Cyclotomic KLR induction and restriction functors

Use the standard graded cyclotomic quotient of The cyclotomic KLR quotient at a dominant integral weight, and keep the conventions fixed there: the cyclotomic relation is imposed on the FIRST strand and cyclotomic induction appends strands on the RIGHT. Fix $\beta \in Q_{+}$ and a color $i$; put $A:=R^{\mathrm{\Lambda }}(\beta ),B:=R^{\mathrm{\Lambda }}(\beta +{\alpha }_i),e:=e(\beta ,i),$ where $e(\beta ,i)={\sum }_{\nu \in I^{\beta }}e(\nu i)\in R(\beta +{\alpha }_i)$ is the idempotent of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum supported on the sequences ending in $i$.

The right-append corner homomorphism. Right append defines a $k$-algebra homomorphism ${\iota }_i:A⟶eBe,1_A⟼e,e(\nu )⟼e(\nu i),x_a⟼x_{a}e,{\tau }_a⟼{\tau }_{a}e,$ sending each old dot or crossing to the same-named generator multiplied by the endpoint idempotent $e$, at the same position in the first $\mathrm{\mid }\beta \mathrm{\mid }$ strands of $R^{\mathrm{\Lambda }}(\beta +{\alpha }_i)$ and each old idempotent to the longer idempotent obtained by appending $i$. It is a graded homomorphism of degree zero: each generic relation (1)-(7) of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum is preserved under right append, and the first-strand cyclotomic generator of $A$ maps to the first-strand cyclotomic generator of $B$, so the cyclotomic relation is preserved and ${\iota }_i$ is well defined on the quotient. The unit is sent to the corner unit $e$, not to a global unit of $B$; it is not asserted that ${\iota }_i$ is injective, and $e$ is allowed to be zero.

The functors. On graded left modules define $F_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}(M):=Be{\otimes }_{A}M,E_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}(N):=eN,$ the tensor over the graded balanced tensor product of Graded balanced tensor product and homogeneous Hom and the idempotent restriction of the graded $B$-module $N$ (Associative graded algebras, bimodules, and internal shifts). Both are additive $k$-linear functors between the graded module categories, they preserve degree-zero maps, and they extend to direct sums over the root degrees $\beta \in Q_{+}$.

Adjunction, weights and shifts. Evaluation at $e$ identifies ${\mathrm{HOM}}_B(Be,N)$ with $eN$ by the degree-zero tensor--Hom adjunction, giving $F_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}⊣E_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}$ with its actual grading. The functor $F_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}$ increases ROOT degree by ${\alpha }_i$, taking modules at root $\beta$ to modules at root $\beta +{\alpha }_i$, and decreases the module weight $\mathrm{\Lambda }-\beta$ by ${\alpha }_i$; restriction does the reverse. Both composites are endofunctors at a fixed root degree. Grading shifts satisfy $F_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}(M\{r\})\cong F_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}(M)\{r\}$ and $E_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}(N\{r\})\cong E_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}(N)\{r\}$, using the internal shifts of Associative graded algebras, bimodules, and internal shifts.

Ungraded monic deformations. For a general monic specialization $a_j(u)$ the quotient $R_a^{\mathrm{\Lambda }}(\beta )$ of The cyclotomic KLR quotient at a dominant integral weight carries no grading, and the same formulas define the ungraded functors $F_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}$ and $E_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}$; no graded statement is asserted on that branch.

Status of further structure. Exactness, preservation of projectives and a shifted second adjunction are theorems below, not part of this definition; in particular no unshifted right adjunction $E_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}⊣F_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}$ is asserted here, and the explicit shift of the second adjunction is supplied by the coherent-biadjunction lemma of this page. This definition uses no choice.