Cyclotomic KLR Algebras and Highest-Weight Categorification

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.

Fix a symmetrizable Cartan datum with real simple roots (so $a_{ii}=2$ for every color $i$), a field $k$ placed in degree zero, and weighted-homogeneous KLR polynomials $Q_{jk}$ satisfying (Q1)-(Q4) as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum; write $b_{jk}=d_j{a}_{jk}=b_{kj}$. Fix a color $i$. The minimal-parabolic KLR diagram proof category ${\mathcal{P}}_i$ is the additive graded $k$-linear 2-category of weighted oriented diagrams described by the generators, objects and equations below.

Generators and composition. The objects are the weights $\mu \in P$. The generating $1$-morphisms are the downward generators $G_j:\mu \to \mu -{\alpha }_j$, one for every color $j$, and the single upward generator $H_i:\mu \to \mu +{\alpha }_i$. Horizontal composition reads chronological negative words from the vacuum outward. Take additive sums of objects and grading shifts, and use idempotent completion when speaking of word projectives. The generator $H_i$ denotes normalized projective restriction: in the polynomial flag realization at input weight $h=⟨h_i,\mu ⟩$ it is basic reversed flag restriction shifted by $-d_i(1+h)$; it is distinct from unshifted basic restriction.

The graded 2-morphism spaces. The weights are 0-objects. A 1-morphism is a finite composable word in the $G_j,H_i$, with its specified source and target weights; identities are empty words. For parallel words $u,v$, define ${\mathrm{Hom}}^n(u,v)$ as the degree-$n$ part of the $k$-span of formal oriented string diagrams generated by the dots, downward crossings and four units/counits below, modulo the displayed local relations. Every strand is labeled by its color, and its direction is downward for $G_j$ and upward for $H_i$. The top boundary is the source word and the bottom boundary the target word. Regions carry the unique weights obtained from the outside source weight by the indicated simple-root changes. Diagrams are generic: transverse double crossings, isolated dots and nondegenerate cups/caps, with no triple crossing or dot at a crossing/extremum. Oppositely oriented crossings denote the specified adjunction mates of a downward crossing, not new generators. Use planar isotopies preserving the ordered oriented local events; moving a crossing past an extremum uses the mate relations below. Vertical composition is stacking and horizontal composition is juxtaposition at a common weight; degrees add under either operation. The interchange relation and bilinearity are imposed. A degree-$n$ map from $u$ to $v$ is a degree-zero map $u\{n\}\to v$ with the internal-shift convention of the grading supplier. Additive sums, shifts and idempotent completion have their usual graded 2-category meanings applied to these Hom spaces.

Degrees and the native crossing. Dots have degree $2d_j$ and downward crossings have degree $-b_{jk}$, matching the grading of The KLR grading and the diagrammatic multiplication. The native crossing is $\tau$, with native equal-color divided difference $(f-sf)\mathrm{/}(x_1-x_2)$; for expressing the source diagram equations put $\kappa =-\tau$ for every color pair, retaining dots, idempotents and $Q$. Thus the native cubic braid numerator is globally opposite to the source numerator, while the quadratic $Q$ is unchanged. Impose the full downward KLR relations of the cited Definition.

Adjunctions and the rank-one matrix. At input $i$-weight $h$, the first unit $1\to H_i{G}_i$ and the second counit $H_i{G}_i\to 1$ have degree $d_i(1-h)$; the first counit $G_i{H}_i\to 1$ and the second unit $1\to G_i{H}_i$ have degree $d_i(1+h)$. Impose both adjunction triangles with their adjacent input-weight shifts and cyclicity of the dot. For $h\ge 0$ prescribe the degree-zero rank-one matrix $H_i{G}_i\cong G_i{H}_i\oplus {⨁}_{r=0}^{h-1}1\{d_i(1-h+2r)\};$ for $h<0$ prescribe $G_i{H}_i\cong H_i{G}_i\oplus {⨁}_{r=0}^{-h-1}1\{d_i(1+h+2r)\}\mathrm{.}$ The off-diagonal crossing component is the cup--source-crossing--cap transfer, and the identity columns and rows are the indicated dotted units/counits. Adjoin its actual inverse blocks, requiring both matrix inverse equations; the dotted dual rows and columns are the finite triangular coefficient convolution determined by bubble inversion, and the crossing block satisfies the local cross-and-cap equations. In particular, at $h<0$, if $r_h:G_i{H}_i\to H_i{G}_i\oplus D_{-h}$ is this matrix, then $r_h^{-1}{j}_H$ and $p_H{r}_h$ are the specified natural inclusion and projection with product $1_{H_i{G}_i}$.

Explicit rank-one inverse blocks. Put $P=G_i{H}_i$, $V=H_i{G}_i$ at the common outside weight $\mu$, and write ${\eta }_{\mu }:1_{\mu }\to H_i{G}_i$, ${\epsilon }_{\mu }:G_i{H}_i\to 1_{\mu }$ for the first adjunction and ${\eta }_{\mu }^{\mathrm{\prime }}:1_{\mu }\to G_i{H}_i$, ${\epsilon }_{\mu }^{\mathrm{\prime }}:H_i{G}_i\to 1_{\mu }$ for the second. Let ${\tau }^{GG}$ be the native equal-color downward crossing and ${\tau }^{HH}$ its twofold adjunction mate. Fix the native sideways mates by $T=(H_i{G}_i{\epsilon }_{\mu })(H_i{\tau }^{GG}{H}_i)({\eta }_{\mu }{G}_i{H}_i),S=(G_i{H}_i{\epsilon }_{\mu }^{\mathrm{\prime }})(G_i{\tau }^{HH}{G}_i)({\eta }_{\mu }^{\mathrm{\prime }}{H}_i{G}_i)\mathrm{.}$ Their middle words are respectively $H_i{G}_i{G}_i{H}_i$ and $G_i{H}_i{H}_i{G}_i$; the displayed final counits consume their rightmost $G_i{H}_i$ and $H_i{G}_i$. Thus $T:P\to V$ and $S:V\to P$ are typed, with degree $d_i(1-h)-2d_i+d_i(1+h)=0$. The source-crossing component in the preceding rank-one matrix is $-T$. For $h\ge 0$, its identity columns are $C_k=x^k{\eta }_{\mu }$, and its identity rows are $R_k={\sum }_{a+j=h-1-k}{A}_{\mu ,j}{\epsilon }_{\mu }^{\mathrm{\prime }}{x}^a,0\le k<h,$ where $a,j\ge 0$. The matrix is $\rho =(-T,(C_k{)}_k):P\oplus D_h\to V$ and its specified inverse is $\sigma =(S,(R_k{)}_k):V\to P\oplus D_h$. Impose both equations $\sigma \rho =1$ and $\rho \sigma =1$; in components these say $-ST=1_P$, $R_{k}T=0$, $S{C}_k=0$, $R_k{C}_{\mathrm{\ell }}={\delta }_{k\mathrm{\ell }}$ and $-TS+{\sum }_k{C}_k{R}_k=1_V$. For $h<0$, take $C_k^{-}=x^k{\eta }_{\mu }^{\mathrm{\prime }}$ and $R_k^{-}={\sum }_{a+j=-h-1-k}{\mathsf{H}}_{\mu ,j}{\epsilon }_{\mu }{x}^a,0\le k<-h,$ again with $a,j\ge 0$. Now $\rho =(-T,(R_k^{-}{)}_k):P\to V\oplus D_{-h}$ and $\sigma =(S,(C_k^{-}{)}_k):V\oplus D_{-h}\to P$; impose $\rho \sigma =1$, $\sigma \rho =1$. Equivalently $-TS=1_V$, $T{C}_k^{-}=0$, $R_k^{-}S=0$, $R_k^{-}{C}_{\mathrm{\ell }}^{-}={\delta }_{k\mathrm{\ell }}$ and $-ST+{\sum }_k{C}_k^{-}{R}_k^{-}=1_P$. The shift summands $D_h,D_{-h}$ are exactly those displayed above. The row degrees are respectively $d_i(h-1-2k)$ and $-d_i(1+h+2k)$, opposite the column degrees, so every matrix component is degree zero after the indicated shifts. Empty identity sums at $h=0$ give the same inverse equations. These are imposed relations in the presentation; a claim that the concrete flag bimodules satisfy them still requires their local realization proof.

Bubble series and slides. The normalized far-left bubble series $A_{\mu }(t),{\mathsf{H}}_{\mu }(t)$ have constant coefficient $1$ and satisfy $A_{\mu }(t){\mathsf{H}}_{\mu }(t)=1$. Genuine negative-degree bubbles vanish. The $A$ coefficient with index $m$ is the clockwise bubble with $-h-1+m$ dots when this count is nonnegative, extended by formal inverse coefficients otherwise; its degree is $2d_{i}m$. The other family has dot count $h-1+\mathrm{\ell }$ and degree $2d_i\mathrm{\ell }$. Impose the finite-coefficient slides verified on the monic flag model: adjoining an $i$ root $z$ divides $A$ by $(1-zt{)}^2$ and multiplies $\mathsf{H}$ by that factor; adjoining a spectator root $y$ of color $j\mathrm{\ne }i$ multiplies $A$ by $q_{ij}^{\mathrm{r}\mathrm{e}\mathrm{v}}(y,t)=t^{-a_{ij}}{Q}_{ij}(t^{-1},y)\mathrm{/}{t}_{ij}$, where $t_{ij}$ is the nonzero extremal coefficient, and divides $\mathsf{H}$ by it. Every reciprocal coefficient is defined by the finite recursion with constant coefficient $1$.

Four oriented same-color equations. The following notation specifies actual local diagrams, all with color $i$. The two-crossing equations have common outside weight $\mu$; the single curls have the adjacent region weights specified individually below. Read tangent directions in the page. Let $C_L,C_R$ be the cap arch (two bottom endpoints and no top endpoints), whose tangent at its peak points left or right, respectively. Their bottom arrows, from left to right, are $(\downarrow ,\uparrow )$ and $(\uparrow ,\downarrow )$. Let $U_L,U_R$ be the cup valley (two top endpoints and no bottom endpoints), whose tangent at its trough points left or right; their top arrows are $(\uparrow ,\downarrow )$ and $(\downarrow ,\uparrow )$. An exponent on $C$ or $U$ counts dots on its open arc. Let $B_{\mathrm{c}\mathrm{c}\mathrm{w}}^b$ and $B_{\mathrm{c}\mathrm{w}}^b$ be the counterclockwise and clockwise bubbles with literal dot count $b$, located to the right of the cap/cup in the outside region of weight $\mu$. Their negative dot counts are exactly the fake bubbles defined by the coefficient conventions above, not zero by convention. Write ${\eta }_{\mu }:1_{\mu }\to H_i{G}_i$ and ${\epsilon }_{\mu }^{\mathrm{\prime }}:H_i{G}_i\to 1_{\mu }$ for the first unit and second counit, and ${\eta }_{\mu }^{\mathrm{\prime }}:1_{\mu }\to G_i{H}_i$, ${\epsilon }_{\mu }:G_i{H}_i\to 1_{\mu }$ for the second unit and first counit. Let ${\tau }^{HH}$ be the twofold adjunction mate of the native equal-color downward crossing. Define $L_{\mu }=(H_i{\epsilon }_{\mu }^{\mathrm{\prime }})({\tau }^{HH}{G}_i)(H_i{\eta }_{\mu }),R_{\mu }=({\epsilon }_{\mu }{H}_i)(G_i{\tau }^{HH})({\eta }_{\mu }^{\mathrm{\prime }}{H}_i)\mathrm{.}$ The first formula starts and ends at $H_i$ with input $\mu$; the second starts and ends at $H_i$ with input $\mu -{\alpha }_i$. Thus both units and counits in $L_{\mu }$ have degree $d_i(1-h)$, while those in $R_{\mu }$ have degree $d_i(1+h)$. The crossing degree $-2d_i$ makes their degrees $-2d_{i}h$ and $2d_{i}h$ respectively. Products in these formulas apply the rightmost map first, with adjacent weights determined by composition. Geometrically $L_{\mu }$ is an upward-strand curl with its loop bulging LEFT and its left-adjacent region of weight $\mu$. This is an endomorphism of $H_i$ at input $\mu$, and its bubble on the right-hand side is in the left region $\mu$. Let $R_{\mu }$ be the upward-strand curl with its loop bulging RIGHT and its right-adjacent region of weight $\mu$. Its left region is $\mu -{\alpha }_i$, so it is an endomorphism of $H_i$ at input $\mu -{\alpha }_i$, and its bubble is in the right region $\mu$. Both have one native crossing and the two extremal turns of the loop. Let $D_{\downarrow \uparrow }$ and $D_{\uparrow \downarrow }$ be the two opposite crossing mates composed in the two orders on their indicated straight boundaries, with common outside region $\mu$. Impose $L_{\mu }={\sum }_{a+b=-1}{x}^a{B}_{\mathrm{c}\mathrm{c}\mathrm{w},\mu }^b,R_{\mu }=-{\sum }_{a+b=-1}{x}^a{B}_{\mathrm{c}\mathrm{w},\mu }^b,$ $D_{\downarrow \uparrow }=-1_{\downarrow \uparrow }+{\sum }_{a+b+c=-2}{C}_L^a{B}_{\mathrm{c}\mathrm{w},\mu }^b{U}_R^c,$ $D_{\uparrow \downarrow }=-1_{\uparrow \downarrow }+{\sum }_{a+b+c=-2}{C}_R^a{B}_{\mathrm{c}\mathrm{c}\mathrm{w},\mu }^b{U}_L^c\mathrm{.}$ Composition in the last two displays is top-to-bottom: $U$ first, then the outside bubble, then $C$. The single-curl equations are upward-strand endomorphism equations; their cup/cap forms are their specified adjunction mates, not cap drawings with unspecified endpoint arrows. In every sum, $a,c$ on open strands are nonnegative. The clockwise bubble with literal $b$ dots is the $A_{\mu }$ coefficient of index $b+h+1$, and the counterclockwise bubble is the ${\mathsf{H}}_{\mu }$ coefficient of index $b-h+1$, where $h=⟨h_i,\mu ⟩$. Coefficients of negative index are zero and the index-zero coefficient is one. The nonnegative coefficient and open-dot indices make each sum finite. The opposite-crossing equations have degree zero: their correction degrees are $2d_i(1-h)+2d_i(a+b+c+h+1)=0$ and $2d_i(1+h)+2d_i(a+b+c-h+1)=0$. The two curls have degrees $-2d_{i}h$ and $2d_{i}h$ at their respective, different input weights.

The native single-crossing signs follow from $\kappa =-\tau$; two-crossing signs are unchanged. The literal triple-dot sum is $-2$, not the compact $-1$ printed in Webster's Figure 2: in the full positive-weight equation, coefficient indices satisfy $f_1+f_2+f_3=h-1$ and the bubble has literal count $b=-h-1+f_2$, giving $a+b+c=-2$; the negative-weight equation uses $g_1+g_2+g_3=-h-1$ and $b=h-1+g_2$, giving the same value. Reflection exchanges the two bubble orientations and preserves region weights. The single-curl coefficient sums instead have total $h$ or $-h$, yielding literal $a+b=-1$. At $h=0$ the triple sums are empty and the single curls are respectively $+1$ and $-1$ on their specified shifted input weights. At the $h=1$, $p(u)=u$ vacuum, $H_i{G}_i=k$, $G_i{H}_i=0$, $A_{\mu }=1$ and the only surviving clockwise bubble has $b=-2$; thus the first cancellation is $0=-1_k+1_k$, with no degree or vacuum contradiction.

Distinct-color crossing mates. For $j\mathrm{\ne }i$, at input weight $\mu$ put $\xi =\mu +{\alpha }_i-{\alpha }_j$, so $X=G_j{H}_i$ and $Y=H_i{G}_j$ both go from $\mu$ to $\xi$. Let ${\tau }^{G_i{G}_j}:G_i{G}_j\to G_j{G}_i$ and ${\tau }^{G_j{G}_i}:G_j{G}_i\to G_i{G}_j$ denote the two native downward crossings. Specify the handed pair by the actual typed mate composites $a=(H_i{G}_j{\epsilon }_{\mu })(H_i{\tau }^{G_i{G}_j}{H}_i)({\eta }_{\xi }{G}_j{H}_i):X\to Y,$ $b=({\epsilon }_{\xi }^{\mathrm{\prime }}{G}_j{H}_i)(H_i{\tau }^{G_j{G}_i}{H}_i)(H_i{G}_j{\eta }_{\mu }^{\mathrm{\prime }}):Y\to X\mathrm{.}$ The intermediate words are respectively $H_i{G}_i{G}_j{H}_i$ and $H_i{G}_j{G}_i{H}_i$; in $a$ the final first counit consumes the rightmost $G_i{H}_i$, while in $b$ the final second counit consumes the leftmost $H_i{G}_i$. Impose $ba=t_{ij}{1}_X$ and $ab=t_{ij}{1}_Y$ for this same handed pair. Thus $a$ and $t_{ij}^{-1}b$ are mutually inverse and give $H_i{G}_j\cong G_j{H}_i$, commuting with the $j$ dot. If $h=⟨h_i,\mu ⟩$, both composites have degree $d_i(1+h)+d_i(-1-h+a_{ij})-b_{ij}=0$. Replacing both native crossings by the source crossings $\kappa =-\tau$ negates $a$ and $b$ separately and leaves both cancellation scalars unchanged.

There is no $H_j$ in this minimal category. The analogous calculation in the separately distinguished category ${\mathcal{P}}_j$ has scalar $t_{ji}$ in both orders for its own handed pair. That does not assign $t_{ji}$ to the other composition of the fixed pair $a,b$ above. Comparing the two partial actions, or asserting a full double-right-mate ratio, requires the separate coherent assembly proof; it is not an additional relation imposed here.

Status of the relations. These equations are relations of this proof-tool category. Their consistency and homogeneous realization are established by the polynomial flag model (The minimal-parabolic polynomial flag model realizes the graded diagram relations ↗), rather than assumed for the actual cyclotomic induction/restriction action; no claim about the latter is made here. This definition uses no choice: it is a presentation by generators and homogeneous equations, and every sl(2) infinite family appearing in the matrix and bubble clauses is a finite coefficient recursion from a constant coefficient $1$.

Let $V={⨁}_{\mu \in P}{V}_{\mu }$ be a label-weight-graded $\mathbb{Q}(q)$-vector space with toral operators $K_h$ ($h\in P^{\vee }$) acting on $V_{\mu }$ by $q^{⟨\mu ,h⟩}$, and operators $E_i,F_i$ of weights ${\alpha }_i,-{\alpha }_i$ mapping $V_{\mu }$ to $V_{\mu +{\alpha }_i}$ and $V_{\mu -{\alpha }_i}$. Assume the toral conjugation and mixed relations, and local nilpotence of each $E_i$ and $F_i$; vectors have finite label support. Then these operators uniquely define an integrable type-one $U_q(\mathfrak{g})$-module structure on $V$ for EVERY real symmetrizable Cartan datum. Every vector has a finite rank-one $U_i$ orbit and the admissible Kashiwara $i$-string decomposition applies. Finite weight spaces do not imply finite-dimensional full-$\mathfrak{g}$ orbits; no such conclusion is asserted.