By Antonio Sartori

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As ) on the top line. 1. The category Web is the category whose objects are sequences a = (a1 , . . , a ) of strictly positive integers; a morphism from a to a is a C(q)–linear combination of web diagrams with source a and target a . Composition of morphisms corresponds to vertical concatenation of web diagrams. Horizontal concatenation of web diagrams gives, on the other side, a monoidal structure on Web , whose unit is the empty sequence (). 33 Chapter 3. 2. 2e) 1 1 1 1 1 by the diagrams ai + ai+1 ai+2 ··· a ··· a1 ai−1 ai ai+1 ai+2 a a1 ai−1 ai ai+1 ai+2 a = ··· a1 ··· ai−1 ai + ai+1 ai+2 a and notice that the category Web is generated by such elementary web diagrams.

Let gln = gln (C) be the general Lie algebra of n × n matrices with the standard Cartan decomposition gln = n− ⊕ h ⊕ n+ into strictly lower diagonal, diagonal and strictly upper diagonal matrices respectively. Let also b = h⊕n+ be its standard Borel subalgebra. We will indicate by Λ ⊂ h∗ the set of integral weights of gln and by Λ+ ⊂ Λ the set of integral dominant weights (with our choice of Borel subalgebra b). We let ρ ∈ Λ be half the sum of the positive roots. 1 ([BGG76]). The BGG category O = O(gln ) = O(gln , b) is the full subcategory of U (gln )–modules which are (O1) finitely generated as U (gln )–modules, (O2) weight modules for the action of h with integral weights, and (O3) locally n+ –finite.

Gmod−R such that A module M ∈ mod−R will be called gradable if it has a graded lift M ˜ f(M ) = M . If S is another graded module, then a functor F : mod−R → mod−S will be called gradable if it has a graded lift, that is a functor F˜ : gmod−R → gmod−S such that fF˜ = F f. In general, not every module is gradable, see [Str03a] for an example. Given an abelian category A which is equivalent to mod−R for some graded ring R, we will say that Z A = gmod−R is a graded version of A. 2. 2]). Let R and S be any rings.