Geometric realizations of Lusztig's symmetries on the whole quantum groups

TL;DR

This paper explores the geometric realization of Lusztig's symmetries across entire quantum groups by analyzing the Grothendieck group of Lusztig's perverse sheaves and establishing a decomposition theorem.

Contribution

It introduces a decomposition theorem for the Grothendieck group and extends Lusztig's symmetries from the positive part to the entire quantum group using geometric methods.

Findings

Decomposition theorem for the Grothendieck group

Geometric realization of Lusztig's symmetries on the whole quantum group

Extension of symmetries from positive part to entire quantum group

Abstract

In this paper, we shall study the structure of the Grothendieck group of the category consisting of Lusztig's perverse sheaves and give a decomposition theorem of it. By using this decomposition theorem and the geometric realizations of Lusztig's symmetries on the positive part of a quantum group, we shall give geometric realizations of Lusztig's symmetries on the whole quantum group.

Equations

$$\mathbf{U}\cong {\mathbf{U}}^{-}\otimes {\mathbf{U}}^0\otimes {\mathbf{U}}^{+}.$$

$$K(\mathcal{Q})=\underset{\nu }{⨁}K({\mathcal{Q}}_{\mathbf{V}}),$$

$$DK(\mathcal{Q})\cong K(\mathcal{Q}{)}^{-}\otimes \mathbf{K}\otimes K(\mathcal{Q}{)}^{+}.$$

$$K(\mathcal{Q})\cong \underset{r⩾0}{⨁}[{\mathcal{L}}_{ri}]\ast ({{}_{i}j}_0{)}_{!}(K({{}_i\mathcal{Q}}_0)).$$

$$\mathbf{f}=\underset{t⩾0}{⨁}{\theta }_i^{(t)}{}_i\mathbf{f}.$$

$${\tilde{T}}_i:DK(\mathcal{Q})(Q)\to DK(\mathcal{Q})(Q^{\prime }).$$

$$DK(\mathcal{Q})(Q)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathbf{U}\cong {\mathbf{U}}^{-}\otimes {\mathbf{U}}^0\otimes {\mathbf{U}}^{+},$$

$$\mathbf{f}=\underset{t⩾0}{⨁}{\theta }_i^{(t)}{}_i\mathbf{f}=\underset{t⩾0}{⨁}{}^i\mathbf{f}{\theta }_i^{(t)}.$$

$$E_{\mathbf{V}}=E_{\mathbf{V},Q}=\underset{\rho \in H}{⨁}{\text{Hom}}_{\mathbb{K}}(V_{s(\rho )},V_{t(\rho )}).$$

$$Y_{\nu }=\{\mathbf{y}=(\mathbf{i},\mathbf{a})|\sum _{l=1}^k{a}_l{i}_l=\nu \},$$

$${\tilde{F}}_{\mathbf{y}}=\{(x,\varphi )\in E_{\mathbf{V}}\times F_{\mathbf{y}}|\text{ϕ is x-stable}\}$$

$$K(\mathcal{Q})=\underset{\nu \in \mathbb{N}I}{⨁}K({\mathcal{Q}}_{\mathbf{V}}).$$

$${\text{Ind}}_{{\mathbf{V}}^{\prime },{\mathbf{V}}^{\prime \prime }}^{\mathbf{V}}:{\mathcal{Q}}_{{\mathbf{V}}^{\prime }}\times {\mathcal{Q}}_{{\mathbf{V}}^{\prime \prime }}\to {\mathcal{Q}}_{\mathbf{V}},$$

$$\ast ={\text{ind}}_{{\mathbf{V}}^{\prime },{\mathbf{V}}^{\prime \prime }}^{\mathbf{V}}:K({\mathcal{Q}}_{{\mathbf{V}}^{\prime }})\times K({\mathcal{Q}}_{{\mathbf{V}}^{\prime \prime }})$$

$$([{\mathcal{L}}^{\prime }],[{\mathcal{L}}^{\prime \prime }])$$

$${\text{Res}}_{{\mathbf{V}}^{\prime },{\mathbf{V}}^{\prime \prime }}^{\mathbf{V}}:{\mathcal{Q}}_{\mathbf{V}}\to {\mathcal{Q}}_{{\mathbf{V}}^{\prime }}\times {\mathcal{Q}}_{{\mathbf{V}}^{\prime \prime }},$$

$${\text{res}}_{{\mathbf{V}}^{\prime },{\mathbf{V}}^{\prime \prime }}^{\mathbf{V}}:K({\mathcal{Q}}_{\mathbf{V}})$$

$$[\mathcal{L}]$$

$${\mathrm{Res}}_{{\mathbf{V}}^{\prime },{\mathbf{V}}^{\prime \prime }}^{\mathbf{V}}({\mathrm{Ind}}_{{\mathbf{V}}_1,{\mathbf{V}}_2}^{\mathbf{V}}({\mathcal{L}}_1,{\mathcal{L}}_2))=\underset{{\nu }_1^{\prime },{\nu }_1^{\prime \prime },{\nu }_2^{\prime },{\nu }_2^{\prime \prime }}{⨁}\hat{{\mathrm{Ind}}_{{\mathbf{V}}_1^{\prime },{\mathbf{V}}_2^{\prime }}^{{\mathbf{V}}^{\prime }}\otimes {\mathrm{Ind}}_{{\mathbf{V}}_1^{\prime \prime },{\mathbf{V}}_2^{\prime \prime }}^{{\mathbf{V}}^{\prime \prime }}}({\mathrm{Res}}_{{\mathbf{V}}_1^{\prime },{\mathbf{V}}_1^{\prime \prime }}^{{\mathbf{V}}_1}({\mathcal{L}}_1),{\mathrm{Res}}_{{\mathbf{V}}_2^{\prime },{\mathbf{V}}_2^{\prime \prime }}^{{\mathbf{V}}_2}({\mathcal{L}}_2)),$$

$${\lambda }_{\mathcal{A}}:K(\mathcal{Q})\to {\mathbf{f}}_{\mathcal{A}}$$

$${\lambda }_{\mathcal{A}}\otimes {\lambda }_{\mathcal{A}}(\mathrm{res}(x))=r({\lambda }_{\mathcal{A}}(x))$$

$$\{\mathcal{L},{\mathcal{L}}^{\prime }\}=\sum _{t\in \mathbb{Z}}\dim H_{G_{\mathbf{V}}}^t(\mathcal{L}\otimes {\mathcal{L}}^{\prime },E_{\mathbf{V}})v^t.$$

$$\phi ({\mathbf{k}}_{\alpha }[\mathcal{L}{]}^{+},{\mathbf{k}}_{\beta }[{\mathcal{L}}^{\prime }{]}^{-})=v^{-(\alpha ,\beta )-\nu (\beta )+{\nu }^{\prime }(\alpha )}\{\mathcal{L},{\mathcal{L}}^{\prime }\}$$

$$DK(\mathcal{Q})\cong K(\mathcal{Q}{)}^{-}\otimes \mathbf{K}\otimes K(\mathcal{Q}{)}^{+}.$$

$${\lambda }_{\mathcal{A}}:DK(\mathcal{Q})\to {\mathbf{U}}_{\mathcal{A}}$$

$${{}_{i}E}_{\mathbf{V},r}=\{x\in E_{\mathbf{V}}|Im(\underset{\rho \in H,t(\rho )=i}{⨁}{x}_{\rho })\text{ has codimension r in Vi}\}.$$

$${{}_i\tilde{F}}_{\mathbf{y},r}=\{(x,\varphi )\in {{}_{i}E}_{\mathbf{V},r}\times F_{\mathbf{y}}|\text{ϕ is x-stable}\}$$

$$K({{}_i\mathcal{Q}}_r)=\underset{\nu \in \mathbb{N}I}{⨁}K({{}_i\mathcal{Q}}_{\mathbf{V},r}).$$

$${{}_{i}j}_{\mathbf{V},r}^{\ast }({\mathcal{Q}}_{\mathbf{V}})={{}_i\mathcal{Q}}_{\mathbf{V},r}.$$

$${{}_i\tilde{F}}_{\mathbf{y},r}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

This work was supported by the National Natural Science Foundation of China [No. 11526037]

1. 1 Introduction
2. 2 Quantum groups and Lusztig’s symmetries
3. 3 Geometric realization of the Lusztig’s algebra $\mathbf{f}$
4. 4 Geometric realization of the quantum group $\mathbf{U}$
5. 5 Structure of the $\mathcal{A}$-module $K(\mathcal{Q})$
6. 6 Geometric realization of Lusztig’s symmetry $T_{i}:\mathbf{U}\rightarrow\mathbf{U}$
7. 7 Braid group relations

1. Introduction

1.1.

Let $\mathbf{U}$ be the quantum group associated to a Cartan datum, which is introduced by Drinfeld ([4]) and Jimbo ([6]) respectively in the study of quantum Yang-Baxter equations. As a quantization of the universal enveloping algebra of a Kac-Moody Lie algebra, the quantum group $\mathbf{U}$ is a Hopf algebra and has the following triangular decomposition

[TABLE]

In [14], Lusztig also introduced an algebra $\mathbf{f}$ (called the Lusztig’s algebra) associated to a Cartan datum, satisfying that there are two monomorphisms of algebras ${{}^{+}}:\mathbf{f}\rightarrow\mathbf{U}$ and ${{}^{-}}:\mathbf{f}\rightarrow\mathbf{U}$ with images $\mathbf{U}^{+}$ and $\mathbf{U}^{-}$ respectively.

Lusztig introduced some operators $T_{i}$ on the quantum group $\mathbf{U}$ satisfying the braid group relations, which are called Lusztig’s symmetries ([9, 11]). By the definition of Lusztig’s symmetries, the image of $\mathbf{U}^{+}$ under $T_{i}$ is not contained in $\mathbf{U}^{+}$ for any $i$. So Lusztig introduced two subalgebras ${{}_{i}\mathbf{f}}=\{x\in\mathbf{f}\,\,|\,\,T_{i}(x^{+})\in\mathbf{U}^{+}\}$ and ${{}^{i}\mathbf{f}}=\{x\in\mathbf{f}\,\,|\,\,T^{-1}_{i}(x^{+})\in\mathbf{U}^{+}\}$ of $\mathbf{f}$ ([14]). Note that there exists a unique $T_{i}:{{}_{i}\mathbf{f}}\rightarrow{{}^{i}\mathbf{f}}$ such that $T_{i}(x^{+})=T_{i}(x)^{+}$.

For any finite quiver $Q=(I,H)$, Ringel introduced the Ringel-Hall algebra as an algebraic model of the positive part of the corresponding quantum group ([15]). Green ([5]) introduced the comultiplication on the Ringel-Hall algebra and Xiao ([18]) introduced the antipode. Under these operators, the Ringel-Hall algebra has a Hopf algebra structure. Xiao also considered the Drinfeld double of the Ringel-Hall algebra (called the double Ringel-Hall algebra), the composition subalgebra of which gives a realization of the whole quantum group ([18]).

By this algebraic realization of a quantum group, Ringel applied the BGP reflection functors ([1]) to give realizations of Lusztig’s symmetries $T_{i}:{{}_{i}\mathbf{f}}\rightarrow{{}^{i}\mathbf{f}}$ ([16]). Then Xiao-Yang ([20]) and Sevenhant-Van den Bergh ([17]) realized Lusztig’s symmetries $T_{i}:\mathbf{U}\rightarrow\mathbf{U}$ also by using the BGP reflection functors. Indeed, this method is also available to give precise constructions of Lusztig’s symmetries on a double Ringel-Hall algebra ([2, 3]).

1.2.

In [10, 12], Lusztig gave a geometric realization of the Lusztig’s algebra $\mathbf{f}$. Let $Q=(I,H)$ be the quiver corresponding to $\mathbf{f}$. Inspired by the algebraic realization of $\mathbf{f}$ given by Ringel, Lusztig considered the variety $E_{\mathbf{V}}$ consisting of representations with dimension vector $\nu$ of the quiver $Q$ and the category $\mathcal{Q}_{\mathbf{V}}$ of some semisimple complexes on $E_{\mathbf{V}}$. Let $K(\mathcal{Q}_{\mathbf{V}})$ be the Grothendieck group of $\mathcal{Q}_{\mathbf{V}}$. Considering all dimension vectors, let

[TABLE]

which is isomorphic to the Lusztig’s algebra $\mathbf{f}$.

In [19], Xiao, Xu and Zhao considered a larger category of Weil complexes on the variety $E_{\mathbf{V}}$ for any dimension vector $\nu$. They showed that the direct sum of the Grothendieck groups of these categories gives a realization of the generic Ringel-Hall algebra via Lusztig’s geometric method. They also considered the Drinfeld double of the direct sum and gave a realization of the generic double Ringel-Hall algebra.

By using the method of Lusztig, Kato gave geometric realizations of Lusztig’s symmetries $T_{i}:{{}_{i}\mathbf{f}}\rightarrow{{}^{i}\mathbf{f}}$ in the case of finite type for all $i$ ([7]). Then his constructions were generalized by Xiao and Zhao to all symmetrizable Cartan datum ([21, 22]).

Assume that $i$ is a sink (resp. source) of the quiver $Q$. Xiao and Zhao considered a subvariety ${{}_{i}{E_{\mathbf{V},0}}}$ (resp. ${{}^{i}{E_{\mathbf{V},0}}}$) of $E_{\mathbf{V}}$ and a category ${{}_{i}\mathcal{Q}}_{\mathbf{V},0}$ (resp. ${{}^{i}\mathcal{Q}}_{\mathbf{V},0}$) of some semisimple complexes on ${{}_{i}{E_{\mathbf{V},0}}}$ (resp. ${{}^{i}{E_{\mathbf{V},0}}}$). They showed that $K({{}_{i}\mathcal{Q}}_{0})=\bigoplus_{\nu}K({{}_{i}\mathcal{Q}}_{\mathbf{V},0})$ (resp. $K({{}^{i}\mathcal{Q}}_{0})=\bigoplus_{\nu}K({{}^{i}\mathcal{Q}}_{\mathbf{V},0})$) realizes ${{}_{i}\mathbf{f}}$ (resp. ${{}^{i}\mathbf{f}}$).

Let $i\in I$ be a sink of the quiver $Q$. Then $i$ is a source of $Q^{\prime}=\sigma_{i}Q$, which is the quiver by reversing the directions of all arrows in $Q$ containing $i$. They defined a map $\tilde{\omega}_{i}:K({{}_{i}\mathcal{Q}}_{0})\rightarrow K({{}^{i}\mathcal{Q}}_{0})$, which gives a realization of Lusztig’s symmetry $T_{i}:{{}_{i}\mathbf{f}}\rightarrow{{}^{i}\mathbf{f}}$.

1.3.

In this paper, we shall give a geometric realization of Lusztig’s symmetry $T_{i}:\mathbf{U}\rightarrow\mathbf{U}$ for any $i$.

Let $Q$ be the quiver corresponding to $\mathbf{U}$. First, we shall construct a skew-Hopf pairing $(\tilde{K}(\mathcal{Q})^{+},\tilde{K}(\mathcal{Q})^{-},\varphi)$, where $\tilde{K}(\mathcal{Q})^{+}$ and $\tilde{K}(\mathcal{Q})^{-}$ are two Hopf algebras by adding torus algebra $\mathbf{K}=\bigoplus_{\mu}\mathcal{A}\mathbf{k}_{\mu}$ to ${K}(\mathcal{Q})$. Let $DK(\mathcal{Q})=DK(\mathcal{Q})(Q)$ be the quotient of the Drinfeld double of this skew-Hopf pairing module the Hopf ideal generated by $\mathbf{k}_{\mu}\otimes\mathbf{1}-\mathbf{1}\otimes\mathbf{k}_{\mu}$. Then $DK(\mathcal{Q})$ is isomorphic to the whole quantum group $\mathbf{U}$ and has the following triangular decomposition

[TABLE]

Then, we shall study the structure of ${K}(\mathcal{Q})$. Assume that $i$ is a sink of the quiver $Q$. We have the following theorem.

Theorem 1**.**

The $\mathcal{A}$-module $K(\mathcal{Q})$ has the following direct sum decomposition

[TABLE]

This theorem is a geometric interpretation of the following direct sum decomposition

[TABLE]

When $i$ is a source, we have a similar result.

Assume that $i$ is also a sink of the quiver $Q$. So $i$ is a source of $Q^{\prime}=\sigma_{i}Q$. By using Theorem 1 and the map $\tilde{\omega}_{i}:K({{}_{i}\mathcal{Q}}_{0})\rightarrow K({{}^{i}\mathcal{Q}}_{0})$, we can define a map

[TABLE]

Then we have the following main theorem in this paper.

Theorem 2**.**

The map $\tilde{T}_{i}:DK(\mathcal{Q})(Q)\rightarrow DK(\mathcal{Q})(Q^{\prime})$ is an isomorphism of Hopf algebras satisfying that the following diagram is commutative

[TABLE]

2. Quantum groups and Lusztig’s symmetries

In this section, we shall recall the definitions of quantum groups and Lusztig’s symmetries. We shall follow the notations in [14].

2.1.

Let $I$ be a finite index set with $|I|=n$, $A=(a_{ij})_{i,j\in I}$ be a symmetric generalized Cartan matrix, and $(A,\Pi,\Pi^{\vee},P,P^{\vee})$ be a Cartan datum associated with $A$, where $\Pi=\{\alpha_{i}\,\,|\,\,i\in I\}$ is the set of simple roots, $\Pi^{\vee}=\{h_{i}\,\,|\,\,i\in I\}$ is the set of simple coroots, $P$ is the weight lattice and $P^{\vee}$ is the dual weight lattice. There is a symmetric bilinear form $(-,-)$ on $\mathbb{Z}I$ induced by $(i,j)=\alpha_{j}(h_{i})=a_{ij}$. In this paper, assume that $P^{\vee}=\mathbb{Z}\Pi^{\vee}$ and the symmetric bilinear form on $P^{\vee}$ induced by $(h_{i},h_{j})=(i,j)$ is also denoted by $(-,-)$.

The quantum group $\mathbf{U}$ associated with a Cartan datum $(A,\Pi,\Pi^{\vee},P,P^{\vee})$ is an associative algebra over $\mathbb{Q}(v)$ with unit element $\mathbf{1}$, generated by the elements $E_{i}$, $F_{i}(i\in I)$ and $K_{\mu}(\mu\in P^{\vee})$ subject to the following relations

1. (1)

$K_{0}=\mathbf{1}$ and $K_{\mu}K_{\mu^{\prime}}=K_{\mu+\mu^{\prime}}$ for all $\mu,\mu^{\prime}\in P^{\vee}$, 2. (2)

$K_{\mu}E_{i}K_{-\mu}=v^{\alpha_{i}(\mu)}E_{i}$ for all $i\in I$, $\mu\in P^{\vee}$, 3. (3)

$K_{\mu}F_{i}K_{-\mu}=v^{-\alpha_{i}(\mu)}F_{i}$ for all $i\in I$, $\mu\in P^{\vee}$, 4. (4)

$E_{i}F_{j}-F_{j}E_{i}=\delta_{ij}\frac{K_{i}-K_{-i}}{v-v^{-1}}$ for all $i,j\in I$, 5. (5)

$\sum_{k=0}^{1-a_{ij}}(-1)^{k}E_{i}^{(k)}E_{j}E_{i}^{(1-a_{ij}-k)}=0$ for all $i\neq j\in I$, 6. (6)

$\sum_{k=0}^{1-a_{ij}}(-1)^{k}F_{i}^{(k)}F_{j}F_{i}^{(1-a_{ij}-k)}=0$ for all $i\neq j\in I$,

where $K_{i}=K_{h_{i}}$, $[n]_{v}=\frac{v^{n}-v^{-n}}{v-v^{-1}}$, $E_{i}^{(n)}=E_{i}^{n}/[n]_{v}!$ and $F_{i}^{(n)}=F_{i}^{n}/[n]_{v}!$.

The comultiplication $\Delta:\mathbf{U}\rightarrow\mathbf{U}\otimes\mathbf{U}$ is an algebra homomorphism satisfying that

1. (1)

$\Delta(E_{i})=E_{i}\otimes\mathbf{1}+K_{i}\otimes E_{i}$ for all $i\in I$, 2. (2)

$\Delta(F_{i})=F_{i}\otimes K_{-i}+\mathbf{1}\otimes F_{i}$ for all $i\in I$, 3. (3)

$\Delta(K_{\mu})=K_{\mu}\otimes K_{\mu}$ for all $\mu\in P^{\vee}$.

The antipode $S:\mathbf{U}\rightarrow\mathbf{U}^{op}$ is an algebra homomorphism satisfying that

1. (1)

$S(E_{i})=-K_{-i}E_{i}$ for all $i\in I$, 2. (2)

$S(F_{i})=-F_{i}K_{-i}$ for all $i\in I$, 3. (3)

$S(K_{\mu})=K_{-\mu}$ for all $\mu\in P^{\vee}$.

The counit $\mathbf{e}:\mathbf{U}\rightarrow\mathbb{Q}(v)$ is also an algebra homomorphism satisfying that

1. (1)

$\mathbf{e}(E_{i})=0$ for all $i\in I$, 2. (2)

$\mathbf{e}(F_{i})=0$ for all $i\in I$, 3. (3)

$\mathbf{e}(K_{\mu})=1$ for all $\mu\in P^{\vee}$.

Theorem 2.1** ([14]).**

The algebra $(\mathbf{U},\Delta,S,\mathbf{e})$ is a Hopf algebra.

The quantum group $\mathbf{U}$ has the following triangular decomposition

[TABLE]

where $\mathbf{U}^{-}$, $\mathbf{U}^{+}$ and $\mathbf{U}^{0}$ are the subalgebras $\mathbf{U}$ generated by $F_{i}$, $E_{i}$ and $K_{\mu}$ for all $i\in I$ and $\mu\in P^{\vee}$ respectively.

2.2.

In [14], Lusztig also introduced an associative $\mathbb{Q}(v)$-algebra $\mathbf{f}$, which is generated by $\theta_{i}(i\in I)$ subject to the quantum Serre ralations $\sum_{k=0}^{1-a_{ij}}(-1)^{k}\theta_{i}^{(k)}\theta_{j}\theta_{i}^{(1-a_{ij}-k)}=0$ for all $i\not=j\in I$, where $\theta_{i}^{(n)}=\theta_{i}^{n}/[n]_{v}!$. There are two well-defined $\mathbb{Q}(v)$-algebra monomorphisms ${{}^{+}}:\mathbf{f}\rightarrow\mathbf{U}$ and ${{}^{-}}:\mathbf{f}\rightarrow\mathbf{U}$ satisfying $E_{i}=\theta_{i}^{+}$ and $F_{i}=\theta_{i}^{-}$ for all $i\in I$ and the images of + and - are $\mathbf{U}^{+}$ and $\mathbf{U}^{-}$ respectively.

Define $|\theta_{i}|=i\in\mathbb{N}I$ for any $i\in I$ and $|xy|=|x|+|y|$ by induction. There exists a unique algebra homomorphism $r:\mathbf{f}\rightarrow\mathbf{f}\otimes\mathbf{f}$ such that $r(\theta_{i})=\theta_{i}\otimes\mathbf{1}+\mathbf{1}\otimes\theta_{i}$ for any $i\in I$, where the multiplication on $\mathbf{f}\otimes\mathbf{f}$ is defined as $(x\otimes y)(x^{\prime}\otimes y^{\prime})=v^{(|y|,|x^{\prime}|)}xx^{\prime}\otimes yy^{\prime}$.

2.3.

Denote by $T_{i}:\mathbf{U}\rightarrow\mathbf{U}$ the Lusztig’s symmetries for all $i\in I$ ([9, 11, 14]). The formulas of $T_{i}$ on the generators are

1. (1)

$T_{i}(E_{i})=-F_{i}K_{i}$ and $T_{i}(F_{i})=-K_{-i}E_{i}$, 2. (2)

$T_{i}(E_{j})=\sum_{r+s=-a_{ij}}(-1)^{r}v^{-r}E_{i}^{(s)}E_{j}E_{i}^{(r)}$ for $i\neq j\in I$, 3. (3)

$T_{i}(F_{j})=\sum_{r+s=-a_{ij}}(-1)^{r}v^{r}F_{i}^{(r)}F_{j}F_{i}^{(s)}$ for $i\neq j\in I$, 4. (4)

$T_{i}(K_{\mu})=K_{\mu-\alpha_{i}(\mu)h_{i}}$ for all $\mu\in P^{\vee}$.

Let ${{}_{i}\mathbf{f}}=\{x\in\mathbf{f}\,\,|\,\,T_{i}(x^{+})\in\mathbf{U}^{+}\}$ and ${{}^{i}\mathbf{f}}=\{x\in\mathbf{f}\,\,|\,\,T^{-1}_{i}(x^{+})\in\mathbf{U}^{+}\}$, which are subalgebras of $\mathbf{f}$ ([14]). By the definitions, there exists a unique $T_{i}:{{}_{i}\mathbf{f}}\rightarrow{{}^{i}\mathbf{f}}$ such that $T_{i}(x^{+})=T_{i}(x)^{+}$.

Theorem 2.2** ([14]).**

The algebra $\mathbf{f}$ has the following direct sum decompositions

[TABLE]

3. Geometric realization of the Lusztig’s algebra $\mathbf{f}$

In this section, we shall recall the geometric realization of the algebra $\mathbf{f}$ given by Lusztig ([10, 12, 14]).

3.1.

A quiver $Q=(I,H,s,t)$ consists of a vertex set $I$, an arrow set $H$, and two maps $s,t:H\rightarrow I$ such that an arrow $\rho\in H$ starts at $s(\rho)$ and terminates at $t(\rho)$. Let $h_{ij}=\#\{i\rightarrow j\}$, $a_{ij}=h_{ij}+h_{ji}$ and $\mathbf{f}$ be the Lusztig’s algebra corresponding to $A=(a_{ij})$. Let $p$ be a prime and $q$ be a power of $p$. Denote by $\mathbb{F}_{q}$ the finite field with $q$ elements and $\mathbb{K}=\overline{\mathbb{F}}_{q}$.

For a finite dimensional $I$-graded $\mathbb{K}$-vector space $\mathbf{V}=\bigoplus_{i\in I}V_{i}$, define

[TABLE]

The dimension vector of $\mathbf{V}$ is defined as $\underline{\mbox{\rm dim}\,}\mathbf{V}=\sum_{i\in I}(\mbox{\rm dim}\,_{\mathbb{K}}V_{i})i\in\mathbb{N}I$, which can also be viewed as an element in the weight lattice $P$. The algebraic group $G_{\mathbf{V}}=\prod_{i\in I}GL_{\mathbb{K}}(V_{i})$ acts on $E_{\mathbf{V}}$ naturally.

Fix a nonzero element $\nu\in\mathbb{N}I$. Let

[TABLE]

where $\mathbf{i}=(i_{1},i_{2},\ldots,i_{k}),\,\,i_{l}\in I$, $\mathbf{a}=(a_{1},a_{2},\ldots,a_{k}),\,\,a_{l}\in\mathbb{N}$. Fix a finite dimensional $I$-graded $\mathbb{K}$-vector space $\mathbf{V}$ such that $\underline{\mbox{\rm dim}\,}\mathbf{V}=\nu$. For any element $\mathbf{y}=(\mathbf{i},\mathbf{a})$, a flag of type $\mathbf{y}$ in $\mathbf{V}$ is a sequence $\phi=(\mathbf{V}=\mathbf{V}^{k}\supset\mathbf{V}^{k-1}\supset\dots\supset\mathbf{V}^{0}=0)$ of $I$-graded $\mathbb{K}$-vector spaces such that $\underline{\mbox{\rm dim}\,}\mathbf{V}^{l}/\mathbf{V}^{l-1}=a_{l}i_{l}$.

Let $F_{\mathbf{y}}$ be the variety of all flags of type $\mathbf{y}$ in $\mathbf{V}$. For any $x=(x_{\rho})_{\rho\in H}\in E_{\mathbf{V}}$, a flag $\phi$ is called $x$-stable if $x_{\rho}(V^{l}_{s(\rho)})\subset{V}^{l}_{t(\rho)}$ for all $l$ and all $\rho\in H$. Let

[TABLE]

and $\pi_{\mathbf{y}}:\tilde{F}_{\mathbf{y}}\rightarrow E_{\mathbf{V}}$ be the projection to $E_{\mathbf{V}}$.

Let $\overline{\mathbb{Q}}_{l}$ be the $l$-adic field and $\mathcal{D}_{G_{\mathbf{V}}}(E_{\mathbf{V}})$ be the bounded $G_{\mathbf{V}}$-equivariant derived category of ${\overline{{\mathbb{Q}}}_{l}}$-constructible complexes on $E_{\mathbf{V}}$. For each $\mathbf{y}\in{Y}_{\nu}$, $\mathcal{L}_{\mathbf{y}}=(\pi_{\mathbf{y}})_{!}(\mathbf{1}_{\tilde{F}_{\mathbf{y}}})[d_{\mathbf{y}}](\frac{d_{\mathbf{y}}}{2})\in\mathcal{D}_{G_{\mathbf{V}}}(E_{\mathbf{V}})$ is a semisimple perverse sheaf, where $d_{\mathbf{y}}=\mbox{\rm dim}\,\tilde{F}_{\mathbf{y}}$ and $\mathcal{L}(d)$ is the Tate twist of $\mathcal{L}$. Let $\mathcal{P}_{\mathbf{V}}$ be the set of simple perverse sheaves $\mathcal{L}$ on $E_{\mathbf{V}}$ such that $\mathcal{L}[r]$ appears as a direct summand of $\mathcal{L}_{\mathbf{y}}$ for some $\mathbf{y}\in Y_{\nu}$ and $r\in\mathbb{Z}$. Let $\mathcal{Q}_{\mathbf{V}}$ be the full subcategory of $\mathcal{D}_{G_{\mathbf{V}}}(E_{\mathbf{V}})$ consisting of all complexes which are isomorphic to finite direct sums of complexes in the set $\{\mathcal{L}[r]\,\,|\,\,\mathcal{L}\in\mathcal{P}_{\mathbf{V}},r\in\mathbb{Z}\}$.

Let $K(\mathcal{Q}_{\mathbf{V}})$ be the Grothendieck group of $\mathcal{Q}_{\mathbf{V}}$ and define $v^{\pm}[\mathcal{L}]=[\mathcal{L}[\pm 1](\pm\frac{1}{2})]$ for any $\mathcal{L}\in\mathcal{Q}_{\mathbf{V}}$. Then, $K(\mathcal{Q}_{\mathbf{V}})$ is a free $\mathcal{A}$-module, where $\mathcal{A}=\mathbb{Z}[v,v^{-1}]$. Define

[TABLE]

Let $\mathbf{B}_{\nu}=\{[\mathcal{L}]\,\,|\,\,\mathcal{L}\in\mathcal{P}_{\mathbf{V}}\}$ and $\mathbf{B}=\sqcup_{\nu\in\mathbb{N}I}\mathbf{B}_{\nu}$. Then $\mathbf{B}$ is the canonical basis of $K(\mathcal{Q})$ introduced by Lusztig in [10, 12].

3.2.

For $\nu,\nu^{\prime},\nu^{\prime\prime}\in\mathbb{N}I$ such that $\nu=\nu^{\prime}+\nu^{\prime\prime}$ and three $I$-graded $\mathbb{K}$-vector spaces $\mathbf{V}$, $\mathbf{V}^{\prime}$, $\mathbf{V}^{\prime\prime}$ such that $\underline{\mbox{\rm dim}\,}\mathbf{V}=\nu$, $\underline{\mbox{\rm dim}\,}\mathbf{V}^{\prime}=\nu^{\prime}$, $\underline{\mbox{\rm dim}\,}\mathbf{V}^{\prime\prime}=\nu^{\prime\prime}$, Lusztig introduced the induction functor

[TABLE]

which induces the following $\mathcal{A}$-bilinear operator

[TABLE]

Under these operators, $K(\mathcal{Q})$ becomes an associative $\mathcal{A}$-algebra.

For $\nu,\nu_{1},\nu_{2},\dots,\nu_{s}\in\mathbb{N}I$ such that $\nu=\nu_{1}+\nu_{2}+\cdots+\nu_{s}$ and $I$-graded $\mathbb{K}$-vector spaces $\mathbf{V},\mathbf{V}_{1},\mathbf{V}_{2},\dots,\mathbf{V}_{s}$ such that $\underline{\mbox{\rm dim}\,}\mathbf{V}=\nu$, $\underline{\mbox{\rm dim}\,}\mathbf{V}_{l}=\nu_{l}$, we can define the $s$-fold version of the induction functor ${\textrm{ind}}_{\mathbf{V}_{1},\mathbf{V}_{2},\dots,\mathbf{V}_{s}}^{\mathbf{V}}$ by induction ([19]).

3.3.

For $\nu,\nu^{\prime},\nu^{\prime\prime}\in\mathbb{N}I$ such that $\nu=\nu^{\prime}+\nu^{\prime\prime}$ and three $I$-graded $\mathbb{K}$-vector spaces $\mathbf{V}$, $\mathbf{V}^{\prime}$, $\mathbf{V}^{\prime\prime}$ such that $\underline{\mbox{\rm dim}\,}\mathbf{V}=\nu$, $\underline{\mbox{\rm dim}\,}\mathbf{V}^{\prime}=\nu^{\prime}$, $\underline{\mbox{\rm dim}\,}\mathbf{V}^{\prime\prime}=\nu^{\prime\prime}$, Lusztig introduced the restriction functor

[TABLE]

which induces the following $\mathcal{A}$-bilinear operator

[TABLE]

Under these operators, we have an operator $\textrm{res}:K(\mathcal{Q})\rightarrow K(\mathcal{Q})\otimes K(\mathcal{Q})$.

For $\nu,\nu_{1},\nu_{2},\dots,\nu_{s}\in\mathbb{N}I$ such that $\nu=\nu_{1}+\nu_{2}+\cdots+\nu_{s}$ and $I$-graded $\mathbb{K}$-vector spaces $\mathbf{V},\mathbf{V}_{1},\mathbf{V}_{2},\dots,\mathbf{V}_{s}$ such that $\underline{\mbox{\rm dim}\,}\mathbf{V}=\nu$, $\underline{\mbox{\rm dim}\,}\mathbf{V}_{l}=\nu_{l}$, we can define the $s$-fold version of the restriction functor ${\textrm{res}}_{\mathbf{V}_{1},\mathbf{V}_{2},\dots,\mathbf{V}_{s}}^{\mathbf{V}}$ by induction ([19]).

3.4.

Fix $\nu,\nu_{1},\nu_{2},\nu^{\prime},\nu^{\prime\prime}\in\mathbb{N}I$ such that $\nu=\nu_{1}+\nu_{2}=\nu^{\prime}+\nu^{\prime\prime}$ and $I$-graded $\mathbb{K}$-vector spaces $\mathbf{V},\mathbf{V}_{1},\mathbf{V}_{2},\mathbf{V}^{\prime},\mathbf{V}^{\prime\prime}$ such that $\underline{\mbox{\rm dim}\,}\mathbf{V}=\nu,\underline{\mbox{\rm dim}\,}\mathbf{V}_{1}=\nu_{1},\underline{\mbox{\rm dim}\,}\mathbf{V}_{2}=\nu_{2},\underline{\mbox{\rm dim}\,}\mathbf{V}^{\prime}=\nu^{\prime},\underline{\mbox{\rm dim}\,}\mathbf{V}^{\prime\prime}=\nu^{\prime\prime}$.

Proposition 3.1** ([12, 14, 19]).**

For any $\mathcal{L}_{1}\in\mathcal{Q}_{\mathbf{V}_{1}}$ and $\mathcal{L}_{2}\in\mathcal{Q}_{\mathbf{V}_{2}}$, we have

[TABLE]

where $\mathbf{V}^{\prime}_{1},\mathbf{V}^{\prime\prime}_{1},\mathbf{V}^{\prime}_{2},\mathbf{V}^{\prime\prime}_{2}$ are $I$-graded $\mathbb{K}$-vector spaces with dimension vectors $\nu^{\prime}_{1},\nu^{\prime\prime}_{1},\nu^{\prime}_{2},\nu^{\prime\prime}_{2}$ such that $\nu^{\prime}_{1}+\nu^{\prime\prime}_{1}=\nu_{1}$, $\nu^{\prime}_{2}+\nu^{\prime\prime}_{2}=\nu_{2}$, $\nu^{\prime}_{1}+\nu^{\prime}_{2}=\nu^{\prime}$, $\nu^{\prime\prime}_{1}+\nu^{\prime\prime}_{2}=\nu^{\prime\prime}$ and the functor $\widehat{\mathrm{Ind}^{\mathbf{V}^{\prime}}_{\mathbf{V}^{\prime}_{1},\mathbf{V}^{\prime}_{2}}\otimes\mathrm{Ind}^{\mathbf{V}^{\prime\prime}}_{\mathbf{V}^{\prime\prime}_{1},\mathbf{V}^{\prime\prime}_{2}}}$ is just the twist of $\mathrm{Ind}^{\mathbf{V}^{\prime}}_{\mathbf{V}^{\prime}_{1},\mathbf{V}^{\prime}_{2}}\otimes{\mathrm{Ind}}^{\mathbf{V}^{\prime\prime}}_{\mathbf{V}^{\prime\prime}_{1},\mathbf{V}^{\prime\prime}_{2}}$.

As a corollary, we have

Corollary 3.2**.**

The operator $\mathrm{res}:K(\mathcal{Q})\rightarrow K(\mathcal{Q})\otimes K(\mathcal{Q})$ is an algebra homomorphism with respect to the twisted multiplication on $K(\mathcal{Q})\otimes K(\mathcal{Q})$.

Theorem 3.3** ([12, 14]).**

There is a unique $\mathcal{A}$-algebra isomorphism

[TABLE]

such that

[TABLE]

and $\lambda_{\mathcal{A}}(\mathcal{L}_{\mathbf{y}})=\theta_{\mathbf{y}}$ for all $\mathbf{y}\in Y_{\nu}$, where $\theta_{\mathbf{y}}=\theta_{i_{1}}^{(a_{1})}\theta_{i_{2}}^{(a_{2})}\cdots\theta_{i_{k}}^{(a_{k})}$ and $\mathbf{f}_{\mathcal{A}}$ is the integral form of $\mathbf{f}$.

4. Geometric realization of the quantum group $\mathbf{U}$

In this section, we shall define a skew-Hopf pairing and show that a quotient of the Drinfeld double of this skew-Hopf pairing is isomorphic to the quantum group $\mathbf{U}$.

Let $Q$ be a quiver and fix a Cartan datum $(A,\Pi,\Pi^{\vee},P,P^{\vee})$ with $A=(a_{ij})$, where $a_{ij}=\#\{i\rightarrow j\}+\#\{j\rightarrow i\}$. Let $\mathbf{U}$ be the quantum group corresponding to this Cartan datum.

Let $\mathbf{K}=\bigoplus_{\mu\in P^{\vee}}\mathcal{A}\mathbf{k}_{\mu}$ be the torus algebra and set $\tilde{K}(\mathcal{Q})^{+}$ be the free $\mathcal{A}$-module with the basis $\{\mathbf{k}_{\mu}[\mathcal{L}]^{+}\mid\mu\in P^{\vee},[\mathcal{L}]\in\mathbf{B}\}.$ The Hopf algebra structure of $\tilde{K}(\mathcal{Q})^{+}$ is given by the following operations ([18, 19]).

1. (a)

The multiplication is defined as

1. (1)

$[\mathcal{L}_{1}]^{+}[\mathcal{L}_{2}]^{+}=[\mathcal{L}_{1}\ast\mathcal{L}_{2}]^{+}$ for $\mathcal{L}_{i}\in\mathcal{Q}_{\mathbf{V}_{i}}$, 2. (2)

$\mathbf{k}_{\mu}[\mathcal{L}]^{+}\mathbf{k}_{-\mu}=v^{\alpha(\mu)}[\mathcal{L}]^{+}$ for $\mu\in P^{\vee}$ and $\mathcal{L}\in\mathcal{Q}_{\mathbf{V}}$ with $\underline{\mbox{\rm dim}\,}\mathbf{V}=\alpha$, 3. (3)

$\mathbf{k}_{\mu}\mathbf{k}_{\mu^{\prime}}=\mathbf{k}_{\mu+\mu^{\prime}}$ for all $\mu,\mu^{\prime}\in P^{\vee}$. 2. (b)

The comultiplication is defined as

1. (1)

$\tilde{\Delta}([\mathcal{L}]^{+})=\sum_{\nu^{\prime}+\nu^{\prime\prime}=\nu}\mathrm{res}^{\mathbf{V}}_{\mathbf{V}^{\prime},\mathbf{V}^{\prime\prime}}[\mathcal{L}]^{+}(\mathbf{k}_{\mu^{\prime\prime}}\otimes\mathbf{1})$ for $\mathcal{L}\in\mathcal{Q}_{\mathbf{V}}$, where $\underline{\mbox{\rm dim}\,}\mathbf{V}=\nu$, $\underline{\mbox{\rm dim}\,}\mathbf{V}^{\prime}=\nu^{\prime}$ and $\underline{\mbox{\rm dim}\,}\mathbf{V}^{\prime\prime}=\nu^{\prime\prime}$, 2. (2)

$\tilde{\Delta}(\mathbf{k}_{\mu})=\mathbf{k}_{\mu}\otimes\mathbf{k}_{\mu}$ for all $\mu\in P^{\vee}$. 3. (c)

The antipode is defined as

1. (1)

$\tilde{S}([\mathcal{L}]^{+})=\sum_{r\geqslant 1}(-1)^{r}\sum_{\nu_{1}+\cdots+\nu_{r}=\nu}\mathbf{k}_{-\nu}\mathrm{ind}^{\mathbf{V}}_{\mathbf{V}_{1},\dots,\mathbf{V}_{r}}\mathrm{res}^{\mathbf{V}}_{\mathbf{V}_{1},\dots,\mathbf{V}_{r}}[\mathcal{L}]^{+}$ for $0\not\cong\mathcal{L}\in\mathcal{Q}_{\mathbf{V}}$, where $\underline{\mbox{\rm dim}\,}\mathbf{V}=\nu$, $\underline{\mbox{\rm dim}\,}\mathbf{V}_{l}=\nu_{l}$, 2. (2)

$\tilde{S}(\mathbf{k}_{\mu})=\mathbf{k}_{-\mu}$ for all $\mu\in P^{\vee}$.

Set $\tilde{K}(\mathcal{Q})^{-}$ be the free $\mathcal{A}$-module with the basis $\{\mathbf{k}_{\mu}[\mathcal{L}]^{-}\mid\mu\in P^{\vee},[\mathcal{L}]\in\mathbf{B}\}.$ The Hopf algebra structure of $\tilde{K}(\mathcal{Q})^{-}$ is given by the following operations.

1. (a)

The multiplication is defined as

1. (1)

$[\mathcal{L}_{1}]^{-}[\mathcal{L}_{2}]^{-}=[\mathcal{L}_{1}\ast\mathcal{L}_{2}]^{-}$ for $\mathcal{L}_{i}\in\mathcal{Q}_{\mathbf{V}_{i}}$, 2. (2)

$\mathbf{k}_{\mu}[\mathcal{L}]^{-}\mathbf{k}_{-\mu}=v^{-\alpha(\mu)}[\mathcal{L}]^{-}$ for $\mu\in P^{\vee}$ and $\mathcal{L}\in\mathcal{Q}_{\mathbf{V}}$ with $\underline{\mbox{\rm dim}\,}\mathbf{V}=\alpha$, 3. (3)

$\mathbf{k}_{\mu}\mathbf{k}_{\mu^{\prime}}=\mathbf{k}_{\mu+\mu^{\prime}}$ for all $\mu,\mu^{\prime}\in P^{\vee}$. 2. (b)

The comultiplication is defined as

1. (1)

$\tilde{\Delta}([\mathcal{L}]^{-})=\sum_{\nu^{\prime}+\nu^{\prime\prime}=\nu}(\mathbf{1}\otimes\mathbf{k}_{-\mu^{\prime\prime}})(\mathrm{res}^{\mathbf{V}}_{\mathbf{V}^{\prime},\mathbf{V}^{\prime\prime}})^{op}[\mathcal{L}]^{-}$ for $\mathcal{L}\in\mathcal{Q}_{\mathbf{V}}$, where $\underline{\mbox{\rm dim}\,}\mathbf{V}=\nu$, $\underline{\mbox{\rm dim}\,}\mathbf{V}^{\prime}=\nu^{\prime}$ and $\underline{\mbox{\rm dim}\,}\mathbf{V}^{\prime\prime}=\nu^{\prime\prime}$, 2. (2)

$\tilde{\Delta}(\mathbf{k}_{\mu})=\mathbf{k}_{\mu}\otimes\mathbf{k}_{\mu}$ for all $\mu\in P^{\vee}$. 3. (c)

The antipode is defined as

1. (1)

$\tilde{S}([\mathcal{L}]^{-})=\sum_{r\geqslant 1}(-1)^{r}\sum_{\nu_{1}+\cdots+\nu_{r}=\nu}\mathrm{ind}^{\mathbf{V}}_{\mathbf{V}_{1},\dots,\mathbf{V}_{r}}(\mathrm{res}^{\mathbf{V}}_{\mathbf{V}_{1},\dots,\mathbf{V}_{r}})^{op}[\mathcal{L}]^{-}\mathbf{k}_{\nu}$ for $0\not\cong\mathcal{L}\in\mathcal{Q}_{\mathbf{V}}$, where $\underline{\mbox{\rm dim}\,}\mathbf{V}=\nu$, $\underline{\mbox{\rm dim}\,}\mathbf{V}_{l}=\nu_{l}$, 2. (2)

$\tilde{S}(\mathbf{k}_{\mu})=\mathbf{k}_{-\mu}$ for all $\mu\in P^{\vee}$.

Fix an $I$-graded $\mathbb{K}$-vector space ${\mathbf{V}}$ with dimension vector $\nu\in{\mathbb{N}}I$. Given ${\mathcal{L}},{\mathcal{L}}^{\prime}\in\mathcal{Q}_{\mathbf{V}}$, let

[TABLE]

This definition can be extended to define a bilinear form $\varphi:\tilde{K}(\mathcal{Q})^{+}\times\tilde{K}(\mathcal{Q})^{-}\rightarrow\mathbb{Q}(v)$ by setting

[TABLE]

for ${\mathcal{L}}\in\mathcal{Q}_{\mathbf{V}}$ and ${\mathcal{L}}^{\prime}\in\mathcal{Q}_{\mathbf{V}^{\prime}}$ such that $\underline{\mbox{\rm dim}\,}\mathbf{V}=\nu$ and $\underline{\mbox{\rm dim}\,}\mathbf{V}^{\prime}=\nu^{\prime}$.

Proposition 4.1** ([19]).**

The triple $(\tilde{K}(\mathcal{Q})^{+},\tilde{K}(\mathcal{Q})^{-},\varphi)$ is a skew-Hopf pairing.

Let $D(\tilde{K}(\mathcal{Q})^{+},\tilde{K}(\mathcal{Q})^{-})$ be the Drinfeld double of this skew-Hopf pairing and $DK(\mathcal{Q})=DK(\mathcal{Q})(Q)$ be the quotient of $D(\tilde{K}(\mathcal{Q})^{+},\tilde{K}(\mathcal{Q})^{-})$ module the Hopf ideal generated by $\mathbf{k}_{\mu}\otimes\mathbf{1}-\mathbf{1}\otimes\mathbf{k}_{\mu}$ for all $\mu\in P^{\vee}$. It is clear that $DK(\mathcal{Q})$ has the following triangular decomposition

[TABLE]

Theorem 3.3 and the construction of $DK(\mathcal{Q})$ imply the following theorem.

Theorem 4.2**.**

There exists an isomorphism of Hopf algebras

[TABLE]

such that $\lambda_{\mathcal{A}}([\mathcal{L}_{\mathbf{y}}]^{+})=\theta_{\mathbf{y}}^{+}$, $\lambda_{\mathcal{A}}([\mathcal{L}_{\mathbf{y}}]^{-})=\theta_{\mathbf{y}}^{-}$ and $\lambda_{\mathcal{A}}(\mathbf{k}_{\mu})=K_{\mu}$ for all $\mathbf{y}\in Y_{\nu}$ and $\mu\in P^{\vee}$, where $\mathbf{U}_{\mathcal{A}}$ is the integral form of $\mathbf{U}$.

5. Structure of the $\mathcal{A}$-module $K(\mathcal{Q})$

For the geometric definition of Lusztig’s symmetries, we shall study the structure of the $\mathcal{A}$-module $K(\mathcal{Q})$ in this section.

5.1.

Let $Q=(I,H,s,t)$ be a quiver. Assume that $i\in I$ is a sink. Let ${\mathbf{V}}$ be a finite dimensional $I$-graded ${\mathbb{K}}$-vector space such that $\underline{\mbox{\rm dim}\,}{\mathbf{V}}=\nu\in{\mathbb{N}}I$. For any $r\in\mathbb{Z}_{\geqslant 0}$, consider a subvariety ${{}_{i}E}_{{\mathbf{V}},r}$ of $E_{\mathbf{V}}$

[TABLE]

Denote by ${{}_{i}j}_{{\mathbf{V}},r}:{{}_{i}E}_{{\mathbf{V}},r}\rightarrow E_{{\mathbf{V}}}$ the natural embedding. Let ${\mathcal{D}}_{G_{\mathbf{V}}}^{b}({{}_{i}E}_{{\mathbf{V}},r})$ be the $G_{\mathbf{V}}$-equivariant bounded derived category of ${\overline{{\mathbb{Q}}}_{l}}$-constructible complexes on ${{}_{i}E}_{{\mathbf{V}},r}$. Naturally, we have two functors $({{}_{i}j}_{\mathbf{V},r})_{!}:\mathcal{D}_{G_{\mathbf{V}}}({{}_{i}E}_{{\mathbf{V}},r})\rightarrow\mathcal{D}_{G_{\mathbf{V}}}(E_{\mathbf{V}})$ and ${{}_{i}j}^{\ast}_{\mathbf{V},r}:\mathcal{D}_{G_{\mathbf{V}}}(E_{\mathbf{V}})\rightarrow\mathcal{D}_{G_{\mathbf{V}}}({{}_{i}E}_{{\mathbf{V}},r})$.

Note that the variety $E_{\mathbf{V}}$ is a disjoint union of ${{}_{i}E}_{{\mathbf{V}},r}$ for all $r\geqslant 0$. For any $r\geqslant 0$, let ${{}_{i}E}_{{\mathbf{V}},\geqslant r}=\cup_{r^{\prime}\geqslant r}\,\,{{}_{i}E}_{{\mathbf{V}},r^{\prime}}$ and ${{}_{i}E}_{{\mathbf{V}},\leqslant r}=\cup_{r^{\prime}\leqslant r}\,\,{{}_{i}E}_{{\mathbf{V}},r^{\prime}}$. Let $i_{{\mathbf{V}},\geqslant r}:{{}_{i}E}_{{\mathbf{V}},\geqslant r}\rightarrow E_{{\mathbf{V}}}$ be the natural closed embedding and ${{}_{i}j}_{{\mathbf{V}},\leqslant r}:{{}_{i}E}_{{\mathbf{V}},\leqslant r}\rightarrow E_{{\mathbf{V}}}$ be the natural open embedding.

For any $\mathbf{y}=(\mathbf{i},\mathbf{a})\in{Y}_{\nu}$, let

[TABLE]

and ${{}_{i}\pi}_{\mathbf{y},r}:{{}_{i}\tilde{F}}_{\mathbf{y},r}\rightarrow{{}_{i}E}_{{\mathbf{V}},r}$ be the projection to ${{}_{i}E}_{{\mathbf{V}},r}$.

For any $\mathbf{y}\in{Y}_{\nu}$, ${{}_{i}\mathcal{L}}_{\mathbf{y},r}=({{}_{i}\pi}_{\mathbf{y},r})_{!}(\mathbf{1}_{{{}_{i}\tilde{F}}_{\mathbf{y},r}})[d_{\mathbf{y}}](\frac{d_{\mathbf{y}}}{2})\in\mathcal{D}_{G_{\mathbf{V}}}({{}_{i}E}_{{\mathbf{V}},r})$ is a semisimple perverse sheaf. Let ${{}_{i}\mathcal{P}}_{\mathbf{V},r}$ be the set of simple perverse sheaves $\mathcal{L}$ on ${{}_{i}E}_{{\mathbf{V}},r}$ such that $\mathcal{L}[s]$ appears as a direct summand of ${{}_{i}\mathcal{L}}_{\mathbf{y},r}$ for some $\mathbf{y}\in{Y}_{\nu}$ and $s\in\mathbb{Z}$. Let ${{}_{i}\mathcal{Q}}_{\mathbf{V},r}$ be the full subcategory of $\mathcal{D}_{G_{\mathbf{V}}}({{}_{i}E}_{{\mathbf{V}},r})$ consisting of all complexes which are isomorphic to finite direct sums of complexes in the set $\{\mathcal{L}[s]\,\,|\,\,\mathcal{L}\in{{}_{i}\mathcal{P}}_{\mathbf{V},r},s\in\mathbb{Z}\}$.

Let $K({{}_{i}\mathcal{Q}}_{\mathbf{V},r})$ be the Grothendieck group of ${{}_{i}\mathcal{Q}}_{\mathbf{V},r}$ and define $v^{\pm}[\mathcal{L}]=[\mathcal{L}[\pm 1](\pm\frac{1}{2})]$ for any $\mathcal{L}\in{{}_{i}\mathcal{Q}}_{\mathbf{V},r}$. Then, $K({{}_{i}\mathcal{Q}}_{\mathbf{V},r})$ is a free $\mathcal{A}$-module. Let

[TABLE]

Let ${{}_{i}\mathbf{B}}_{\mathbf{V},r}=\{[\mathcal{L}]\,\,|\,\,\mathcal{L}\in{{}_{i}\mathcal{P}}_{\mathbf{V},r}\}$ and ${{}_{i}\mathbf{B}}_{r}=\bigcup_{\nu}{{}_{i}\mathbf{B}}_{\mathbf{V},r}$, which is an $\mathcal{A}$-basis of $K({{}_{i}\mathcal{Q}}_{r})$.

Lemma 5.1**.**

For any $I$-graded $\mathbb{K}$-vector space $\mathbf{V}$ and $0\leqslant r\in\mathbb{Z}$, we have

[TABLE]

Proof.

For any $\mathbf{y}\in{Y}_{\nu}$, we have the following fiber product

[TABLE]

Hence

[TABLE]

That is ${{}_{i}j}^{\ast}_{\mathbf{V},r}(\mathcal{Q}_{\mathbf{V}})={{}_{i}\mathcal{Q}}_{\mathbf{V},r}$. ∎

Hence, we get an $\mathcal{A}$-linear map ${{}_{i}j}^{\ast}_{r}:K(\mathcal{Q})\rightarrow K({{}_{i}\mathcal{Q}}_{r})$.

Theorem 5.2**.**

For any $\mathcal{L}\in\mathcal{Q}_{\mathbf{V}}$ with $\mbox{\rm dim}\,V_{i}=s$, let $\mathcal{L}_{\geqslant r}=(i_{{\mathbf{V}},\geqslant r})_{\ast}i^{\ast}_{{\mathbf{V}},\geqslant r}\mathcal{L}$ and $\mathcal{L}_{r}={{}_{i}j}^{\ast}_{\mathbf{V},r}\mathcal{L}\in{{}_{i}\mathcal{Q}}_{\mathbf{V},r}$. Then there exists a distinguished triangle

[TABLE]

for any $0\leqslant r\leqslant s$.

Proof.

By Lemma 5.1, we have $\mathcal{L}_{r}={{}_{i}j}^{\ast}_{\mathbf{V},r}\mathcal{L}\in{{}_{i}\mathcal{Q}}_{\mathbf{V},r}$.

Then, we shall prove this theorem by induction.

By definition, $\mathcal{L}_{\geqslant 0}=(i_{{\mathbf{V}},\geqslant 0})_{\ast}i^{\ast}_{{\mathbf{V}},\geqslant 0}\mathcal{L}=\mathcal{L}$. Since ${{}_{i}E}_{{\mathbf{V}},\geqslant 1}$ is closed in ${E}_{{\mathbf{V}}}$, we have the following distinguished triangle

[TABLE]

That is

[TABLE]

Hence this theorem is true for $r=0$.

Since ${{}_{i}E}_{{\mathbf{V}},\geqslant 2}$ is closed in ${E}_{{\mathbf{V}}}$, we have the following distinguished triangle

[TABLE]

Consider the following distinguished triangle

[TABLE]

Applying the functor $i^{\ast}_{{\mathbf{V}},\geqslant 2}$ to this distinguished triangle, we have $i^{\ast}_{{\mathbf{V}},\geqslant 2}\mathcal{L}_{\geqslant 1}=i^{\ast}_{{\mathbf{V}},\geqslant 2}\mathcal{L}$. Hence

[TABLE]

Applying the functor ${{}_{i}j}^{\ast}_{\mathbf{V},1}$ to this distinguished triangle, we have

[TABLE]

The complex ${{}_{i}j}^{\ast}_{\mathbf{V},\leqslant 1}\mathcal{L}_{\geqslant 1}$ is a complex on ${{}_{i}E}_{\mathbf{V},\leqslant 1}$ and the support is in ${{}_{i}E}_{\mathbf{V},1}$. Hence ${{}_{i}j}^{\ast}_{\mathbf{V},\leqslant 1}\mathcal{L}_{\geqslant 1}=({{{}_{i}\hat{j}}}_{\mathbf{V},1})_{!}{{}_{i}j}^{\ast}_{\mathbf{V},1}\mathcal{L}_{\geqslant 1}$, where ${{{}_{i}\hat{j}}}_{\mathbf{V},1}$ is the embedding from ${{}_{i}E}_{\mathbf{V},1}$ to ${{}_{i}E}_{\mathbf{V},\leqslant 1}$. Then

[TABLE]

By (3), we have

[TABLE]

Hence the distinguished triangle (2) can be rewrote as

[TABLE]

and this theorem is true for $r=1$.

Assume that this theorem is true for $r=k-1$.

Since ${{}_{i}E}_{{\mathbf{V}},\geqslant k+1}$ is closed in ${E}_{{\mathbf{V}}}$, we have the following distinguished triangle

[TABLE]

Consider the following distinguished triangle

[TABLE]

Applying the functor $i^{\ast}_{{\mathbf{V}},\geqslant k+1}$ to this distinguished triangle, we have $i^{\ast}_{{\mathbf{V}},\geqslant k+1}\mathcal{L}_{\geqslant k}=i^{\ast}_{{\mathbf{V}},\geqslant k+1}\mathcal{L}$. Hence

[TABLE]

Applying the functor ${{}_{i}j}^{\ast}_{\mathbf{V},k}$ to this distinguished triangle, we have

[TABLE]

The complex ${{}_{i}j}^{\ast}_{\mathbf{V},\leqslant k}\mathcal{L}_{\geqslant k}$ is a complex on ${{}_{i}E}_{\mathbf{V},\leqslant k}$ and the support is in ${{}_{i}E}_{\mathbf{V},k}$. Hence ${{}_{i}j}^{\ast}_{\mathbf{V},\leqslant k}\mathcal{L}_{\geqslant k}=({{{}_{i}\hat{j}}}_{\mathbf{V},k})_{!}{{}_{i}j}^{\ast}_{\mathbf{V},k}\mathcal{L}_{\geqslant k}$, where ${{{}_{i}\hat{j}}}_{\mathbf{V},k}$ is the embedding from ${{}_{i}E}_{\mathbf{V},k}$ to ${{}_{i}E}_{\mathbf{V},\leqslant k}$. Then

[TABLE]

By (5), we have

[TABLE]

Hence the distinguished triangle (4) can be rewrote as

[TABLE]

and this theorem is true for $r=k$.

Hence, we have proved this theorem. ∎

Lemma 5.3**.**

Fix an $I$-graded $\mathbb{K}$-vector space $\mathbf{V}$ and $0\leqslant r\in\mathbb{Z}$. For any $\mathcal{L}\in{{}_{i}\mathcal{Q}}_{\mathbf{V},r}$, we have $[({{}_{i}j}_{\mathbf{V},r})_{!}(\mathcal{L})]\in K(\mathcal{Q}_{\mathbf{V}}).$

For the proof of this lemma, we need to review Lusztig’s constructions of Hall algebras via functions ([13]). For any $I$-graded $\mathbb{K}$-vector space $\mathbf{V}$ and $1\leqslant n\in\mathbb{Z}$, let $E^{F^{n}}_{\mathbf{V}}$ and $G^{F^{n}}_{\mathbf{V}}$ be the sets consisting of the $F^{n}$-fixed points in $E_{\mathbf{V}}$ and $G_{\mathbf{V}}$ respectively, where $F$ is the Frobenius morphism.

Lusztig defined $\underline{\mathcal{F}}^{n}_{\mathbf{V}}$ as the set of all $G^{F^{n}}_{\mathbf{V}}$-invariant $\overline{\mathbb{Q}}_{l}$-functions on $E^{F^{n}}_{\mathbf{V}}$ and we can give a multiplication on $\underline{\mathcal{F}}^{n}=\bigoplus_{\nu\in\mathbb{N}I}\underline{\mathcal{F}}^{n}_{\mathbf{V}}$ to obtain the Hall algebra. For any $i\in I$, let $\mathbf{V}_{ti}$ be the $I$-graded $\mathbb{K}$-vector space with dimension vector $ti$ and $f_{i}$ be the constant function on $E^{F^{n}}_{\mathbf{V}_{i}}$ with value $1$. Denote by $\mathcal{F}^{n}$ the composition subalgebra of $\underline{\mathcal{F}}^{n}$ generated by $f_{i}$ and $\mathcal{F}^{n}_{\mathbf{V}}=\underline{\mathcal{F}}^{n}_{\mathbf{V}}\cap\mathcal{F}^{n}$.

For any $\mathcal{L}\in\mathcal{D}_{G_{\mathbf{V}}}(E_{\mathbf{V}})$, there is a function $\chi^{n}_{\mathcal{L}}:E^{F^{n}}_{\mathbf{V}}\rightarrow\overline{\mathbb{Q}}_{l}$ (Section III.12 in [8]). Hence, we have the trace map

[TABLE]

Lusztig proved that $\chi^{n}(\mathcal{Q}_{\mathbf{V}})=\mathcal{F}^{n}_{\mathbf{V}}$ in [13].

The proof of Lemma 5.3.

Similarly to the definition of $\chi^{n}$, we can get a function $f_{n}={{}_{r}\chi}^{n}_{\mathcal{L}}$ on ${{}_{i}E}^{F^{n}}_{{\mathbf{V}},r}$ for any $\mathcal{L}\in{{}_{i}\mathcal{Q}}_{\mathbf{V},r}$ and $n\in\mathbb{Z}_{\geqslant 1}$. The function $f_{n}$ can be viewed as a function on $E^{F^{n}}_{\mathbf{V}}$ and ${\chi}^{n}_{({{}_{i}j}_{\mathbf{V},r})_{!}\mathcal{L}}=f_{n}$.

By Lemma 5.1, there exists a complex $\hat{\mathcal{L}}\in\mathcal{Q}_{\mathbf{V}}$ such that ${{}_{i}j}^{\ast}_{\mathbf{V},r}\hat{\mathcal{L}}=\mathcal{L}$. Note that the function $f_{n}$ is the restriction of ${\chi}^{n}_{\hat{\mathcal{L}}}$ on ${{}_{i}E}^{F^{n}}_{{\mathbf{V}},r}$. Since ${\chi}^{n}_{\hat{\mathcal{L}}}\in\mathcal{F}^{n}_{\mathbf{V}}$, we have $f_{n}\in\mathcal{F}^{n}_{\mathbf{V}}$.

By Theorem III.12.1 in [8], there is an injective map

[TABLE]

Hence, we have $[({{}_{i}j}_{\mathbf{V},r})_{!}(\mathcal{L})]\in K(\mathcal{Q}_{\mathbf{V}}).$ ∎

Hence, we get an $\mathcal{A}$-linear map $({{}_{i}j}_{r})_{!}:K({{}_{i}\mathcal{Q}}_{r})\rightarrow K(\mathcal{Q})$.

Theorem 5.4**.**

For any $I$-graded $\mathbb{K}$-vector space $\mathbf{V}$ with $\mbox{\rm dim}\,V_{i}=s$, we have the following isomorphism of $\mathcal{A}$-modules

[TABLE]

where $\mathcal{L}_{r}={{}_{i}j}^{\ast}_{\mathbf{V},r}\mathcal{L}$.

Proof.

By Lemma 5.1, this map is well-defined. Consider the following map

[TABLE]

By Theorem 5.2, we have ${{}_{i}\Psi}_{\mathbf{V}}\circ{{}_{i}\Phi}_{\mathbf{V}}([\mathcal{L}])=[\mathcal{L}]$. Hence the map ${{}_{i}\Phi}_{\mathbf{V}}$ is injective. By Lemma 5.3, the map ${{}_{i}\Phi}_{\mathbf{V}}$ is also surjective. ∎

Theorem 5.5**.**

For any $I$-graded $\mathbb{K}$-vector space $\mathbf{V}$ and integer $0\leqslant r\leqslant\mbox{\rm dim}\,V_{i}$, fix an $I$-graded vector space $\mathbf{W}$ with $\mbox{\rm dim}\,W_{i}=\mbox{\rm dim}\,V_{i}-r$ and $\mbox{\rm dim}\,W_{j}=\mbox{\rm dim}\,V_{j}$ for all $j\neq i$. Then we have the following isomorphism of $\mathcal{A}$-modules

[TABLE]

where ${\mathcal{L}}_{ri}$ is the constant sheaf on $E_{\mathbf{V}_{ri}}$.

Proof.

By Lemma 5.3, this map is well-defined. Consider the following map

[TABLE]

where $\mathrm{Res}_{\mathbf{V}_{ri}\mathbf{W}}^{\mathbf{V}}(({{}_{i}j}_{\mathbf{V},r})_{!}\mathcal{L})={\mathcal{L}}_{ri}\otimes{\mathcal{L}}^{\prime}$. By definition, we have ${{}_{i}\mathrm{res}}_{\mathbf{W}}^{\mathbf{V}}\circ{{}_{i}\mathrm{ind}}_{\mathbf{W}}^{\mathbf{V}}=\mathrm{id}$ and ${{}_{i}\mathrm{ind}}_{\mathbf{W}}^{\mathbf{V}}\circ{{}_{i}\mathrm{res}}_{\mathbf{W}}^{\mathbf{V}}=\mathrm{id}$. ∎

By Theorem 5.4 and 5.5, we have the following decomposition of $K(\mathcal{Q})$.

Theorem 5.6**.**

The $\mathcal{A}$-module $K(\mathcal{Q})$ has the following direct sum decomposition

[TABLE]

Proposition 5.7** ([21, 22]).**

There exists an isomorphism of $\mathcal{A}$-algebras ${{{}_{i}\lambda}_{0,\mathcal{A}}}:K({{}_{i}\mathcal{Q}}_{0})\rightarrow{{}_{i}\mathbf{f}}_{\mathcal{A}}$ such that the following diagram is commutative

[TABLE]

Hence, Theorem 5.6 is a geometric interpretation of Theorem 2.2.

5.2.

When $i\in I$ is a source, we have similar constructions. Let ${\mathbf{V}}$ be a finite dimensional $I$-graded ${\mathbb{K}}$-vector space such that $\underline{\mbox{\rm dim}\,}{\mathbf{V}}=\nu\in{\mathbb{N}}I$. For any $r\in\mathbb{Z}_{\geqslant 0}$, consider a subvariety ${{}^{i}E}_{{\mathbf{V}},r}$ of $E_{\mathbf{V}}$

[TABLE]

Denote by ${{}^{i}j}_{{\mathbf{V}},r}:{E}_{{\mathbf{V}},r}\rightarrow E_{{\mathbf{V}}}$ be the natural embedding. Let ${\mathcal{D}}_{G_{\mathbf{V}}}^{b}({{}^{i}E}_{{\mathbf{V}},r})$ be the $G_{\mathbf{V}}$-equivariant bounded derived category of ${\overline{{\mathbb{Q}}}_{l}}$-constructible complexes on ${{}^{i}E}_{{\mathbf{V}},r}$. Naturally, we have two functors $({{}^{i}j}_{\mathbf{V},r})_{!}:\mathcal{D}_{G_{\mathbf{V}}}({{}^{i}E}_{\mathbf{V},r})\rightarrow\mathcal{D}_{G_{\mathbf{V}}}(E_{\mathbf{V}})$ and ${{}^{i}j}^{\ast}_{\mathbf{V},r}:\mathcal{D}_{G_{\mathbf{V}}}(E_{\mathbf{V}})\rightarrow\mathcal{D}_{G_{\mathbf{V}}}({{}^{i}E}_{\mathbf{V},r})$.

Similarly, we can define ${{}^{i}\mathcal{P}}_{\mathbf{V},r}$, ${{}^{i}\mathcal{Q}}_{\mathbf{V},r}$, $K({{}^{i}\mathcal{Q}}_{\mathbf{V},r})$, $K({{}^{i}\mathcal{Q}}_{r})$, $({{}^{i}j}_{r})_{!}$ and ${{}^{i}j}_{r}^{\ast}$. We also have the following results.

Theorem 5.8**.**

For any $I$-graded $\mathbb{K}$-vector space $\mathbf{V}$ with $\mbox{\rm dim}\,V_{i}=s$, we have the following isomorphism

[TABLE]

where $\mathcal{L}_{r}={{}^{i}j}^{\ast}_{\mathbf{V},r}\mathcal{L}$. The inverse of ${{}^{i}\Phi}_{\mathbf{V}}$ is

[TABLE]

Theorem 5.9**.**

For any $I$-graded $\mathbb{K}$-vector space $\mathbf{V}$ and integer $0\leqslant r\leqslant\mbox{\rm dim}\,V_{i}$, fix an $I$-graded vector space $\mathbf{W}$ with $\mbox{\rm dim}\,W_{i}=\mbox{\rm dim}\,V_{i}-r$ and $\mbox{\rm dim}\,W_{j}=\mbox{\rm dim}\,V_{j}$ for all $j\neq i$. Then we have the following isomorphism

[TABLE]

whose inverse is

[TABLE]

where $\mathrm{Res}_{\mathbf{W}\mathbf{V}_{ri}}^{\mathbf{V}}(({{}^{i}j}_{\mathbf{V},r})_{!}\mathcal{L})={\mathcal{L}}^{\prime}\otimes{\mathcal{L}}_{ri}$.

Theorem 5.10**.**

The $\mathcal{A}$-module $K(\mathcal{Q})$ has the following direct sum decomposition

[TABLE]

Proposition 5.11** ([21, 22]).**

There exists an isomorphism of $\mathcal{A}$-algebras ${{}^{i}\lambda}_{0,\mathcal{A}}:K({{}^{i}\mathcal{Q}}_{0})\rightarrow{{}^{i}\mathbf{f}}_{\mathcal{A}}$ such that the following diagram is commutative

[TABLE]

6. Geometric realization of Lusztig’s symmetry $T_{i}:\mathbf{U}\rightarrow\mathbf{U}$

In this section, we shall recall the geometric realization of $T_{i}:{{}_{i}\mathbf{f}}\rightarrow{{}^{i}\mathbf{f}}$ in [21, 22]. By using this geometric realization and the structure of $K(\mathcal{Q})$ in last section, we shall give a geometric realization of Lusztig’s symmetry $T_{i}:\mathbf{U}\rightarrow\mathbf{U}$.

6.1.

Assume that $i$ is a sink of $Q=(I,H,s,t)$. So $i$ is a source of $Q^{\prime}=\sigma_{i}Q=(I,H^{\prime},s,t)$, where $\sigma_{i}Q$ is the quiver by reversing the directions of all arrows in $Q$ containing $i$. For any $\nu,\nu^{\prime}\in\mathbb{N}I$ such that $\nu^{\prime}=s_{i}\nu$ and $I$-graded $\mathbb{K}$-vector spaces $\mathbf{V}$, $\mathbf{V}^{\prime}$ such that $\underline{\mbox{\rm dim}\,}\mathbf{V}=\nu$, $\underline{\mbox{\rm dim}\,}\mathbf{V}^{\prime}=\nu^{\prime}$, there exists a functor ([21, 22])

[TABLE]

which induces the following map

[TABLE]

Theorem 6.1** ([21, 22]).**

We have the following commutative diagram

[TABLE]

6.2.

By the triangular decomposition (1) of $DK(\mathcal{Q})(Q)$, the following set is an $\mathcal{A}$-basis of $DK(\mathcal{Q})(Q)$

[TABLE]

where ${\mathcal{L}}\in\mathcal{P}_{\mathbf{V}}$ for some $I$-graded $\mathbb{K}$-vector space $\mathbf{V}$ with $\underline{\mbox{\rm dim}\,}{\mathbf{V}}=\nu\in{\mathbb{N}}I$, ${\mathcal{L}}^{\prime}\in\mathcal{P}_{\mathbf{V}^{\prime}}$ for some $I$-graded $\mathbb{K}$-vector space $\mathbf{V}^{\prime}$ with $\underline{\mbox{\rm dim}\,}{\mathbf{V}}^{\prime}=\nu^{\prime}\in{\mathbb{N}}I$ and $\mu\in P^{\vee}$.

By Theorem 5.6, we have

[TABLE]

and

[TABLE]

where $[{\mathcal{L}}_{r}]={{}_{i}\mathrm{res}}_{\mathbf{W}_{r}}^{\mathbf{V}}[{{}_{i}j}^{\ast}_{\mathbf{V},r}\mathcal{L}]$, $[{\mathcal{L}}^{\prime}_{r}]={{}_{i}\mathrm{res}}_{\mathbf{W}_{r}}^{\mathbf{V}}[{{}_{i}j}^{\ast}_{\mathbf{V},r}\mathcal{L}^{\prime}]$ and $\mathbf{W}_{r}$ is an $I$-graded $\mathbb{K}$-vector space with $\mbox{\rm dim}\,W_{i}=\mbox{\rm dim}\,V_{i}-r$ and $\mbox{\rm dim}\,W_{j}=\mbox{\rm dim}\,V_{j}$ for all $j\neq i$.

Define

[TABLE]

[TABLE]

and

[TABLE]

Hence, we get a map

[TABLE]

Proposition 6.2**.**

The map $\tilde{T}_{i}$ is a bijection.

For the proof of this proposition, we should construct the inverse of $\tilde{T}_{i}$. The algebra $DK(\mathcal{Q}))(Q^{\prime})$ also has the following $\mathcal{A}$-basis

[TABLE]

where ${\mathcal{L}}\in\mathcal{P}_{\mathbf{V}}$ for some $I$-graded $\mathbb{K}$-vector space $\mathbf{V}$ with $\underline{\mbox{\rm dim}\,}{\mathbf{V}}=\nu\in{\mathbb{N}}I$, ${\mathcal{L}}^{\prime}\in\mathcal{P}_{\mathbf{V}^{\prime}}$ for some $I$-graded $\mathbb{K}$-vector space $\mathbf{V}^{\prime}$ with $\underline{\mbox{\rm dim}\,}{\mathbf{V}}^{\prime}=\nu^{\prime}\in{\mathbb{N}}I$ and $\mu\in P^{\vee}$.

By Theorem 5.10, we have

[TABLE]

and

[TABLE]

where $[{\mathcal{L}}_{r}]={{}^{i}\mathrm{res}}_{\mathbf{W}_{r}}^{\mathbf{V}}[{{}^{i}j}^{\ast}_{\mathbf{V},r}\mathcal{L}]$, $[{\mathcal{L}}^{\prime}_{r}]={{}^{i}\mathrm{res}}_{\mathbf{W}_{r}}^{\mathbf{V}}[{{}^{i}j}^{\ast}_{\mathbf{V},r}\mathcal{L}^{\prime}]$.

Define

[TABLE]

[TABLE]

and

[TABLE]

Hence, we get a map

[TABLE]

By the definitions of $\tilde{T}_{i}$ and $\tilde{T}^{\prime}_{i}$, the map $\tilde{T}^{\prime}_{i}$ is the inverse of $\tilde{T}_{i}$. Hence, we have proved Proposition 6.2.

The following theorem is the main result in this paper.

Theorem 6.3**.**

The map $\tilde{T}_{i}:DK(\mathcal{Q})(Q)\rightarrow DK(\mathcal{Q})(Q^{\prime})$ is an isomorphism of Hopf algebras satisfying that the following diagram is commutative

[TABLE]

Proof.

Consider the $\mathcal{A}$-basis ${{}_{i}\mathbf{B}}_{0}$ of $K({{}_{i}\mathcal{Q}}_{0})$. By Theorem 5.6, the following set is an $\mathcal{A}$-basis of $DK(\mathcal{Q})(Q)$

[TABLE]

Similarly, by Theorem 2.2, the following set is an $\mathcal{A}$-basis of $\mathbf{U}_{\mathcal{A}}$

[TABLE]

Note that these two basis are identified under the isomorphism $\lambda_{\mathcal{A}}:DK(\mathcal{Q})(Q)\cong\mathbf{U}_{\mathcal{A}}$. By Theorem 6.1 and the definition of $\tilde{T}_{i}$, we have the desired commutative diagram. Since $T_{i}:\mathbf{U}_{\mathcal{A}}\rightarrow\mathbf{U}_{\mathcal{A}}$ is an isomorphism of Hopf algebras, so is the map $\tilde{T}_{i}:DK(\mathcal{Q})(Q)\rightarrow DK(\mathcal{Q})(Q^{\prime})$. ∎

7. Braid group relations

In this section, we shall consider the braid group relations of Lusztig’s symmetries.

7.1.

First, we shall recall the Fourier-Deligne transform ([8][14]). Let $Q=(I,H,s,t)$ be a quiver. Let $E$ be a subset of $H$ and denote by $Q^{\prime}=\sigma_{E}Q$ the quiver obtained from $Q$ by reversing all the arrows in $E$. Given $\nu\in{\mathbb{N}}I$, let ${\mathbf{V}}$ be an $I$-graded $\mathbb{K}$-vector space with dimension vector $\nu$. The Fourier-Deligne transform is denoted by ([14])

[TABLE]

Lemma 7.1** ([14]).**

The Fourier-Deligne transform $\Theta_{Q,Q^{\prime}}:{\mathcal{D}}^{b}_{G_{{\mathbf{V}}}}(E_{{\mathbf{V}},Q})\rightarrow{\mathcal{D}}^{b}_{G_{{\mathbf{V}}}}(E_{{\mathbf{V}},Q^{\prime}})$ is an equivalence of triangulated categories and $\Theta_{Q,Q^{\prime}}(\mathcal{Q}_{\mathbf{V},Q})=\mathcal{Q}_{\mathbf{V},Q^{\prime}}$.

By Lemma 7.1, we have the following proposition.

Proposition 7.2**.**

The Fourier-Deligne transform $\Theta_{Q,Q^{\prime}}:\mathcal{Q}_{\mathbf{V},Q}\rightarrow\mathcal{Q}_{\mathbf{V},Q^{\prime}}$ for various dimension vectors induce an isomorphism of Hopf algebras

[TABLE]

sending $[{\mathcal{L}}]^{-}\mathbf{k}_{\mu}[{\mathcal{L}}^{\prime}]^{+}$ to $[\Theta_{Q,Q^{\prime}}({\mathcal{L}})]^{-}\mathbf{k}_{\mu}[\Theta_{Q,Q^{\prime}}({\mathcal{L}}^{\prime})]^{+}.$

7.2.

For any quiver $Q=(I,H,s,t)$, choose a subset $E$ of $H$ such that $i$ is a sink of $\sigma_{E}Q$. Consider the quiver $\sigma_{i}\sigma_{E}Q=(I,H^{\prime},s,t)$ with $i$ as a source. Choose a subset $E^{\prime}$ of $H^{\prime}$ such that $\sigma_{E^{\prime}}\sigma_{i}\sigma_{E}Q=Q$.

Define

[TABLE]

as the composition of

[TABLE]

By using Fourier-Deligne transform, Theorem 6.3 can be rewrote as following.

Theorem 7.3**.**

The map $\hat{T}_{i}:DK(\mathcal{Q})(Q)\rightarrow DK(\mathcal{Q})(Q)$ is an isomorphism of Hopf algebras satisfying that the following diagram is commutative

[TABLE]

Since ${T}_{i}:\mathbf{U}\rightarrow\mathbf{U}$ satisfies the braid group relations, the isomorphism $\hat{T}_{i}:DK(\mathcal{Q})(Q)\rightarrow DK(\mathcal{Q})(Q)$ also satisfies the braid group relations, that is, we have the following commutative diagrams

[TABLE]

for any $i\neq j\in I$ such that $a_{ij}=-1$, and

[TABLE]

for any $i\neq j\in I$ such that $a_{ij}=0$.