(M_0(V,W))$).

Define the (possibly empty) open set 
\[
M_0^{\rm reg}(V,W) = \{(B, \alpha, \beta) \in M_0(V,W) \mid {\rm Stab}_{G_V}(B, \alpha, \beta) = 1 \} \subset M_0(V,W).
\]

\begin{definition}
A pair $(V,W)$ is {\bf $\ell$-dominant} if 
\[
\dim W_i(a) - \dim V_i(a + \frac{1}{2}) - \dim V_i(a - \frac{1}{2}) + \sum_{\substack{h \in \Omega\\ {\rm out}(h) = i}} \dim V_{{\rm in}(h)}(a + \frac{1}{2}) + \sum_{\substack{h \in \Omega\\ {\rm in}(h) = i}} \dim V_{{\rm out}(h)}(a- \frac{1}{2}) \ge 0
\]
for all $(i,a) \in \hat{I}_0$. Denote this quantity $W - C_qV \in \bZ_{\ge 0}^{\hat{I}_0}$.
\end{definition}

\begin{theorem}[Qin]
\begin{compactenum}[\rm 1.]
\item $M_0^{\rm reg}(V,W) \ne \emptyset$ if and only if $M(V,W) \ne \emptyset$ and $(V,W)$ is $\ell$-dominant.
\item $M_0(W) = \coprod_{(V,W)\ \ell\mathrm{-dominant}} M_0^{\rm reg}(V,W)$
\item $\cP_W = \{{\rm IC}_V(W) := {\rm IC}(\ol{M_0^{\reg}(V,W)}; \bC) \mid (V,W)\ \ell\mathrm{-dominant}\}$.
\end{compactenum}
\end{theorem}

\section{(Deformed) monoidal categorification.}

Let $K_W = K_0(Q_W)$ and $K_W^* = \hom_{\bZ[t,t^{-1}]}(K_W, \bZ[t,t^{-1}])$ which contains $\{L_W(V)\}$ which is the dual of $\cP_W$. We define $\cR_t = \{(f_W) \in \prod K_W^* \mid \langle f_W, {\rm IC}_W(V) \rangle = \langle f_{W - C_q V}, {\rm IC}_{W - C_q V}(0) \rangle\}$.

$\{L_W(0)\}$ gives a basis of $\cR_t$ (dual canonical basis). 

\begin{compactitem}
\item We can construct a product on $\cR_t$ (analogue of Lusztig's restriction functor).

\item $L$ is a $\bZ[t,t^{-1}]$-basis with positive structure constants.
\end{compactitem}

$W$ satisfies $(*)_V$ if $\dim W_i(a) = 0$ unless $a \in \{0, 1,\dots, r\}$. This gives a subalgebra $\cR_{t,r} \subset \cR_t$ whose basis is $L_r = \{L_W(0) \mid W \text{ satisfies } (*)_r\}$.

\begin{conjecture}
\begin{compactenum}[\rm 1.]
\item $\cR_{t,r}$ has the structure of a quantum cluster algebra.
\item $\cM \subset L_r$
\end{compactenum}
\end{conjecture}

\begin{theorem}[Kimura--Qin]
The above conjecture holds in the case $r=1$.
\end{theorem}

\end{document}