数理解析研究所講究録
第 1998 巻 2016 年 46-52

46

Quantum groups, quiver varieties, andLusztig’s symmetries
Fan QIN
Abstract
In this talk, I will give a geometric construction of the quantizedenveloping algebras of type ADE and their bases via cyclic quivervarieties. The construction respects BGP reflections, which turns out
to be Lusztigs symmetries acting on these algebras.

1Introduction

1.1

Quantum group$U_{t}(g)$We take the following notations:
I is the set of vertices 1, 2,$\{$$\bullet$$\bullet$$\bullet$$\bullet$$C=(C_{ij})_{i,j\in I}$

$g$

$t$

$\cdots$,$n\}.$is a symmetric generalized Cartan matrix.

is complex Kac-Moody Lie algebra associated with $C.$

is an indeterminate.

The quantum group
generators
other relations :-algebra generated by the Chevalley
is the$i\in I$ , subject to the quantum Serre relations and$U_{t}(g)$$E_{i},$ $K_{i}^{\pm},$ $F_{i},$$\mathbb{Q}(t)$$\sim$$U_{t}(g)=\langle E_{i}, K_{i}^{\pm}, F_{i}\rangle/\sim.$

It has the triangular decomposition into the sub-algebras$U_{t}(h)=\langle K_{i}^{\pm}\rangle/(K_{i}K_{i}^{-1}=1, K_{i}K_{j}=K_{j}K_{i})$ ,, generated by
Now let us slightly enlarge the quantum group into
, subject to similar relations. Then this algebra has the triangular decomposition into the sub-algebras$U_{t}(n^{+})=\langle E_{i}\rangle/\sim,$$U_{t}(n^{-})=\langle F_{i})/\sim.$$U_{t}(g)$$E_{i},$ $K_{i},$ $K_{i}’,$$F_{i}$$U_{t}(n^{+})=\langle E_{i}\rangle/\sim,$$\langle K_{i},$$K_{i}’\rangle/(K_{i}K_{j}’=K_{j}’K_{i})$,$\tilde{U}_{t}(h)=$$U_{t}(n^{-})=\langle F_{i}\rangle/\sim.$

Taking the reduction ofobtain the usual quantum group$\tilde{U}_{t}(g)$by imposing the relation $K_{i}K_{i}’=1$ , we

$U_{t}(g)$.

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1.2

Categorification of$U_{t}(n^{+})$We let denote the diagram of , namely, it has the vertex set , andedges between any two diﬀerent vertices $i,j.$
By choosing an orientation on the diagram , we obtain an oriented$C$$\Gamma$$I$$\Omega$$-C_{ij}$$\Gamma$graph (called quiver).
. Then we have the path algebra
We work over the base field
whose category of left modules will be denoted by
Recall that, by naturally viewing
as a category, its representations
are the functors from the category to the category of -vector spaces.
The category of the representations of , which we denote by Rep (Q) , is. For any $d=(d_{i})\in N^{I}$ , letequivalent to the module category
Rep $(Q, d)$ denote the vector space of representations which sends to$Q=(\Gamma, \Omega)$$k=\mathbb{C}$$\mathbb{C}Q,$$\mathbb{C}Q-mod.$

$Q$

$Q$

$\mathbb{C}$$Q$$\mathbb{C}Q-mod$$\mathbb{C}^{d_{i}}.$$i$

Theorem 1.1 (Ringel [Rin90], Green [Gre95]). Assume is acyclic, $namely_{f}$it has no oriented cycles. Let the base field be a finite field and take to. Let $H(Rep(Q))$ denote the Hall algebra of the abelian category
beRep (Q) . Then we have an embedding of algebra$Q$

$k$

$t$

$\sqrt{|k|}$$U_{t}(n^{+})c\prec H(Rep(Q))$

This embedding oe an isomorphism when$g$.

is of type $ADE.$

Here, the Hall algebra $H(Rep(Q\rangle)$ has the natural basis $\{[M]\}$ , where$|M]$ denote the isoclass of an object $M$ in Rep (Q) .
Its multiplication isdetermined by counting the short exact sequences.

Theorem 1.2 (Lusztig $|Lus90]$ [Lus91]). Let the base field$k$

be$\mathbb{C}.$ofperverse
1. There is an embeddingfrom
to the Grothendieck$d\inN^{I}$. This embedding issheaves over the vector spaces Rep $(Q, d)$ ,an isomorphism if is of type $ADE.$$U_{l}(n^{+})$$r\acute{\iota}ng$$g$

2. Via this embedding, we obtain the canonical basis ofsists of perverse sheaves and whose structure constants are in$U_{t}(n^{+})w\backslash hich$con-$N[t^{\pm}].$Theorem 1.3 (Hernandez-Leclerc [HLII]). Let be of type $ADE$ Thenis isomorphic to the dual of Grothendieck ring of perverse sheaves
are some dimension vectors
over graded quiver varietieswhere$d\in N^{I}.$associated with$g$,

$U_{t}(n^{+})$$\mathcal{M}(w^{d})_{f}$$w^{d}$

Proof They prove that the graded quiver varieties
the vector space Rep $(Q, d)$ .

$\mathcal{M}(w^{d})$are isomorphic to$\square$

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1.3

Categorification of$U_{t}(g)$Letbe the idempotended form of
introduced by Lusztig [Lus93].
It can be categorified by using quiver Hecke algebras (Khovanov-Lauda [KL09],Rouquier [Rou08]).
Let $C_{2}(Rep(Q))$ denote the abelian category of 2-periodic complexes of$Let\sim$ denote the quasi-isomorphisms.
-representations:$\dot{U}_{t}(g)$$U_{t}(g)$$Q$$M^{0}rightarrow M^{1}.$$M^{\cdot}$

Theorem 1.4 (Bridgeland[Bri13]). Let be a finite field and specialize. Assume to be acyclic. Then there is an algebra embedding
toto the localized Hall algebrafrom localized quantum algebra$H(C_{2}(Rep(Q)))[[M^{\cdot}] : H\cdot(M^{\cdot})=0]/\sim$ , such that
andcorrespond torespectively.
andThis embedding is an isomorphism if is of type $ADE.$$k$$\sqrt{|k|}$

$t$

$Q$

$\tilde{U}_{t}(g)[K_{i}^{-1}, K_{i}^{J-1}]$$K_{i}$$[S_{i}arrow 1S_{i}]$$K_{i}’$$[S_{i}arrow 1S_{i}]$

$g$

1.4

Main result

We take the base field. Letof type $ADE.$ the Coxeter number.
. Let
Take the complex number
denote the cyclic quivervariety introduced by Nakajima, for any function
from the cyclic groupN.
ontoThis variety depends
the orientation of the associated quiver $Q,$
which we always take to be acyclic.
$k=\mathbb{C}$$h$$?^{be}$$q=e^{\frac{\pi}{2h}}$$\mathcal{M}_{0}(w)$$w$$\langle q\rangle$

Theorem 1.5 (Main Theorem [Qin13]). (1) After the field extension to
we have the $isomo7phism$ of algebras$\mathbb{Q}(\sqrt{t})_{f}$$R_{t}(Q)\otimes \mathbb{Q}(\sqrt{t})arrow^{Q}\kappa\tilde{U}_{i}(g)\otimes \mathbb{Q}(\sqrt{t})$where$R_{t}(Q)=\oplus_{specia1w}K_{0}^{*}(w)$$K_{0}(w)$,is the Grothendieck ring of some perverse sheaves over the cyclic
its dual.
andquiver variety$\mathcal{M}_{0}(w)$$K_{0}^{*}(w)$As a consequence, the natural geometric basis $L(Q)$ in $R_{4}(Q)$ gives us abasis
in $R_{t}(Q)$ . Moreover, it has the following property by construction.
$\kappa_{Q}(L(Q))$

Theorem 1.6 ([HLII]). $L(Q)$ contains the dual canonical basis of(dual to the canonical basis with respect to Lusztig’s bilinear form).

$U_{t}(Q)$Corollary 1.7. Let $g(Q)$ be the reduction of $R_{t}(Q)$ by taking reduction$K_{i}K_{i}’=1$ .
Then we obtain corresponding claims for the quantum groupand the Grothendieck ring $g(Q)$ .
$U_{t}(g)$49
Remark 1.8.
indenote the shift functor on complexes. Then
$C_{2}(Rep(Q)\rangle$ .
This gives the indication of our choice of such that$1$$\bullet$Let$\Sigma^{2}=$$\Sigma$$q$$q^{2h}=1.$$\bullet$Our choice of special is inspired from the choice of Hernandez Leclerc.
.
We generalize their result fromto$vJ$$U_{t}(g)$$U_{t}(n^{+})$$0$In $Bridgetand’s$ work, the Cartan partis realized by contractible
complexes, which are redundant infor ation in the study of triangulated
categories. In our approach, the Cartan part are associated with somestrata of. Their counterparts for genericchoice are
redundant information, by Nakajima, in the study of finite dimensional
representations of quantum aﬃne algebras.
$U_{t}(h)$$\mathcal{M}_{0}(w)$$q\in \mathbb{C}^{*}$

2

Construction

2.1

Cyclic quiver variety$\mathcal{M}_{0}(w)$We use the language of Keller-Scherotzke [KS13] to define quiver varieties[NakOl].
Let $D^{b}(Q)$ denote the bounded derived category of Rep (Q) , $\Sigma=[1]$ its
shift functor, the Auslander-Reiten translation.
We choose a representative for each isoclass of an indecomposable object.
Let $IndD^{b}(Q)$ denote the corresponding full subcategory.
$\tau$

Example 2.1. We take the quiverQ to be the graph $2arrow 1.$
and its i-th
simple and injective respectively. Then $IndD^{b}(Q)$ is drawn in Figure 1 whereeach arrow denotes an irreducible (minimat non-isomorphic) morphism. The
functor is the horizontal one-step shift to the left.
$S_{i}$$P_{i}$$\tau$$P_{2} \Sigma S_{1} \Sigma S_{2} \Sigma^{2}P_{2}$$\cdots$$\nearrow$$\nearrow$$\backslash _{x}$$\nearrow$$\backslash _{\searrow}$$\nearrow$$\backslash \backslash$$\cdots$$S_{1} S_{2} \Sigma P_{2} \Sigma^{2}S_{X}$Figure 1: $IndD^{b}(Q)$ for a type

$A_{2}$quiver $Q.$

We deform $(IndD^{b}(Q))^{op}$ into $R(Q)$ the regular Nakajima category by:

1. inserting a vertex$\sigma x$

between

2. adding horizontal arrows from$\tau x$$x$

andto$x,$ $\forall x\in IndD^{b}(Q)$$\sigma x$

and from$\sigma x$

to,$\tau x,$50
3. imposing the mesh relations on this category (namely, sums of trianglesvanish).

see Figure 2 for an example.
$\sigma P_{2}arrow P_{2}arrow\sigma\Sigma S_{1}arrow\Sigma S_{1}\sim\sigma\SigmaS_{2}arrow\Sigma S_{2}+\sigma\Sigma^{2}P_{2}+\Sigma^{2}P_{2}$’$\cdots$$\nearrow$$\backslash$$\nearrow$$\backslash$$\backslash$$\cdots$$\swarrow^{/}$$\sigma S_{1}arrow S_{1}arrow\sigma S_{2}arrow S_{2}arrow\sigma\Sigma P_{2}arrow\Sigma P_{2}+\sigma\Sigma^{2}S_{1}+\Sigma^{2}S_{1}$Figure 2: $R(Q)$ for a type

quiver $Q.$

$A_{2}$

Define the operator such that
Define the singular Nakajima category $S(Q)$ to be the full subcategory of$\sigma^{2}=\tau.$$\sigma$$x\in IndD^{b}(Q)$generated on
respectively.
andFold the categories $R(Q)$ and $S(Q)$ to(called the -degree) to each object inWe assign the an element insuch that the arrows decrease the degrees by
By standard
We take dimension vectors
argument in Nakajima’s work, we obtain cyclic quiver varieties$R(Q)$$\sigma x,$$S(Q)/\Sigma^{2}$$R(Q)/\Sigma^{2}$$\langle q\rangle$

$u$

$q$

$R(Q)/\Sigma^{2}$$q.$$v\in N^{IndD^{b}(Q)/\Sigma^{2}},$$w\in N^{\sigma S\langle Q)/\Sigma^{2}}$$\mathcal{M}(v, w)=Rep(R(Q)/\Sigma^{2}, v,w)//GL(v)$, (GITquotient)and$\mathcal{M}_{0}(w)=Rep(S(Q)/\Sigma^{2}, w)$.

. The derived push
toThere is a natural proper map fromforward on the constant perverse sheaf gives us the decomposition into$\pi$$\mathcal{M}(v, w)$$\mathcal{M}_{0}(w)$$\pi_{1}$perverse sheaves$\pi_{!}1_{\mathcal{M}(v,w)}=\oplus_{v’\leq v:(v’,w)l-do\min ant}IC(\mathcal{M}_{0}(v’, w$(1)where -dominant is some combinatorial condition on the pair $(v, w)$ ,, $IC()$ denote the intersection cohomology
is a closed subvariety insheaf.
The ring
acts on the Grothendieck group of derived categories ofsuch that acts as the shift functor. Let $K_{0}(w)$ denote
sheafs overthe submodule spanned by the IC sheaves appearing in Equation (1).

$\mathcal{M}_{0}(v’, w)$$l$$\mathcal{M}_{0}(w)$$\mathbb{Z}[t^{\pm}]$$\mathcal{M}_{0}(w)$

$t$

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2.2Special$w$

We define
Define $R_{4}(Q)$ as. Its multiplication is defined geometrically via restriction functors on quiver varieties.
Define$W^{S}=N^{\{\sigma \mathcal{S}_{i}\}},$$W^{\Sigma S}=N^{\{\sigma\Sigma S_{i}\}}.$$\oplus_{w\in W}s_{\oplus W}\Sigma sK_{0}^{*}(w)$$R_{t}^{+}(Q)=\oplus_{w\in W}sK_{0}^{*}(w)$$R_{\overline{t}}(Q)=\oplus_{w\epsilon W}\Sigma sK_{0}^{*}(w)$$R_{t}^{0}(Q)=\langle K_{i}, K_{i}$where$K_{i},$ $K_{i}’$are special central element in$R_{t}(Q)$.

Proof of Theorem X. 5. We show the triangular decomposition$R_{t}^{0}(Q)\otimes R_{r}^{-}(Q)$implies. It is then easy to verify the quantum Serre relations which,,
are isomorphic to$R_{t}^{+}(Q)\otimes \mathbb{Q}(\sqrt{t})$$U_{t}(n^{+})\otimes \mathbb{Q}(\sqrt{t})$$R_{\gamma}(Q)=Rf(Q)\otimes$,$R_{t}^{0}(Q)\otimes \mathbb{Q}(\sqrt{t})$$\tilde{U}_{t}(h)\otimes \mathbb{Q}(\sqrt{t})$,$R_{y}^{-}(Q)\otimes \mathbb{Q}(\sqrt{t})$$U_{t}(n^{-})\otimes \mathbb{Q}(\sqrt{t})$respectively.

Cl3

Reflection

Theorem 3.1. Letj be a sink point (no outgoing arrow) in Q. Let $Q’$ denote
Lusztig’s
andthe quiver obtained from by a reflection at andsymmetries. Then we can construct an isomorphism such that the diagramin Figure 3 is commutative.
$Q$

$j$

$T_{j,-1}"$$\theta$

Figure 3: Changing quiver orientations.

$T_{j,1}’$

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[KS13] Bernhard Keller and Sarah Scherotzke. Graded quiver varieties andderived categories. to appear in Journal f\"ur die reine und angewandte Mathematik, 2013, 1303.2318.

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