Quiver Hecke Algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 27 July 2012

Quiver Hecke Algebras $R_{\alpha }$

Let $C=(⟨{\alpha }_i,{\alpha }_j^{\vee }⟩)$ be a symmetrizable Cartan matrix so that $$⟨{\alpha }_i,{\alpha }_i^{\vee }⟩=2,⟨{\alpha }_i,{\alpha }_j^{\vee }⟩\in {\mathbb{Z}}_{\le 0},\text{and}⟨{\alpha }_i,{\alpha }_j^{\vee }⟩=0\iff ⟨{\alpha }_j,{\alpha }_i^{\vee }⟩=0.$$ Let $${\alpha }_i^{\vee }=\frac{2}{⟨{\alpha }_i,{\alpha }_i⟩}{\alpha }_i,\text{so that}⟨,⟩:{𝔥}^{*}\times {𝔥}^{*}\to ℂ,$$ is a symmetric bilinear form on ${𝔥}^{*}=\mathrm{span}\{{\alpha }_1,\dots ,{\alpha }_n\}$.

The parameters $t_{ij}$ and $t_{ijrs}$ for the quiver Hecke algebras $R_{\alpha }$ are specified by a choice of polynomials $Q_{ij}(u,v)$such that $$Q_{ji}(u,v)=Q_{ij}(v,u),Q_{ii}(u,v)=0,\text{and}{Q}_{ij}(u,v)=t_{ij},\text{if}i\ne j\text{and}⟨{\alpha }_i,{\alpha }_j^{\vee }⟩=0,$$ and $$Q_{ij}(u,v)=t_{ij}{u}^{-⟨{\alpha }_i,{\alpha }_j^{\vee }⟩}+t_{ij}{v}^{-⟨{\alpha }_j,{\alpha }_i^{\vee }⟩}+\sum _{\begin{array}{l}0\le r\le -⟨{\alpha }_i,{\alpha }_j^{\vee }⟩-1\\ 0\le s\le -⟨{\alpha }_j,{\alpha }_i^{\vee }⟩-1\end{array}}{t}_{ijrs}{u}^r{v}^s,\text{if}i\ne j\text{and}⟨{\alpha }_i,{\alpha }_j^{\vee }⟩\ne 0.$$

Let $ℱ$ be the free algebra generated by $f_1,\dots ,f_n$. The set of words in the letters $f_1,\dots ,f_n$,

$${\Gamma }^{*}=\left\{f_{i_1}\cdots f_{i_d}\right|1\le i_1,\dots ,i_d\le n\},\text{is a basis of}ℱ$$

and the algebra $ℱ$ is graded by

$$Q^{+}=\sum _{i=1}^n{\mathbb{Z}}_{\le 0}{\alpha }_i\text{with}\mathrm{deg}\left(f_i\right)={\alpha }_{\mathrm{i}}.$$

Fix $\alpha \in Q^{+}$ and let $${\Gamma }^{\alpha }=\{u=u_1\cdots u_d|u\in {\Gamma }^{*},\mathrm{deg}\left(u\right)=\alpha \},\text{so that}d=l(u)$$ is the length of the words in ${\Gamma }^{\alpha }$. The symmetric group $S_d$, generated by the simple transpositions $s_1,\dots ,s_{d-1}$, acts on the set ${\Gamma }^{\alpha }$ by permuting the positions of the letters in the words.

The quiver Hecke algebra the $\mathbb{Z}$-graded algebra $R_{\alpha }$ given by generators $$e_u,x_1{e}_u,\dots ,x_d{e}_u,{\tau }_1{e}_u,\dots ,{\tau }_{d-1}{e}_u,\text{for}u\in {\Gamma }^{\alpha },$$ with degrees $$\mathrm{deg}\left(e_u\right)=0,\mathrm{deg}\left(x_i{e}_u\right)=⟨u_i,u_i⟩,\mathrm{deg}\left({\tau }_i{e}_u\right)=-⟨u_i,u_{i+1}⟩$$ (where $u_i$ denotes the $i^{\mathrm{th}}$ letter of the word $u$ and we write $⟨u,v⟩$ for $⟨\mathrm{deg}\left(u\right),\mathrm{deg}\left(v\right)⟩$),   and relations $$e_u{e}_v={\delta }_{uv}{e}_u,x_i{e}_u=e_u{x}_i,{\tau }_i{e}_u=e_{s_{i}u}{\tau }_i,x_i{x}_j=x_j{x}_i,$$ $${\tau }_i^2{e}_u=Q_{u_i{u}_{i+1}}(x_i,x_{i+1})e_u,{\tau }_i{\tau }_j={\tau }_j{\tau }_i\text{if}j\ne i,i\pm 1,$$ $$({\tau }_{i+1}{\tau }_i{\tau }_{i+1}-{\tau }_i{\tau }_{i+1}{\tau }_i)e_u=\{\begin{array}{ll}\frac{1}{x_{i+2}-x_i}\left(Q_{u_{i+2},u_{i+1}}\right(x_{i+2},x_{i+1})-Q_{u_i,u_{i+1}}(x_i,x_{i+1}),& \text{if}{u}_i=u_{i+2},\\ 0,& \text{otherwise,}\end{array}$$ and $${\tau }_i{x}_j{e}_u=\{\begin{array}{ll}{x}_{s_i\left(j\right)}{\tau }_i{e}_u-e_u,& \text{if}{u}_i=u_{i+1}\text{and}j=i,\\ x_{s_i\left(j\right)}{\tau }_i{e}_u+e_u,& \text{if}{u}_i=u_{i+1}\text{and}j=i+1,\\ x_{s_i\left(j\right)}{\tau }_i{e}_u,& \text{otherwise,}\end{array}$$ where $$x_i=x_{i}1=\sum _{u\in {\Gamma }^{\alpha }}{x}_i{e}_u,\text{and}{\tau }_i={\tau }_{i}1=\sum _{u\in {\Gamma }^{\alpha }}{\tau }_i{e}_u,\text{since}1=\sum _{u\in {\Gamma }^{\alpha }}{e}_u\text{in}{R}_{\alpha }.$$

[KL, thm. 2.5], [Ro, thm. 3.7] As in (1.2) fix $\alpha \in Q^{+}$, let ${\Gamma }^{\alpha }$ be the set of words of degree $\alpha$, and let $d$ be the length of words in ${\Gamma }^{\alpha }$. The algebra $R_{\alpha }$ has a basis $$\left\{{\tau }_{\sigma }{x}_1^{n_1}\cdots x_d^{n_{\mathrm{d}}}{e}_u\right|u\in {\Gamma }^{\alpha },\sigma \in S_d,n_1,\dots ,n_d\in {\mathbb{Z}}_{\ge 0}\}$$ where for each $\sigma \in S_d$ we fix a reduced word $$\sigma =s_{i_1}\cdots s_{i_l}\text{and set}{\tau }_{\sigma }={\tau }_{i_1}\cdots {\tau }_{i_l}.$$

The Quiver Hecke algebra for type $A_{\infty }$

$\begin{array}{c}\text{The }\text{KLR Quiver Hecke algebra }{R}_{\alpha }\text{ for type }{A}_{\infty }\text{ is given by generators}\\ y_1,\dots ,y_d,e_u,\text{for}u\in {\mathbb{Z}}^d,{\psi }_1,\dots ,{\psi }_{d-1}\\ \text{with relations}\\ {\displaystyle y_r{y}_s=y_s{y}_r,e_u{e}_v={\delta }_{uv}{e}_u,1=\sum _{u\in {\mathbb{Z}}^d}{e}_u}\\ {\displaystyle e_u{y}_r=y_r{e}_u,e_u{\psi }_r={\psi }_r{e}_u,}\\ {\psi }_r{y}_s=y_s{\psi }_r,\text{if }s\ne r,r+1\\ {\displaystyle {\psi }_r{y}_r{e}_u=\left\{\begin{array}{ccc}\left(y_{r+1}{\psi }_r-1\right)e_u,& & \text{if}\left(u_r,u_{r+1},=,\left(u_r,u_r\right)\right),\\ y_{r+1}{\psi }_r{e}_u,& & \text{otherwise},\end{array}\right.}\\ {\displaystyle {\psi }_r{y}_{r+1}{e}_u=\left\{\begin{array}{ccc}\left(y_r{\psi }_r+1\right)e_u,& & \text{if}\left(u_r,u_{r+1},=,\left(u_r,u_r\right)\right),\\ y_{r+1}{\psi }_r,& & \text{otherwise},\end{array}\right.}\\ {\psi }_r{\psi }_s={\psi }_s{\psi }_r\text{if}s\ne r,r\pm 1,\\ {\displaystyle {\psi }_r^2{e}_u=\left\{\begin{array}{ccc}\pm \left(y_{r+1}-y_r\right)e_u,& & \text{if}\left(u_r,u_{r+1}\right)=\left(u_r,u_r\pm 1\right),\\ 0,& & \text{if}\left(u_r,u_{r+1}\right)=\left(u_r,u_r\right),\\ e_u,& & \text{otherwise},\end{array}\right.}\\ {\displaystyle {\psi }_r{\psi }_{r+1}{\psi }_r{e}_u=\left\{\begin{array}{ccc}\left({\psi }_{r+1}{\psi }_r{\psi }_{r+1}\pm 1\right)e_u,& & \text{if}\left(u_r,u_{r+1},u_{r+2}\right)=\left(u_r,u_r\pm 1,u_r\right),\\ {\psi }_{r+1}{\psi }_r{\psi }_{r+1}{e}_u,& & \text{otherwise},\end{array}\right.}\end{array}$

In this case the Cartan matrix, Dynkin diagram and quiver Hecke parameters are

$C=\left(\begin{array}{ccccccc}\multicolumn{6}{c}{}& 0\\ & \ddots & \multicolumn{5}{c}{}\\ & -1& 2& -1& \multicolumn{3}{c}{}\\ \multicolumn{2}{c}{}& -1& 2& -1& \multicolumn{2}{c}{}\\ \multicolumn{3}{c}{}& -1& 2& -1& \\ \multicolumn{5}{c}{}& \ddots & \\ 0& \multicolumn{6}{c}{}\end{array}\right),$

$Q=\left(\begin{array}{ccccccc}\multicolumn{6}{c}{}& 1\\ & \ddots & \multicolumn{5}{c}{}\\ & u-v& 0& v-u& \multicolumn{3}{c}{}\\ \multicolumn{2}{c}{}& u-v& 0& v-u& \multicolumn{2}{c}{}\\ \multicolumn{3}{c}{}& u-v& 0& v-u& \\ \multicolumn{5}{c}{}& \ddots & \\ 1& \multicolumn{6}{c}{}\end{array}\right)$

These formulas have been checked against [KR1, (2.9) and (2.11)], [KR2, (3.1) and (3.2)] and [MH, §3.1].

The Quiver Hecke algebra for $C=\left(\begin{array}{cc}2& -m\\ -m& 2\end{array}\right)$

Special cases of this are

$\text{type }{A}_2\text{ with }C=\left(\begin{array}{cc}2& -m\\ -m& 2\end{array}\right)\text{and}\text{type }{A}_1^{\left(1\right)}\text{ with }C=\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right)\text{.}$

The Cartan matrix, Dynkin diagram and quiver Hecke parameters are

$C=\left(\begin{array}{cc}2& -m\\ -m& 2\end{array}\right)$ $Q=\left(\begin{array}{cc}0& {\left(v-u\right)}^m\\ {\left(u-v\right)}^m& 0\end{array}\right)$

$\begin{array}{c}\text{The }\text{KLR Quiver Hecke algebra }{R}_{\alpha }\text{ for }C=\left(\begin{array}{cc}2& -m\\ -m& 2\end{array}\right)\text{ is given by generators}\\ y_1,\dots ,y_d,e_u,\text{for}u\in {\left\{0,1\right\}}^d,{\psi }_1,\dots ,{\psi }_{d-1}\\ \text{with relations}\\ {\displaystyle y_r{y}_s=y_s{y}_r,e_u{e}_v={\delta }_{uv}{e}_u,1=\sum _{u\in {\left\{0,1\right\}}^d}{e}_u}\\ {\displaystyle e_u{y}_r=y_r{e}_u,e_u{\psi }_r={\psi }_r{e}_{s_{r}u},}\\ {\psi }_r{y}_s=y_s{\psi }_r,\text{if }s\ne r,r+1,\\ {\displaystyle {\psi }_r^2{e}_u=\left\{\begin{array}{ccc}\left(y_{r+1}-y_r\right)e_u,& & \text{if}\left(u_r,u_{r+1}\right)=\left(0,1\right),\\ \left(y_r-y_{r+1}\right)e_u,& & \text{if}\left(u_r,u_{r+1}\right)=\left(1,0\right),\\ 0& & \text{otherwise}\end{array}\right.}\\ {\displaystyle {\psi }_r{\psi }_{r+1}{\psi }_r{e}_u=\left\{\begin{array}{ccc}\left({\psi }_{r+1}{\psi }_r{\psi }_{r+1}-\sum _{j=0}^{m-1}{\left(y_{r+1}-y_{r+2}\right)}^{m-1-j}{\left(y_{r+1}-y_r\right)}^j\right)e_u,& & \text{if}\left(u_r,u_{r+1},u_{r+2}\right)=\left(0,1,0\right),\\ \left({\psi }_{r+1}{\psi }_r{\psi }_{r+1}+\sum _{j=0}^{m-1}{\left(y_{r+2}-y_{r+1}\right)}^{m-1-j}{\left(y_r-y_{r+1}\right)}^j\right)e_u,& & \text{if}\left(u_r,u_{r+1},u_{r+2}\right)=\left(1,0,1\right),\\ {\psi }_{r+1}{\psi }_r{\psi }_{r+1}{e}_u,& & \text{otherwise},\end{array}\right.}\\ \text{since}\\ {\displaystyle \frac{Q_{01}\left(y_{r+2},y_{r+1}\right)-Q_{01}\left(y_r,y_{r+1}\right)}{y_{r+2}-y_r}=\frac{{\left(y_{r+1}-y_{r+2}\right)}^m-{\left(y_{r+1}-y_r\right)}^m}{-\left(\left(y_{r+1}-y_{r+2}\right)-\left(y_{r+1}-y_r\right)\right)}}\\ =-\sum _{j=0}^{m-1}{\left(y_{r+1}-y_{r+2}\right)}^{m-1-j}{\left(y_{r+1}-y_r\right)}^j\\ \text{and}\\ {\displaystyle \frac{Q_{10}\left(y_{r+2},y_{r+1}\right)-Q_{10}\left(y_r,y_{r+1}\right)}{y_{r+2}-y_r}=\frac{{\left(y_{r+2}-y_{r+1}\right)}^m-{\left(y_r-y_{r+1}\right)}^m}{\left(y_{r+2}-y_{r+1}\right)-\left(y_r-y_{r+1}\right)}}\\ =\sum _{j=0}^{m-1}{\left(y_{r+2}-y_{r+1}\right)}^{m-1-j}{\left(y_r-y_{r+1}\right)}^j\end{array}$

The KLR quiver Hecke algebra for $C=\left(\begin{array}{cc}2& -m\\ -1& 2\end{array}\right)$

This case comes from the quiver

In this case, with

$\begin{array}{c}⟨{\alpha }_0,{\alpha }_0⟩=2·1=,⟨{\alpha }_0,{\alpha }_1⟩=-m,\\ ⟨{\alpha }_1,{\alpha }_0⟩=-m,⟨{\alpha }_1,{\alpha }_1⟩=2·m=2m\end{array}$

$\begin{array}{c}{d}_{01}=\text{ number of orbits of }a\text{ on edges from }0\text{ to }1=1\\ d_{10}=\text{ number of orbits of }a\text{ on edges from }1\text{ to }0=0\\ \text{and}\\ {\displaystyle Q_{01}={\left(-1\right)}^{d_{01}}{\left(u^{\frac{2m}{2}}-v^{\frac{2m}{2m}}\right)}^{\frac{-2\left(-m\right)}{2m}}=-{\left(u^m-v\right)}^1=v{u}^m}\\ \text{so that}\\ C=\left(\begin{array}{cc}2& -m\\ -1& 2\end{array}\right),A=\left(\begin{array}{cc}2& -m\\ -m& 2m\end{array}\right),Q=\left(\begin{array}{cc}0& v-u^m\\ u-v^m& 0\end{array}\right)\end{array}$

These computations are checked against [KR2, (3.1)] and [R, §3.2.4] and [KR2, (3.2)].

Special cases are

$\begin{array}{c}\text{Type }{A}_2\text{ with}\\ C=\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right),A=\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right),Q=\left(\begin{array}{cc}0& v-u\\ u-v& 0\end{array}\right)\\ \text{Type }{B}_2\text{ with}\\ C=\left(\begin{array}{cc}2& -2\\ -1& 2\end{array}\right),A=\left(\begin{array}{cc}2& -2\\ -2& 4\end{array}\right),Q=\left(\begin{array}{cc}0& v-u^2\\ u-v^2& 0\end{array}\right)\\ \text{Type }{G}_2\text{ with}\\ C=\left(\begin{array}{cc}2& -3\\ -1& 2\end{array}\right),A=\left(\begin{array}{cc}2& -3\\ -3& 6\end{array}\right),Q=\left(\begin{array}{cc}0& v-u^3\\ u-v^3& 0\end{array}\right)\\ \text{Type }{C}_2\text{ with}\\ C=\left(\begin{array}{cc}2& -1\\ -2& 2\end{array}\right),A=\left(\begin{array}{cc}4& -2\\ -2& 2\end{array}\right),Q=\left(\begin{array}{cc}0& v^2-u\\ u^2-v& 0\end{array}\right)\end{array}$

$\begin{array}{c}\text{Rouquier [R, §3.2.4] follows the step of quivers with compatible automorphism as presented in [Lu, §12.1.1] so that}\\ \left(\tilde{I},\tilde{E}\right)\text{ is a directed graph with an automorphism }a\text{.}\\ \text{Set }\left(I,E\right)=\left(\tilde{I},\tilde{E}\right)/a\text{and,}\\ \text{for vertices }i\text{ and }j\text{ of }I,\\ ⟨{\alpha }_i,{\alpha }_i⟩=2·\text{(number of elements of }i\text{),}\\ ⟨{\alpha }_i,{\alpha }_j⟩=-\text{(number of edges of }\tilde{E}\text{ between }i\text{ and }j\text{),}\\ d_{ij}=-\text{number of orbits of }a\text{ on edges from }i\text{ to }j,\\ \text{and}\\ {\displaystyle Q_{ij}={\left(-1\right)}^{d^{ij}}{\left(u^{\frac{\mathrm{lcm}\left(⟨{\alpha }_i,{\alpha }_i⟩,⟨{\alpha }_j,{\alpha }_j⟩\right)}{⟨{\alpha }_i,{\alpha }_i⟩}}-v^{\frac{\mathrm{lcm}\left(⟨{\alpha }_i,{\alpha }_i⟩,⟨{\alpha }_j,{\alpha }_j⟩\right)}{⟨{\alpha }_j,{\alpha }_j⟩}}\right)}^{\frac{-2⟨{\alpha }_i,{\alpha }_i⟩}{\mathrm{lcm}\left(⟨{\alpha }_i,{\alpha }_i⟩,⟨{\alpha }_j,{\alpha }_j⟩\right)}}}\end{array}$

Notes and References

These notes are partly based on joint work with A. Kleshchev.

In [Ro, Theorem 3.7], it is explained that $R_{\gamma }$ has basis as given in Theorem 1.1 if and only if $R_{\gamma }$ satisfies PBW if and only if $Q_{ji}(u,v)=Q_{ij}(v,u)$.

References

[KL] M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups I, Representation Theory 13 (2009), 309–347. MR2525917

[Ro] R. Rouquier, 2 Kac-Moody algebras, arXiv:08125023

[MH] A. Mathas and J. Hu, arXiv:0907.2985

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