By Antonio Sartori

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2. Subquotient categories of O A complete set of pairwise non-isomorphic simple objects in SΓ is given by the L(γ)’s for γ ∈ Γ and each of them has a projective cover P SΓ (γ) in SΓ , which is the biggest quotient of P (γ) which lies in SΓ . 3) HomA/S (M, N ) = lim HomA (M , N/N ) −→ where the direct limit is taken over all pairs M ⊆ M , N ⊆ N such that M/M ∈ S and N ∈ S. The quotient category turns out to be an abelian category, and comes with an exact quotient functor Q : A → A/S (see [Gab62]).

Proof. 4) short . Notice that (x0 ) = ai + ai+1 . By ai + ai+1 µ resµλ B λ ⊗B µ • −(ai + ai+1 ) ∼ B ⊗B µ •. = ai Equivalently, it suffices to show that B λ is free as left B µ –module of (graded) rank i+1 i+1 q ai +ai+1 ai +a = ai +a . 2, (1)] (see ai ai 0 also [Dem73, Théorème 2, (c)]). Isotopy and gln so that λ ⊕λ invariance. Let now n , n > 0 with n = n + n , and consider the Lie algebras gln with standard Cartan subalgebras h and h . Consider the inclusion gln ⊕gln ⊂ gln , h = h ⊕ h . Given a weight λ for gln and a weight λ for gln , let us denote by the weight for gln whose restriction to h is λ and to h is λ .

10) ⊗ ··· ⊗ a : n → a. Webs as intertwiners Now we are going to define a monoidal functor T : Web → Rep. On objects we set T (a) = V(a) and T (()) = C(q), where () is the empty sequence. To define T on morphisms, it suffices to consider elementary pieces of webs. 12) a+b T = a b V(a + b) Φa,b V(a) ⊗ V(b) Chapter 3. 3. 13) define a dense full monoidal functor T : Web → Rep. Proof. First, we have to check that T satisfies the relations defining Web. 2d). 7). 9c). The functors T is dense since, by definition, the objects of Rep are exactly the V(a) for all sequences a of positive integer numbers.