This is the third post of a series (see here and there) where I explore the mathematical theory of quivers, following [1]. Today, we will focus on the geometry of the isomorphism classes of quiver representations, and proceed to the definition of the quiver varieties.

The geometry of orbits

In all the following we fix a quiver $\vec{Q}$, and we will drop the $\vec{Q}$ dependence although of course all the construction depends on $\vec{Q}$. Let $V$ be a finite-dimensional representation of $\vec{Q}$, with dimension vector $\mathbf{v}$ (see the first post for definitions and notations). When the dimension is fixed, we can assume the vector spaces at nodes of the quiver are just $\mathbb{C}^{\mathbf{v}_i}$, and the particular representation $V$ is specified by the matrices of the various linear maps corresponding to the edges. Let's call $$R( \mathbf{v}) = \bigoplus\limits_{h \in \Omega} \mathrm{Mat}_{\mathbf{v}_{t(h)} \times \mathbf{v}_{s(h)}}(\mathbb{C})$$ the representation space. The group $$\mathrm{PGL}( \mathbf{v}) = \left( \prod\limits_{i \in I} \mathrm{GL}( \mathbf{v}_i) \right) / \mathbb{C}^{\ast}$$ acts on $R( \mathbf{v})$ by conjugation, and two elements of $R( \mathbf{v})$ define isomorphic representations if and only if they are in the same $\mathrm{PGL}( \mathbf{v})$ orbit. So, in other words, the isomorphism classes of representations $V$ of dimension $\mathbf{v}$ are the $\mathrm{PGL}( \mathbf{v})$ orbits $\mathbb{O}_V$ in $R( \mathbf{v})$.

Now we can state an important result: the orbit $\mathbb{O}_V$ is closed if and only if $V$ is semisimple, and the closure of any $\mathbb{O}_V$ contains a unique closed orbit, namely $\mathbb{O}_{V^{\mathrm{ss}}}$.

These facts should resonate with the GIT theory! Indeed, these can be reformulated as $$R( \mathbf{v})/\mathrm{PGL}( \mathbf{v}) = \{\textrm{Isomorphism classes of reps of dimension } \mathbf{v}\}$$ $$\mathcal{R}_0 ( \mathbf{v}) := R( \mathbf{v})// \mathrm{PGL}( \mathbf{v}) = \{\textrm{Isomorphism classes of semisimple reps of dimension } \mathbf{v}\} \, . $$ Finally, this begs for the introduction of twisted GIT quotients. For that, note that the characters of $\mathrm{PGL}( \mathbf{v})$ are of the form $$\chi_{\theta} : g \mapsto \prod\limits_{i \in I} \det (g_i)^{- \theta_i} \qquad \textrm{with} \qquad \theta \in \mathbb{Z}^I \, , \quad \theta \cdot \mathbf{v} = 0 \, . $$ As we know from GIT theory, the twisted GIT quotient $$\mathcal{R}_\theta ( \mathbf{v}) := R( \mathbf{v})//_\theta \mathrm{PGL}( \mathbf{v})$$ has an interpretation as the set of closed semistable orbits inside the set of semistable elements. The concepts of stability and semistability depend on the character $\chi_\theta$, and by a theorem of King, we have that for any representation $V$ of dimension $\mathbf{v}$, and for any $\theta\in \mathbb{Z}^I$ such that $\theta \cdot \mathbf{v} = 0$, $$V \textrm{ is } \chi_\theta\textrm{-semistable} \quad \Leftrightarrow \quad \textrm{for any subrepresentation } V' \subset V \, , \quad \theta \cdot (\mathbf{dim} \, V') \leq 0$$ $$V \textrm{ is } \chi_\theta\textrm{-stable} \quad \Leftrightarrow \quad \textrm{for any nonzero proper subrepresentation } V' \subset V \, , \quad \theta \cdot (\mathbf{dim} \, V') < 0 \, . $$

Double quivers

We know introduce the intuitive notion of double quiver. If $Q$ is a graph, the double quiver $Q^\sharp$ is the quiver with the vertices of $Q$ and such that each edge connecting vertices $i$ and $j$ in $Q$ give rise to two edges $h : i \rightarrow j$ and $\bar{h} : j \rightarrow i$ in $Q^\sharp$. Because of that, we have $$R(Q^\sharp , \mathbf{v}) = R(\vec{Q} , \mathbf{v}) \oplus R(\vec{Q} , \mathbf{v}) ^{ast} = T^{\ast} ( R(\vec{Q} , \mathbf{v})) \, , $$ and as for any cotangent bundle, this space has a canonical symplectic structure. So we see the interest of considering double quivers: the symplectic geometric aspects of GIT will come into play. In fact, let's be more general and introduce a function $\epsilon : H \rightarrow \mathbb{C}^\ast$ such that $\epsilon (h) +\epsilon (\bar{h}) = 0$ for all $h$. Then we can define the symplecic form $$\omega_\epsilon (z,w) = \mathrm{tr}_V \left( \sum\limits_{h \in H} \epsilon(h) z_{\bar{h}} w_h\right) \qquad \textrm{for} \quad z,w \in R(Q^\sharp , \mathbf{v}) \, . $$ Then the action of $\mathrm{PGL}( \mathbf{v})$ on $R(Q^\sharp , \mathbf{v})$ is Hamiltonian, and the corresponding moment map is \begin{eqnarray*} \mu_{\mathbf{v}} & : & R(Q^\sharp , \mathbf{v}) \longrightarrow \bigoplus\limits_{i \in I} \mathfrak{gl}(\mathbf{v}_i , \mathbb{C}) \\ & & z \mapsto \bigoplus\limits_{i \in I} \sum\limits_{t(h)=i} \epsilon(h) z_h z_{\bar{h}} \, . \end{eqnarray*} We can then define the GIT quotients $$\mathcal{M}_0 (\mathbf{v}) = \mu_{\mathbf{v}}^{-1}(0) // \mathrm{PGL}( \mathbf{v}) \, , \qquad \mathcal{M}_\theta (\mathbf{v}) = \mu_{\mathbf{v}}^{-1}(0) //_{\chi_\theta} \mathrm{PGL}( \mathbf{v}) \, $$ and also, introducing $(\mu_{\mathbf{v}}^{-1}(0))^\mathrm{s} = \{z \in R(Q^\sharp , \mathbf{v}) \mid \mu_{\mathbf{v}}(z) = 0 \textrm{ and } z \textrm{ is } \chi_\theta \textrm{-stable}\}$, $$\mathcal{M}^\mathrm{s}_\theta (\mathbf{v}) = (\mu_{\mathbf{v}}^{-1}(0))^\mathrm{s} //_{\chi_\theta} \mathrm{PGL}( \mathbf{v}) \subset \mathcal{M}_\theta (\mathbf{v}) \, . $$

Framing

We need a last ingredient before coming at last to quiver varieties. This is called framing. Let $\vec{Q}$ be a quiver with set of vertices $I$ and set of edges $\Omega$. Let $W$ be a $I$-graded vector space. A $W$-framed representation of $\vec{Q}$ is a representation $V = (V_h , x_h)$ of $\vec{Q}$ together with a collection of linear maps $j_k : V_k \rightarrow W_k$ for any $k \in I$. A morphism of two $W$-framed representations is a morphism of representations of $\vec{Q}$, $f : V \rightarrow V'$, which commutes with the $j$s, $j'_k \circ f_k = j_k$ for all $k \in I$.

We define in an obvious way the representation space $R(\mathbf{v},\mathbf{w})$ where $\mathbf{v}$ and $\mathbf{v}$ are the graded dimensions of $V$ and $W$. The group $\mathrm{GL}(\mathbf{v})$ acts on this space1, and we define as above the GIT quotients $\mathcal{R}_0 (\mathbf{v},\mathbf{w})$ and2 $\mathcal{R}_\theta (\mathbf{v},\mathbf{w})$

Doubling and Framing : Quiver varieties

It is now time to combine together all the previous ingredients. We take a graph $Q$ (with set of vertices $I$), and consider the double quiver $Q^\sharp$ (with set of edges $H$), and $V$, $W$ two $I$-graded vector spaces. The framed-representation space of the double quiver is $$R(Q^\sharp , V , W) = \left( \bigoplus\limits_{h \in H} \mathrm{Hom}(V_{s(h)} , V_{t(h)})\right) \oplus \left( \bigoplus\limits_{k \in I} \mathrm{Hom}(V_{k} , W_{k})\right)\oplus \left( \bigoplus\limits_{k \in I} \mathrm{Hom}(W_{k} , V_{k})\right) \, . $$ As for any double quiver, the action of $\mathrm{GL}(V)$ is Hamiltonian, with moment map \begin{eqnarray*} \mu_{V,W} & : & R(Q^\sharp , V,W) \longrightarrow \mathfrak{gl}(V) \\ & & (z,i,j)) \mapsto \sum\limits_{h \in H} \epsilon(h) z_h z_{\bar{h}} - \sum\limits_{k \in I} i_k j_k \, . \end{eqnarray*} Finally, we introduce the symplectic (twisted) GIT quotients $$\mathcal{M}_0 (V,W) = \mu_{V,W}^{-1}(0) // \mathrm{GL}(V) \, , \qquad \mathcal{M}_\theta (V,W) = \mu_{V,W}^{-1}(0) //_{\chi_\theta} \mathrm{GL}(V) \, . $$ The variety $\mathcal{M}_\theta (V,W)$ is called a quiver variety. In the next episode, we will explore the properties of these varieties!

Reference

[1] Kirillov, Quiver representations and quiver varieties. Vol. 174. American Mathematical Soc., 2016.



1. Warning! there is no $\mathbf{w}$ involved in this group, and the subgroup of scalars is included, so this is not $\mathrm{PGL}$.
2. Because the group of scalars act, we no longer require $\theta \cdot \mathbf{v} = 0$ here.