KLR categorical action contract for line-changing data

Fix a field $\mathbb{k}$, a finite color set $I$, a symmetrizable Cartan matrix $A=(a_{ij}{)}_{i,j\in I}$, and positive integers $d_i$ satisfying $d_i{a}_{ij}=d_j{a}_{ji}$. Here $a_{ii}=2$, $a_{ij}\le 0$ for $i\mathrm{\ne }j$, and $a_{ij}=0$ exactly when $a_{ji}=0$. Fix a weight lattice $X$ (a free abelian group), simple roots ${\alpha }_i\in X$, and integer-valued pairings $⟨\lambda ,{\alpha }_i^{\vee }⟩$ with $⟨{\alpha }_j,{\alpha }_i^{\vee }⟩=a_{ij}$. Choose polynomials $Q_{ij}(u,v)\in \mathbb{k}[u,v]$ with $Q_{ii}=0$, $Q_{ij}(u,v)=Q_{ji}(v,u)$, and $Q_{ij}\mathrm{\ne }0$ for $i\mathrm{\ne }j$; each $Q_{ij}$ is homogeneous for $\mathrm{deg}u=2d_i$, $\mathrm{deg}v=2d_j$, of weighted degree $-2d_i{a}_{ij}$.

For $\nu ={\sum }_{i\in I}{m}_{i}i\in \mathbb{N}[I]$ of height $r={\sum }_i{m}_i$, a quiver Hecke algebra $R_Q(\nu )$ is the graded $\mathbb{k}$-algebra generated by idempotents $e(\mathbf{i})$ for sequences $\mathbf{i}=(i_1,\dots ,i_r)$ with content $\nu$, dots $y_1,\dots ,y_r$, and crossings ${\psi }_1,\dots ,{\psi }_{r-1}$. Put $s_k\mathbf{i}=(i_1,\dots ,i_{k+1},i_k,\dots ,i_r)$. The defining relations are

The displayed quotient is a polynomial because its numerator vanishes when $y_{k+2}=y_k$. The degrees are $\mathrm{deg}e(\mathbf{i})=0$, $\mathrm{deg}(y_{k}e(\mathbf{i}))=2d_{i_k}$, and $\mathrm{deg}({\psi }_{k}e(\mathbf{i}))=-d_{i_k}{a}_{i_k{i}_{k+1}}$.

For this definition, a small additive $\mathbb{k}$-linear graded abelian category has a set of objects and morphisms, finite biproducts, kernels and cokernels with every monomorphism a kernel and every epimorphism a cokernel, $\mathbb{k}$-vector-space Hom groups with bilinear composition, and a decomposition ${\mathrm{Hom}}_{\mathcal{C}}(M,N)={⨁}_{s\in \mathbb{Z}}{\mathrm{Hom}}_{\mathcal{C}}^s(M,N)$ with identities in degree zero and composition adding internal degrees. This internal degree is separate from any cohomological degree. An exact functor is a $\mathbb{k}$-linear functor preserving short exact sequences in the underlying abelian categories. A functor $G:\mathcal{C}\to \mathcal{D}$ between such graded categories is degree-preserving (degree zero) if

for every $M,N$ and $s\in \mathbb{Z}$. An adjunction $F⊣E$ is specified by natural unit and counit maps satisfying the two triangle identities. A natural transformation is a family of morphisms commuting with every source morphism; homogeneous degree $s$ means each component belongs to the internal degree-$s$ Hom space. If a proposed action uses internal shifts in an adjunction, those shifts and the degrees of its unit and counit maps are part of the specified data. For an abelian category $\mathcal{C}$, its exact Grothendieck group $K_0(\mathcal{C})$ is generated by isomorphism classes $[M]$ with relations $[B]=[A]+[C]$ for each short exact sequence $0\to A\to B\to C\to 0$.

A KLR categorical action contract specifies, for each $\lambda \in X$, such a small additive $\mathbb{k}$-linear graded abelian category ${\mathcal{C}}_{\lambda }$, degree-preserving exact functors $F_i:{\mathcal{C}}_{\lambda }\to {\mathcal{C}}_{\lambda -{\alpha }_i}$ with specified degree-preserving exact right adjoints $E_i:{\mathcal{C}}_{\lambda -{\alpha }_i}\to {\mathcal{C}}_{\lambda }$, and homogeneous natural transformations $x_i:F_i\Rightarrow F_i$ of degree $2d_i$ and ${\tau }_{ij}:F_i{F}_j\Rightarrow F_j{F}_i$ of degree $-d_i{a}_{ij}$. If a proposed adjunction convention uses internal shifts, those shifts and the degrees of its unit and counit maps must be stated explicitly. For each $\nu$ and $\lambda$, the direct sum of composite functors $F_{i_1}\cdots F_{i_r}$ over sequences of content $\nu$ must carry an action of $R_Q(\nu )$: $e(\mathbf{i})$ is projection to the indicated sequence summand, $y_k$ acts by the dot on the $k$th factor, and ${\psi }_k$ acts by the crossing on adjacent factors, all with the relations displayed above. Functor-composition order and grading shifts must be fixed in the proposed action and used consistently.

This contract records required data and checks; it does not assert that any such categories, functors, or transformations exist. A claim of an integrable action additionally states and proves local nilpotence: for every object $M$ and color $i$, some powers $E_i^N(M)$ and $F_i^N(M)$ vanish. A claimed highest-weight action additionally specifies a weight $\lambda$, a chosen object in ${\mathcal{C}}_{\lambda }$ killed by every $E_i$, and the precise subcategory generated from it by the $F_i$, shifts, finite sums, and summands. To assert that the induced operators on $K_0(\mathcal{C}):={⨁}_{\lambda \in X}{K}_0({\mathcal{C}}_{\lambda })$ form a representation of $U_q(\mathfrak{g})$, the contract must identify the exact Grothendieck-group convention, establish every required quantum-group relation, and cite a decategorification theorem whose hypotheses match the specified categories and functors. Merely satisfying the positive KLR relations does not establish those extra conclusions.

A physical interpretation as line-changing defects requires a separately specified defect 2-category, whose objects are declared defect sectors, whose 1-morphisms are declared defects changing sectors, and whose 2-morphisms are declared junction maps. A 2-functor here assigns objects, functors, and natural transformations while preserving identities and both compositions; a weak target additionally requires its coherence isomorphisms and equations. No physical defect realization is part of this mathematical contract.

The diagram depicts the generating 2-morphism that exchanges adjacent functor factors. The lower labels are the source order $(i,j)$ and the upper labels the target order $(j,i)$; time/composition is read upward. The crossing is the formal natural transformation ${\tau }_{ij}$, not an invertible topological braid crossing. The overpass convention is only a drawing convention and carries no extra algebraic relation; the displayed KLR relations, not the picture, define the action.

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  \node at (0.5,0.75) {$\tau_{ij}$};
  \draw[->] (1.7,0.1) -- (1.7,1.4) node[midway,right] {$\text{composition}$};
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The Khovanov–Lauda and Rouquier presentations use convention choices for root data, grading shifts, and polynomial parameters. The data above fixes one symmetrizable Cartan convention explicitly. A proposed action in another convention must provide a checked translation to these formulas before it can be used as a supplier.