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We apply this to the two elements in B We compute: (ρ ∗ μ)(x ⊗ y) = ∑ x⊗y H ⊗ H → H ν : x ⊗ y ↦→ S(y) ∙ S(x) ρ : x ⊗ y ↦→ S(x ∙ y) ρ((x ⊗ y) (1) ) ∙ μ((x ⊗ y) (2) ) [defn. of the convolution ∗] = ∑ ρ(x (1) ⊗ y (1) )μ(x (2) ⊗ y (2) ) [defn. of the coproduct of H ⊗ H] = ∑ S(x (1) y (1) )x (2) y (2) [defn. of ρ and μ] = ∑ (xy) S((xy) (1) )(xy) (2) [Δ is multiplicative] = ηɛ(xy) [defn. of the antipode] = 1 B (x ⊗ y) It is instructive to do such a calculation graphically: S S = = The first equality is the multiplicativity of the coproduct in a bialgebra, the second is the definition of the antipode. On the other hand, we compute μ ∗ ν: S S = = = = S S S ηɛ(x) ∙ ɛ(y) = ηɛ(x ∙ y) where in the last step we used that the counit ɛ is a map of algebras. Finally, the equality defining the antipode can be applied to 1 H and then yields id ∗ S = ηɛ , 1 H ∙ S(1 H ) = id ∗ S(1 H ) = ηɛ(1 H ) = 1 H , where the first equality is unitality of the coproduct Δ and the last identity is the unitality of the counit ɛ. This identity in H implies S(1 H ) = 1 H . The assertions about the coproduct are proven in an analogous way: use the identity Δ ◦ S = Δ −1 = (S ⊗ S) ◦ Δ copp in Hom(H, H ⊗ H) . 34
Finally, apply ɛ to the equality ɛ(x)1 = S(x (1) ) ∙ x (2) to get ɛ(x) = ɛ(x)ɛ(1) = ɛ(S(x (1) )ɛ(x (2) ) = ɛ ◦ S(x) for all x ∈ H . ✷ Proposition 2.5.6. Let H be a Hopf algebra. Then the following identities are equivalent: (a) S 2 = id H (b) ∑ x S(x (2))x (1) = ɛ(x)1 H for all x ∈ H. (c) ∑ x x (2)S(x (1) ) = ɛ(x)1 H for all x ∈ H. Proof. We show (b) ⇒ (a) by first showing from (b) that S ∗ S 2 is the identity for the convolution product. In graphical notation, (b) reads S = Thus S S ∗ S 2 = S = S 2 S = (b) S = = η ◦ ɛ For comparison, we also compute in equations: S ∗ S 2 (x) = ∑ ( ∑ ) S(x (1) )S 2 (x (2) ) = S S(x (2) )x (1) (x) (x) (b) = S(ɛ(x)1) = ɛ(x)S(1) = ɛ(x)1 . Then conclude S 2 = id by the uniqueness of the inverse of S with respect to the convolution product. Conversely, assume S 2 = id H S S = S S S S 2 = id S S S S = = = = η ◦ ɛ where we used S 2 = id, the fact that S is an anticoalgebra map, again S 2 = id and then the defining property of the antipode S. The equivalence of (c) and (a) is proven in complete 35
Page 1 and 2: Hopf algebras, quantum groups and t
Page 3 and 4: 1 Introduction 1.1 Braided vector s
Page 5 and 6: Proposition 1.2.3. Let (V, c) with
Page 7 and 8: In the first identity, the identifi
Page 9 and 10: 3. Similarly, denote by I − (V )
Page 11 and 12: Definition 2.1.7 Let A be a K-algeb
Page 13 and 14: • We want to encode this informat
Page 15 and 16: ι g : g → T (g) ↠ T (g)/I(g) =
Page 17 and 18: We introduce some more language: De
Page 19 and 20: (a) The functor F is essentially su
Page 21 and 22: 3. A coalgebra map is a linear map
Page 23 and 24: It is easy to check that a subspace
Page 25 and 26: in pictures = or in Sweedler notati
Page 27 and 28: Lemma 2.3.5. Let (A, μ, η, Δ, ɛ
Page 29 and 30: 2. Let G be a group and vect G (K)
Page 31 and 32: • Compatibility with the right un
Page 33 and 34: Remark 2.4.8. Let (A, μ, Δ) again
Page 35: Proposition 2.5.5. Let H be a Hopf
Page 39 and 40: 2. Use again a convolution product
Page 41 and 42: 2. A monoidal category is called ri
Page 43 and 44: In the last line, we used the defin
Page 45 and 46: We discuss a final example. Example
Page 47 and 48: 3. Note that a pair of adjoint func
Page 49 and 50: We then conclude, since for the Taf
Page 51 and 52: 1. An element h ∈ H \ {0} of a Ho
Page 53 and 54: Proof. 1. The equation x = (ɛ ⊗
Page 55 and 56: 3 Finite-dimensional Hopf algebras
Page 57 and 58: We discuss a first simple applicati
Page 59 and 60: The transpose is a map m ∗ x : H
Page 61 and 62: 1. Consider H ∗ with the Hopf mod
Page 63 and 64: 2. Note that, unlike in the case of
Page 65 and 66: Proof. From the associativity and b
Page 67 and 68: 1. For every diagram with A-modules
Page 69 and 70: 2.⇒ 1. Trivial, since 1. is a spe
Page 71 and 72: splits. Thus H = ker ɛ ⊕ I with
Page 73 and 74: obeys and for all k, l = 1, . . . n
Page 75 and 76: It follows that the components Λ i
Page 77 and 78: 2. If S 2 = id H , then by proposit
Page 79 and 80: Lemma 3.3.10. Let a ∈ G(H) be the
Page 81 and 82: 2. By corollary 3.1.16, the Hopf al
Page 83 and 84: Since S 2 χ H = χ H and since S 2
Page 85 and 86: It defines a tensor product: given
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3. Hence, in braided tensor categor
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This shows that the vectors (b i )
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The structure of three-dimensional
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- Objects are oriented triangulated
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If the braided category is not stri
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• Conversely, suppose that the ca
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✷ 4.3 Interlude: Yang-Baxter equa
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A vertex model with parameters λ i
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Proof. • We first show that By eq
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1. The invertible element Q := R 21
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We will see below that any factoriz
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and trivial coaction K → A ⊗ K
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• Define an associative multiplic
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We leave the proof of the converse
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while for the right hand side, we f
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Proposition 4.5.22. This defines a
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2. A pivotal Hopf algebra is called
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Proof. Identifying I ∼ = I ⊗ I,
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✷ Definition 5.1.11 A spherical c
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Proposition 5.1.17. 1. Let (H, R,
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a fibre functor in the category of
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Remark 5.3.3. 1. If one projects a
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In the third identity, we used the
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The composition is concatenation of
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Proof. One can show that any tangle
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the boundary points, for example gl
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• D is the dimension of the categ
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5.5 Quantum codes and Hopf algebras
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5.5.2 Classical gates To process in
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• Codes, i.e. interesting subspac
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5.5.6 Topological quantum computing
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Proof. 1. The operators A − (v,
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• The map is an isomorphism of K-
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5.7 Topological field theories of R
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A Facts from linear algebra A.1 Fre
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Remarks A.2.3. 1. This reduces the
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6. For any pair of K-vector spaces
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English quantum circuit quantum cod
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[S] Hans-Jürgen Schneider: Some pr
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invariant, 53 isomorphism, 9 isotop
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