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

Graded R α -modules

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia


Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 2 March 2010

Graded R α -modules

A -graded vector space is a vector space with a decomposition V= i V i andgdim V = i q i dim V is the graded dimension of V . If M is a -graded R α -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 α,β Q + and let k and l be the lengths of the words in Γ α and Γ β , 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 α R β -module, R α+β 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 σ of a coset in S k+l / S k × S l , we fix a reduced word σ= s i 1 s i l and set τ σ = τ i 1 τ i l .

Let R α -mod be the category of -graded R α -modules. For M R α -mod and N R β define MN= Ind R α R β R α+β MN .

Let K R α -mod be the Grothendieck group of -graded R -modules and define a product on α Q + K R α -mod by M . N = MN .

The following theorem says that the categories R α -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 R α -mod then gch M U q 𝔫 - .
  2. If M R α -mod and N R β -mod then gch Ind R α R β R α+β MN =gch M gch N .


  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], gch M satisfies Serre relations. Thus, by Proposition ?, gch M U q 𝔫 - .
  2. (second proof) (sketch) By understanding the intertwiners thoroughly, analyse the R α -modules when α=- α i α j α 1 + α 2 . 111211121112111211121111
  3. This follows from the fact that τ σ 1 αβ | σ S k+l / S k × S l is a basis of R α+β as a right R α R β -module.


[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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