Author: Eiko

Tags: quiver representation, moment map, symplectic reduction, Kleinian singularity

Time: 2024-11-19 20:15:09 - 2025-01-09 11:22:13 (UTC)

Reference: Deomposition of Marsden-Weinstein Reductions for Representations of Quivers by Crawley-Boevey

Infinite Root Systems And Representation Theory

• Indecomposable representations of quivers of finite type (finite number of indecomposable representations) correspond to the positive roots of the root system of the corresponding finite dimensional simple Lie algebra.

  Recall that Gabriel’s theorem says that a quiver is of finite type iff it is Dynkin.

  It can be proved by using reflection functors, constructing all indecomposables from the simples, just like Weyl group produces all positive roots from simple roots.

Root System Explained In Representation Theory

• $R_{\lambda }^{+}$ the set of positive roots is expected to be the set dimensions of indecomposable representations.

• $\mathbb{N}{R}_{\lambda }^{+}$ is expected to be the set of dimension vectors that have representations.

• ${\mathrm{\Sigma }}_{\lambda }$ is expected to be the set of dimension vectors that corresponds to simple representations.

• ${\mu }_{\alpha }^{-1}(\lambda )$ is the space parametrizing representations of ${\mathrm{\Pi }}^{\lambda }$.

• $\mathcal{M}(\lambda ,\alpha )={\mu }_{\alpha }^{-1}(\lambda )//G_{\alpha }$ is the set of isomorphic classes of representations of preprojective algebra ${\mathrm{\Pi }}^{\lambda }$ of dimension $\alpha$.

• $\mathcal{M}(\lambda ,\theta ,\alpha )={\mu }_{\alpha }^{-1}(\lambda {)}^{\theta -ss}/{/}_{\theta }{G}_{\alpha }$ is the set of isomorphic classes of $\theta$-semistable representations of preprojective algebra ${\mathrm{\Pi }}^{\lambda }$ of dimension $\alpha$.

Symplectic (Marsden-Weinstein) Reduction

Recall that given a quiver and a dimension vector $\alpha \in {\mathbb{N}}^I$, the group $G(\alpha )=\prod {\mathrm{GL}}_{{\alpha }_i}/K^{\times }$ acts on $\mathrm{Rep}(Q,\alpha )$ and also its cotangent bundle $\mathrm{Rep}(\bar{Q},\alpha )$. The moment map is

$${\mu }_{\alpha }:\mathrm{Rep}(\bar{Q},\alpha )\to \mathrm{Lie}(G{)}^{\ast }={\mathrm{End}}_0(\alpha )=\{\sum \mathrm{tr}(A_i)=0\}$$

Let $\lambda =({\lambda }_i{1}_{{\alpha }_i}{)}_i$, the reduction in the title is defined to be the GIT quotient

$${\mu }^{-1}(\lambda )//G(\alpha ).$$

For the undoubled $Q$ (we always use $Q$ for the undoubled quiver, to distinguish from the doubled quiver $\bar{Q}$), there is a very useful quantity

$$p_Q(\alpha )=1+\sum {\alpha }_s{\alpha }_t-\sum {\alpha }_i^2=1-⟨\alpha ,\alpha {⟩}_Q.$$

If $\alpha$ is a positive root, which are exactly the dimensions of indecomposables, $p(\alpha )$ is the number of parameters of indecomposable representations of dimension $\alpha$.

Geometry In Terms of Combinatorics Of Decomposition

The folllowing are equivalent:

• ${\mu }_{\alpha }$ is flat

• ${\mu }_{\alpha }^{-1}(0)$ has dimension $\alpha \cdot \alpha -1+2p(\alpha )$.

  how to understand this number? Consider it subtract $\alpha \cdot \alpha -1$, which is what we expect the dimension of ${\mu }_{\alpha }^{-1}(0)//G(\alpha )$ to be, and that equals $2p(\alpha )$. It is $2p(\alpha )$ not $p(\alpha )$ because of the doubling.

• $p(\alpha )\ge \sum p({\beta }^{(i)})$ for all decomposition of $\alpha$ into positive roots.

• $p(\alpha )\ge \sum p({\beta }^{(i)})$ for all decomposition of $\alpha$ into ${\mathbb{N}}^I$.

Simple Representations

Let ${\mathrm{\Sigma }}_{\lambda }$ be the set of positive roots $\alpha \in R_{\lambda }^{+}$ such that $\lambda \cdot \alpha =0$ and $p(\alpha )>\sum p({\beta }^{(i)})$ for all decomposition of $\alpha$ into positive roots ${\beta }^{(i)}$ with $\lambda \cdot {\beta }^{(i)}=0$.

Here the symbol $\mathrm{\Sigma }$ stands for simple, this is the set of dimension vectors that have a simple representation, and the generic representation in the locus ${\mu }_{\alpha }^{-1}(0)$ is simple representation of ${\mathrm{\Pi }}^{\lambda }$.

In this case the locus ${\mu }_{\alpha }^{-1}(0)$, is a reduced and irreducible complete intersection of dimension $\alpha \cdot \alpha -1+2p(\alpha )$ and the quotient ${\mu }_{\alpha }^{-1}(0)//G(\alpha )$ is irreducible of dimension $2p(\alpha )$.

Canonical Decomposition

Any $\alpha \in \mathbb{N}{R}_{\lambda }^{+}$ has a decomposition $\alpha =\sum _{i=1}^n{\sigma }^{(i)}$ into elements of ${\mathrm{\Sigma }}_{\lambda }$, with the property that, any other decomposition of $\alpha$ into ${\mathrm{\Sigma }}_{\lambda }$ is a refinement of this decomposition.

Collecting terms $\alpha =\sum _{i=1}^n{m}_i{\sigma }^{(i)}$ with $m_i>0$, we have

$${\mu }_{\alpha }^{-1}(\lambda )//G_{\alpha }\cong \prod _{i=1}^n{S}^{m_i}({\mu }_{{\sigma }^{(i)}}^{-1}(\lambda )//G_{{\sigma }^{(i)}}).$$

Roots Classification

Recall that roots can be divided into three classes: (recall that $p(\alpha )=1-⟨\alpha ,\alpha ⟩$)

• $p(\beta )=0$ are real roots $⟨\beta ,\beta ⟩=1$.

• $p(\beta )=1$ are isotropic imaginary roots $⟨\beta ,\beta ⟩=0$.

• $p(\beta )>1$ are non-isotropic imaginary roots $⟨\beta ,\beta ⟩<0$.

We have the following properties:

• For $\beta \in \mathrm{\Sigma }$ real root, ${\mu }_{\beta }^{-1}(\lambda )//G(\beta )$ is a point.

• If $\beta \in \mathrm{\Sigma }$ is isotropic imaginary root, then it is indivisible $gcd(\beta )=1$ and $N(\lambda ,\beta )$ is isomorphic to a deformation of a Kleinian singularity.

• If $\beta \in \mathrm{\Sigma }$ is non-isotropic imaginary root, then any positive multiple of $\beta$ is also in $\mathrm{\Sigma }$.

  Therefore in any canonical decomposition, the multiplicity of a non-isotropic imaginary root must be $1$, since otherwise their multiples lie in $\mathrm{\Sigma }$ and is therefore a decomposition into $\mathrm{\Sigma }$ not a refinement.

Extended Dynkin Quiver

For quivers like $\widehat{A_n}$ and $\widehat{D_n}$ which are extended Dynkin quivers, they corresponds to the root systems of the corresponding affine Lie algebras. They will have coordinate vectors $e_i$ as their simple roots, and a positive imaginary root $\delta$. We have ${\mathrm{\Sigma }}_0=\{\delta ,e_0,\dots ,e_n\}$. Note that in this case the $\mathrm{\Sigma }$ is finite and thus the canonical decomposition of any vector is obtained by taking as much $\delta$ as possible, and distribute the remaining part among the $e_i$’s.

The maximal value of $\sum _{i=0}^{n}p({\beta }^{(i)})$ over $\mathrm{Decomp}(\alpha ,R_{\lambda }^{+})$ is denoted by $|\alpha {|}_{\lambda }$, and in fact we can assume ${\beta }^{(i)}\in {\mathrm{\Sigma }}_{\lambda }$, i.e.

$$max[\mathrm{Decomp}(\alpha ,R_{\lambda }^{+})]=max[\mathrm{Decomp}(\alpha ,{\mathrm{\Sigma }}_{\lambda })].$$

Since for terms in $\mathrm{Decomp}(\alpha ,R_{\lambda }^{+})$, the decomposition with maximal number of terms will lie in $\mathrm{Decomp}(\alpha ,{\mathrm{\Sigma }}_{\lambda })$.