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Extra resources for Categorification of tensor powers of the vector representation of Uq (gl(1|1))

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2. Let λ, µ, γ ∈ Λ+ and suppose Sλ ⊆ Sµ ⊆ Sγ . Then Tγµ Tµλ ∼ = Tλ and Tλµ Tµγ ∼ = Tλγ . Proof. 8) of the graded translation γ µ γ functors. First, we have resγµ resµλ ∼ = resλ , hence Tγµ Tλ ∼ = Tλ . 5. Translation functors short short where x0 , y0 are the longest elements of Sµ Sλ and Sγ Sµ , respectively. Notice short that y0 x0 is the longest element of Sγ Sλ and (y0 ) (x0 ) = (y0 ) + (x0 ). Hence λ Tλµ Tµγ ∼ T . = γ Squares. 3. 3) Sµ = Sa1 × · · · × Sai−1 × Sai +ai+1 × Sai+2 × · · · × Sa .

Subquotient categories of O (GP3) direct sum of objects of Apq as a–modules. 3. The categories O{p + q, Apq } fall into a more general family of categories that were first introduced in [FKM02] (called generalized parabolic subcategories of O) and then generalized in [Maz04]. Our definition follows [MS08a], and in particular is a special case of [MS08a, Definition 32]. However, in [MS08a] only the trivial block is studied, while we are interested also in singular blocks. 3]) by [MS08a, Lemma 33].

1 Gradings If R is a ring we will denote from now on by mod−R the category of finitely generated (right) R–modules. If moreover R is graded, then we will denote by gmod−R the category of finitely generated graded R–modules. We stress that by graded we will always mean Z–graded. We denote by f : gmod−R → mod−R the grading forgetting functor. If M ∈ gmod−R then M = i∈Z Mi . e. f(M ) = f(M m ). We will also use the notation qM = M 1 . Given two graded R–modules M and N we denote by HomR (M, N ) the set of non-graded homomorphisms.

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Categorification of tensor powers of the vector representation of Uq (gl(1|1)) by Antonio Sartori

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