The cyclotomic KLR quotient at a dominant integral weight

Fix a symmetrizable Cartan datum with finite color set $I$ and no imaginary simple roots, a KLR datum $Q=(Q_{ij}{)}_{i\mathrm{\ne }j}$ satisfying (Q1)-(Q4), and a field $k$ placed in degree zero (The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum, Symmetrizable Cartan data for quantum groups); let $R={⨁}_{\beta \in Q_{+}}R(\beta )$ be the resulting graded KLR algebra, with its idempotents $e(\nu )$, dots $x_a$, crossings ${\tau }_a$ and degrees $\mathrm{deg}{x}_a=2d_{{\nu }_a}$, $\mathrm{deg}{\tau }_a=-b_{{\nu }_a{\nu }_{a+1}}$ (The KLR grading and the diagrammatic multiplication). Let $\mathrm{\Lambda }$ be a dominant integral weight and put ${\lambda }_i:=⟨h_i,\mathrm{\Lambda }⟩\ge 0$ (Kac moody integral and dominant integral weights).

Cyclotomic data. For every color $i$ choose a monic polynomial $a_i(u)\in k[u]$ of degree exactly ${\lambda }_i$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution); the standard choice made below is $a_i(u)=u^{{\lambda }_i}$. For $n\ge 1$ write $a(x_s)={\sum }_{\nu \in I^n}{a}_{{\nu }_s}(x_s)e(\nu )(1\le s\le n),$ the element of $R_n(Q)$ that acts on the $\nu$-component by the monic polynomial $a_{{\nu }_s}$ in the dot at position $s$.

The cyclotomic quotient. For $\beta \in Q_{+}$ define $J_a(\beta )$ to be the two-sided ideal of $R(\beta )$ generated by the first-strand elements $a_{{\nu }_1}(x_1)e(\nu )$, $\nu \in I^{\beta }$; equivalently $J_a(\beta )={\sum }_{\nu \in I^{\beta }}R(\beta )a_{{\nu }_1}(x_1)e(\nu )R(\beta ),$ and put $R_a^{\mathrm{\Lambda }}(\beta ):=R(\beta )\mathrm{/}{J}_a(\beta ),R_a^{\mathrm{\Lambda }}(0):=k\mathrm{.}$ The cyclotomic KLR algebra is the direct sum $R_a^{\mathrm{\Lambda }}:={⨁}_{\beta \in Q_{+}}{R}_a^{\mathrm{\Lambda }}(\beta )$; it carries the local units $e(\beta )={\sum }_{\nu \in I^{\beta }}e(\nu )$, which act as the identity on $R_a^{\mathrm{\Lambda }}(\beta )$ and annihilate the other summands, and it is a $k$-algebra with these orthogonal idempotents.

Standard choice and grading. The STANDARD choice $a_i(u)=u^{{\lambda }_i}$ is used throughout the graded categorification, duality-normalization and based statements below; for it we write $R^{\mathrm{\Lambda }}:=R_a^{\mathrm{\Lambda }}$. Under the standard choice each individual first-strand generator $x_1^{{\lambda }_{{\nu }_1}}e(\nu )$ is homogeneous of degree $2d_{{\nu }_1}{\lambda }_{{\nu }_1}$ in the grading of The KLR grading and the diagrammatic multiplication, so the ideal $J_a(\beta )$ and the quotient $R^{\mathrm{\Lambda }}(\beta )$ are graded, and $R^{\mathrm{\Lambda }}$ is a $\mathbb{Z}$-graded algebra with local units. The sum $a(x_1)$ itself need not have one common degree across different colors, and no single degree is assigned to it.

Boundary of the definition. Arbitrary monic polynomials with lower coefficients in degree-zero $k$ give an UNGRADED deformation $R_a^{\mathrm{\Lambda }}$; nilpotent dots, homogeneous trace degree and the highest-weight graded $K_0$ identification are not asserted for that branch, and the ungraded branch is used only where a statement explicitly says so. Finite dimension, finite root strings, ordinary projectivity and symmetry of the standard quotient have the separately stated monic conclusions below. The cyclotomic relation is imposed on the FIRST strand, and cyclotomic induction appends strands on the RIGHT (Cyclotomic KLR induction and restriction functors); this convention fixes the side on which the quotient map acts throughout the page.

This definition uses no choice.