Quantum groups and knot algebra by tom Dieck T.

By tom Dieck T.

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4) We use that s is an antihomomorphism of coalgebras, i. e. the relation µs = τ (s ⊗ s)µ. To begin with, Λ is an isomorphism of A ⊗ A0 ⊗ A0 -modules with actions ((a ⊗ b ⊗ c) · ϕ)(x ⊗ y) = aϕ(bx ⊗ cy) left ((a ⊗ b ⊗ c) · ψ)(x)(y) = a(ψ(bx)(cy)) right. The A-module structure on the left side uses (1 ⊗ µs)µ, on the right side ((1 ⊗ s)µ ⊗ 1)(1 ⊗ s)µ = (1 ⊗ τ )(1 ⊗ s ⊗ s)(µ ⊗ 1)µ. Coassociativity of µ and antihomomorphy of s yield the equality. (5) The definitions give T (a · (ϕ ⊗ ψ))(y ⊗ x) = ϕ(s(a1 )x)ψ(s(a2 )y).

32 1 Tensor categories T. 4) Theorem. The left dual f ∗ and the right dual f # coincide. Proof. The morphism f # is characterized by (f # 1) ◦ aV = (1f ) ◦ aV . The morphism f ∗ by a similar equality with the b-morphisms. If we apply (δ1) ◦ z and (1δ) ◦ z to this characterization of f ∗ , we obtain (f ∗ 1 ◦ aV = 1f ◦ aV . We conclude f # = f ∗ . 5) Theorem. The morphisms I bV ✲ VV∗ f 1✲ VV∗ cV ✲ I I aV ✲ V ∗V 1f ✲ V ∗V dV ✲ I coincide for each endomorphism f : V → V . Proof. We insert the definition of aV and cV and use the naturality of z and δ.

10) Example. Let A be cocommutative. Then the dualization H(M, N ) → H(N ∗ , M ∗ ), ϕ → ϕ∗ is A-linear. ♥ T. tom Dieck 4. 11) Example. Let A be cocommutative. Then the tautological map H(M1 , N1 ) ⊗ H(M2 , N2 ) → H(M1 ⊗ M2 , N1 ⊗ N2 ), ϕ⊗ψ →ϕ⊗ψ ♥ is A-linear. 4. The finite dual Let A be a Hopf algebra over a field K. We use finite dimensional A-modules to construct the finite dual of A. The algebra A acts on the left and right on the dual vector space A∗ by (a · f )(x) = f (xa) and (f · a)(x) = f (ax).

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