Graded <math> <msub> <mi>R</mi> <mi>α</mi> </msub> </math>-modules

## Graded ${R}_{\alpha }$-modules

A $ℤ$-graded vector space is a vector space with a decomposition $V= ⊕ i∈ℤ V i andgdim V = ∑ i∈ℤ q i dim V$ is the graded dimension of $V$. If $M$ is a $ℤ$-graded ${R}_{\alpha }$-module then, as graded vector spaces, $M= ⊕ u∈ Γ α eu Mandgdim M = ∑ u∈ Γ α gdim e u M f u$ is the graded character of $M$.

Let $\alpha ,\beta \in {Q}^{+}$ and let $k$ and $l$ be the lengths of the words in ${\Gamma }^{\alpha }$ and ${\Gamma }^{\beta }$, respectively. Then $Γ α+β = ⨆ σ∈ S k+l / S k × S l and R α ⊗ R β ↦ R α+β e u ⊗ e v ↦ e uv x i e u ⊗ e v ↦ x i e uv τ i e u ⊗ e v ↦ τ i e uv e u ⊗ x j e v ↦ x j+k e uv e u ⊗ τ j e v ↦ τ j+k e uv$ defines an injection (of nonunital algebras).

[KL1, Prop 2.16] As a right ${R}_{\alpha }\otimes {R}_{\beta }$-module, ${R}_{\alpha +\beta }$ has basis $τ σ 1 αβ | σ∈ S k+l / S k × S l ,where 1 αβ = ∑ u∈ Γ α ,v∈ Γ β e uv ,$ and, for each minimal length representative $\sigma$ of a coset in ${S}_{k+l}/\left({S}_{k}×{S}_{l}\right),$ we fix a reduced word $σ= s i 1 … s i l and set τ σ = τ i 1 … τ i l .$

Let ${R}_{\alpha }$-mod be the category of $ℤ$-graded ${R}_{\alpha }$-modules. For $M\in {R}_{\alpha }$-mod and $N\in {R}_{\beta }$ define $M∘N= Ind R α ⊗ R β R α+β M⊗N .$

Let $K\left({R}_{\alpha }\text{-mod}\right)$ be the Grothendieck group of $ℤ$-graded $R$-modules and define a $product on ⨁ α∈ Q + K R α -mod by M . N = M∘N .$

The following theorem says that the categories ${R}_{\alpha }$-mod form a categorification of ${U}_{q}{𝔫}^{-}.$ It is the main theorem of this theory.

[KL, Prop 3.4] The graded character map $gch: ⨁ α∈ Q + K R α -mod → U q 𝔫 - M ↦ gch M$ is an algebra ismorphism.

1. If $M\in {R}_{\alpha }$-mod then $\mathrm{gch}\left(M\right)\in {U}_{q}{𝔫}^{-}.$
2. If $M\in {R}_{\alpha }$-mod and $N\in {R}_{\beta }$-mod then $gch Ind R α ⊗ R β R α+β M⊗N =gch M ⊔⊔gch N .$

Proof

1. (sketch only) $gch M = ∑ u∈ Γ α gdim e u M f u = ∑ u∈ Γ α gdim Hom M R α e u f u ,$ and so by Roquier's complex [Ro, Lemma 3.13], $\mathrm{gch}\left(M\right)$ satisfies Serre relations. Thus, by Proposition ?, $\mathrm{gch}\left(M\right)\in {U}_{q}{𝔫}^{-}.$
2. (second proof) (sketch) By understanding the intertwiners thoroughly, analyse the ${R}_{\alpha }$-modules when $\alpha =-⟨{\alpha }_{i},{\alpha }_{j}^{\vee }⟩{\alpha }_{1}+{\alpha }_{2}.$ $1112↔11121↔11211↔12111↔21111$
3. This follows from the fact that $\left\{{\tau }_{\sigma }{1}_{\alpha \beta }\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\sigma \in {S}_{k+l}/{S}_{k}×{S}_{l}\right\}$ is a basis of ${R}_{\alpha +\beta }$ as a right ${R}_{\alpha }\otimes {R}_{\beta }$-module.