The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum

Let $(I,A,D,P,P^{\vee },q)$ be a symmetrizable Cartan datum with no imaginary simple roots, so $a_{ii}=2$, and put $b_{ij}:=d_i{a}_{ij}=({\alpha }_i\mid {\alpha }_j)$, a symmetric matrix with $b_{ii}=2d_i>0$ (Symmetrizable Cartan data for quantum groups, Kac Moody root lattice height and positive cone). Fix a field $k$. A KLR datum on it is a family $Q=(Q_{ij}{)}_{i\mathrm{\ne }j}$ of polynomials $Q_{ij}\in k[u,v]$ such that

(Q1) $Q_{ij}\mathrm{\ne }0$ for $i\mathrm{\ne }j$ and $Q_{ii}:=0$;

(Q2) $Q_{ij}(u,v)=Q_{ji}(v,u)$ for all $i\mathrm{\ne }j$;

(Q3) $Q_{ij}$ is homogeneous of total degree $-2b_{ij}$ for the degree assignments $\mathrm{deg}u:=2d_i$, $\mathrm{deg}v:=2d_j$;

(Q4) the two extremal coefficients of $Q_{ij}$, namely that of $u^{-a_{ij}}$ and that of $v^{-a_{ji}}$, are nonzero.

For a chosen finite quiver with a compatible automorphism having no arrow within a vertex orbit, write $i\cdot j=b_{ij}$ and let $d_{ij}$ be the number of arrow orbits directed from vertex orbit $i$ to vertex orbit $j$; the specialization applies when its induced Cartan datum is the fixed one. Then one takes $Q_{ij}(u,v)=(-1{)}^{d_{ij}}(u^{l\mathrm{/}(i\cdot i)}-v^{l\mathrm{/}(j\cdot j)}{)}^{-2(i\cdot j)\mathrm{/}l}$ with $l=\mathrm{l}\mathrm{c}\mathrm{m}(i\cdot i,j\cdot j)$, and $Q_{ij}=1$ when $a_{ij}=0$; the cyclic affine $A_1^{(1)}$ double edge $a_{ij}=a_{ji}=-2$ is the instance $Q_{ij}(u,v)=-(u-v{)}^2$ up to the fixed normalization of the top coefficients. The simply-laced standard choice is $Q_{ij}(u,v)=u+v$ for $a_{ij}=-1$.

The algebra. For $n\ge 0$ let $I^n$ be the set of sequences, let $e(\nu )$ for $\nu \in I^n$ be idempotents, and let $x_a$ ($1\le a\le n$) and ${\tau }_a$ ($1\le a\le n-1$) be generators. The KLR algebra $R_n(Q)$ is the unital associative $k$-algebra generated by the $e(\nu ),x_a,{\tau }_a$ subject to the relations, for all $\nu \in I^n$ (with $s_a$ the adjacent transposition acting on positions $a,a+1$):

(1) $e(\nu )e({\nu }^{\mathrm{\prime }})={\delta }_{\nu {\nu }^{\mathrm{\prime }}}e(\nu )$ and ${\sum }_{\nu \in I^n}e(\nu )=1$;

(2) $x_{a}e(\nu )=e(\nu )x_a$ and $x_a{x}_b=x_b{x}_a$;

(3) ${\tau }_{a}e(\nu )=e(s_a\nu ){\tau }_a$;

(4) ${\tau }_a^{2}e(\nu )=Q_{{\nu }_a{\nu }_{a+1}}(x_a,x_{a+1})e(\nu )$;

(5) ${\tau }_a{\tau }_b={\tau }_b{\tau }_a$ whenever $\mathrm{\mid }a-b\mathrm{\mid }>1$;

(6) $({\tau }_a{x}_b-x_{s_a(b)}{\tau }_a)e(\nu )=0$ if $b\mathrm{\notin }\{a,a+1\}$ or ${\nu }_a\mathrm{\ne }{\nu }_{a+1}$, while if ${\nu }_a={\nu }_{a+1}$ one has $({\tau }_a{x}_a-x_{a+1}{\tau }_a)e(\nu )=e(\nu )$ and $({\tau }_a{x}_{a+1}-x_a{\tau }_a)e(\nu )=-e(\nu )$ (the sign normalization of the divided difference ${\tau }_a(f)=(f-s_{a}f)\mathrm{/}(x_a-x_{a+1})$; Rouquier's printed relation has the opposite global sign, the two conventions being exchanged by ${\tau }_a\mapsto -{\tau }_a$);

(7) $({\tau }_{a+1}{\tau }_a{\tau }_{a+1}-{\tau }_a{\tau }_{a+1}{\tau }_a)e(\nu )=0$ if ${\nu }_a\mathrm{\ne }{\nu }_{a+2}$, and $({\tau }_{a+1}{\tau }_a{\tau }_{a+1}-{\tau }_a{\tau }_{a+1}{\tau }_a)e(\nu )={\displaystyle \frac{Q_{{\nu }_a{\nu }_{a+1}}(x_a,x_{a+1})-Q_{{\nu }_a{\nu }_{a+1}}(x_{a+2},x_{a+1})}{x_{a+2}-x_a}}e(\nu )$ if ${\nu }_a={\nu }_{a+2}$, the numerator carrying the sign of the $\tau$-normalization of (6) (Rouquier's printed numerator is its negative); the quotient is a polynomial in $x_a,x_{a+1},x_{a+2}$ because $Q_{ij}$ is a polynomial in its first variable and the numerator vanishes at $x_{a+2}=x_a$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, Polynomial rings in finitely many commuting indeterminates by iteration).

For $\beta \in Q_{+}$ put $I^{\beta }:=\{\nu \in I^{\mathrm{\mid }\beta \mathrm{\mid }}:{\sum }_a{\alpha }_{{\nu }_a}=\beta \}$, $e(\beta ):={\sum }_{\nu \in I^{\beta }}e(\nu )$ and $R(\beta ):=e(\beta )R_{\mathrm{\mid }\beta \mathrm{\mid }}(Q)e(\beta )={⨁}_{\nu ,\eta \in I^{\beta }}e(\nu )R_{\mathrm{\mid }\beta \mathrm{\mid }}(Q)e(\eta )$, and put $R:={⨁}_{\beta \in Q_{+}}R(\beta )$. The presentation is the sequence form of Rouquier's $H_n(Q)$ via $1_{\nu }=e(\nu )$, $x_{a,\nu }=x_{a}e(\nu )$, ${\tau }_{a,\nu }={\tau }_{a}e(\nu )$; no sign or orientation beyond $Q$ is part of the datum, and $R(0)=k$.

Well-definedness. $R_n(Q)$ is defined as the quotient of the free unital associative $k$-algebra on the symbols $e(\nu )$ ($\nu \in I^n$), $x_a$, ${\tau }_a$ by the two-sided ideal generated by the displayed relations, so no existence issue arises. Two points deserve record. First, relation (7) is stated as an equality in $k[x_a,x_{a+1},x_{a+2}]e(\nu )$: writing $Q(u,v)={\sum }_a{u}^a{q}_a(v)$, the difference $Q(u,v)-Q(w,v)={\sum }_a(u^a-w^a)q_a(v)$ is divisible by $u-w$ in the polynomial ring, and its quotient evaluated at $(u,w)=(x_a,x_{a+2})$, $v=x_{a+1}$ gives the displayed correction; the special case $a_{ij}=0$ with $Q_{ij}$ a nonzero constant is included, where the correction vanishes and the ordinary braid relation holds on ${\nu }_a={\nu }_{a+2}$. Second, the mixed relations are word-component identities: the final $e(\nu )$ is essential, since a global dot commutator has different components on equal and unequal colors. On an equal-color component they are the identities ${\mathrm{\partial }}_a{x}_a-x_{a+1}{\mathrm{\partial }}_a=1$ and ${\mathrm{\partial }}_a{x}_{a+1}-x_a{\mathrm{\partial }}_a=-1$ for ${\mathrm{\partial }}_a=(1-s_a)\mathrm{/}(x_a-x_{a+1})$. The two global sign conventions for ${\tau }_a$ are exchanged by ${\tau }_a\mapsto -{\tau }_a$, which preserves (4) and flips the signs in (6) and (7); this page fixes the convention of (6) once and for all and every later item uses it.

Choice. The definitional clauses are choice-free: they name a presented algebra and its weight decomposition over a fixed datum, with no selection of representatives or of auxiliary sets.

Source comparison: Rouquier §3.2.1-3.2.4 (presentation, Cartan specialization, quiver-with-automorphism formula); Khovanov–Lauda I §2.1-2.2 and Khovanov–Lauda II §1-§2 (diagrammatic presentation with the dot counts $d_{ij}$); Kleshchev §2.11 Definition 2.14 (the affine presentation). The printed Rouquier sign in relation (6) is the opposite of the one fixed here; the passage is ${\tau }_a\mapsto -{\tau }_a$ as recorded above.