## Quiver Hecke Algebras ${R}_{\alpha }$

Let $C=\left(⟨{\alpha }_{i},{\alpha }_{j}^{\vee }⟩\right)$ be a symmetrizable Cartan matrix so that $⟨αi, αi∨⟩ =2, ⟨αi, αj∨⟩ ∈ℤ≤0, and ⟨αi, αj∨⟩ =0 ⇔ ⟨αj, αi∨⟩ =0.$ Let $αi∨ = 2 ⟨αi, αi⟩ αi, so that ⟨,⟩: 𝔥*×𝔥* →ℂ,$ is a symmetric bilinear form on ${𝔥}^{*}=\mathrm{span}\left\{{\alpha }_{1},\dots ,{\alpha }_{n}\right\}$.

The parameters ${t}_{ij}$ and ${t}_{ijrs}$ for the quiver Hecke algebras ${R}_{\alpha }$ are specified by a choice of polynomials ${Q}_{ij}\left(u,v\right)$such that $Qji(u,v) = Qij(v,u), Qii(u,v) =0, and Qij(u,v) = tij, if i≠j and ⟨αi, αj∨⟩ =0,$ and $Qij(u,v) = tij u-⟨αi, αj∨⟩ + tij v-⟨αj, αi∨⟩ + ∑ 0≤r≤- ⟨αi, αj∨⟩-1 0≤s≤- ⟨αj, αi∨⟩-1 tijrs urvs, if i≠j and ⟨αi, αj∨⟩≠0.$

Let $ℱ$ be the free algebra generated by ${f}_{1},\dots ,{f}_{n}$. The set of words in the letters ${f}_{1},\dots ,{f}_{n}$,

 $Γ*={ fi1⋯ fid | 1≤ i1,…,id ≤n }, is a basis of ℱ$
and the algebra $ℱ$ is graded by
 $Q+ = ∑i=1n ℤ≤0αi with deg(fi) =αi.$

Fix $\alpha \in {Q}^{+}$ and let $Γα= {u=u1⋯ud | u∈Γ*, deg(u) =α}, so that d=l(u)$ 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 $ℤ$-graded algebra ${R}_{\alpha }$ given by generators $eu, x1eu,…, xdeu, τ1eu,…, τd-1eu, for u∈Γα,$ with degrees $deg(eu)=0, deg(xieu) =⟨ui, ui⟩, deg(τieu) = -⟨ui, ui+1⟩$ (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 $euev = δuveu , xieu =euxi , τieu =esiu τi , xixj =xjxi,$ $τi2eu =Qui ui+1 (xi, xi+1) eu, τiτj =τjτi if j≠ i,i±1,$ $( τi+1 τiτi+1 -τiτi+1 τi) eu = { 1 xi+2-xi ( Q ui+2, ui+1 (xi+2, xi+1) - Q ui, ui+1 (xi, xi+1 ), if ui=ui+2, 0, otherwise,$ and $τixjeu = { xsi(j) τieu -eu, if ui=ui+1 and j=i, xsi(j) τieu+eu , if ui=ui+1 and j=i+1, xsi(j) τieu, otherwise,$ where $xi=xi1 =∑u ∈Γα xieu, and τi=τi1 =∑u ∈Γα τieu, since 1=∑u ∈Γα eu in Rα.$

[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 ${ τσ x1n1 ⋯ xdnd eu | u∈Γα, σ∈Sd, n1,…, nd∈ℤ≥0 }$ where for each $\sigma \in {S}_{d}$ we fix a reduced word $σ=si1 ⋯sil and set τσ= τi1⋯ τil.$

## The Quiver Hecke algebra for type ${A}_{\infty }$

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

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

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

The Cartan matrix, Dynkin diagram and quiver Hecke parameters are

 $C=\left(\begin{array}{cc}2& -m\\ -m& 2\end{array}\right)\phantom{\rule{5em}{0ex}}$ $m$ $\phantom{\rule{4em}{0ex}}Q=\left(\begin{array}{cc}0& {\left(v-u\right)}^{m}\\ {\left(u-v\right)}^{m}& 0\end{array}\right)$

## 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

$\begin{array}{c}⟨{\alpha }_{0},{\alpha }_{0}⟩=2·1=,\phantom{\rule{2em}{0ex}}⟨{\alpha }_{0},{\alpha }_{1}⟩=-m,\\ ⟨{\alpha }_{1},{\alpha }_{0}⟩=-m,\phantom{\rule{3em}{0ex}}⟨{\alpha }_{1},{\alpha }_{1}⟩=2·m=2m\end{array}$

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

Special cases are

## 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 ${Q}_{ji}\left(u,v\right)={Q}_{ij}\left(v,u\right)$.

## 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