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We adopt the following usual convention: when we regard VP (x · λ) as graded module, its highest degree is (x). 3) Z Oλ = gmod−Aλ This grading is natural in the sense that it is the unique Koszul grading on Oλ . Details can be found in [Soe00] and [BGS96]. In [Str03a] it is proved that projective, simple and Verma modules are gradable, and a graded shift is unique up to isomorphism and overall shift in the grading. We take their standard graded lifts to be determined by requiring that the simple head is concentrated in degree 0, and by a slight abuse of notation we will denote them again by L(λ), M (λ) and P (λ).

4 are well-known to experts, we include them here since we do not know a good reference for them. 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 .

26) a a a a a F (v1 1 ⊗ v0 2 ⊗ · · · ⊗ v0 r ) = v0 1 ⊗ v0 2 ⊗ · · · ⊗ v0 r . 27)     F      h ··· h ··· ···     =     ··· ··· ··· which is exactly our assertion. 10. Fix some representation V(a) and consider a dual canonical basis element vη♥a . 28) E(vη♥a ) = q a1 +···+a −1  0 otherwise. A faithful calculus The functor T : Web → Rep we constructed is full, but not faithful. In the following we will define a category Web by adding more relations to Web, so that T descends to a faithful functor T : Web → Rep.

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

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