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Definition 3.3.7 The automorphism ρ is called the Nakayama automorphism of A with respect to the Frobenius structure κ. Remark 3.3.8. If A is commutative, the Nakayama involution is the identity. A Frobenius algebra is called symmetric, if the Nakayama automorphism equals the identity, i.e. if κ(x, y) = κ(y, x) for all x, y ∈ A. Group algebras are examples of symmetric algebras. Our strategy will now be to compute the Nakayama automorphism for the Frobenius algebra structure of a finite-dimensional Hopf algebra given by the right cointegral. (One can show that the Nakayama automorphism has, in this case, always finite order.) We need two lemmas. Denote by S −1 the composition inverse of the antipode, S ◦ S −1 = S −1 ◦ S = id H . Lemma 3.3.9. Let γ ∈ H ∗ a non-zero right integral of a finite-dimensional Hopf algebra H and let Γ ∈ H be the left integral such that 〈γ, Γ〉 = 1. Denote t := S(Γ) and α ∈ H ∗ the distinguished group like element which is an algebra morphism H → K. 1. Then (S −1 (t (2) ), t (1) ) are dual bases for γ. 2. We have for the Nakayama automorphism for the Frobenius structure given by the right integral: ρ(h) = 〈α, h (1) 〉 S −2 (h (2) ) . Proof. • We already know that for the Frobenius form given by a left integral λ for H ∗ we have a dual basis (SΛ (1) , Λ (2) ). Applying this to the dual of the Hopf algebra H copp which has antipode S −1 , we find the assertion. • As for any pair of dual bases, we have ρ(h) = S −1 (t (2) ) 〈γ, t (1) ρ(h)〉 = S −1 (t (2) ) 〈γ, ht (1) 〉 where in the second step applied the definition of a Nakayama automorphism. Applying S 2 , we find S 2 ρ(h) = 〈γ, ht (1) 〉S(t (2) ) = 〈γ, h (1) t (1) 〉h (2) t (2) S(t (3) ) [γ right cointegral] = 〈γ, h (1) t〉h (2) = 〈γ, α(h (1) )t〉h (2) [antipode, α distinguished element] = 〈α, h (1) 〉 h (2) [normalization] Applying S −2 yields the claim. Similarly, we have 76 ✷
Lemma 3.3.10. Let a ∈ G(H) be the distinguished group-like element of H and t, γ as before. 1. Then (S(t (1) )a, t (2) ) are dual bases for γ. 2. We have for the Nakayama automorphism for the Frobenius structure given by the right integral: ρ(h) = a −1 S 2 (h (1) )〈α, h (2) 〉a . Proof. • Using the definitions, we find for all h ∈ H: S(t (1) )a〈γ, t (2) h〉 = S(t (1) )t (2) h (1) 〈γ, t (3) h (2) 〉 [γ right cointegral] so that we have dual bases. = h (1) 〈γ, th (2) 〉 = h (1) ɛ(h (2) ) = h • By the fact that we have dual bases, we can write ρ(h) = S(t (1) )a〈γ, t (2) ρ(h)〉 = S(t (1) )a〈γ, ht (2) 〉 , where in the second identity, we applied the definition of the Nakayama automorphism. Applying S −2 and conjugating with a, we find aS −2 (ρ(h))a −1 = a〈γ, ht (2) 〉S −1 (t (1) ) = h (1) t (2) 〈γ, h (2) t (3) 〉S −1 (t (1) ) [γ cointegral] = h (1) 〈γ, h (2) t〉 = h (1) 〈α, h (2) 〉 . Conjugating with a −1 and applying S 2 yields the claim. ✷ Observation 3.3.11. If H is a finite-dimensional Hopf algebra, then H ∗ is a finite-dimensional Hopf algebra as well. We then have the structure of a left and right H ∗ -module on H by h ∗ ⇀ h := h (1) 〈h ∗ , h (2) 〉 and h ↼ h ∗ := 〈h ∗ , h (1) 〉h (2) . This can be checked again graphically. Theorem 3.3.12 (Radford,1976). Let H be a finite-dimensional Hopf algebra over a field K. Let a ∈ G(H) and α ∈ G(H ∗ ) be the distinguished modular elements. Then the following identity holds: S 4 (h) = a(α −1 ⇀ h ↼ α)a −1 = α −1 ⇀ (aha −1 ) ↼ α Proof. 77
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Hopf algebras, quantum groups and t
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1 Introduction 1.1 Braided vector s
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Proposition 1.2.3. Let (V, c) with
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In the first identity, the identifi
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3. Similarly, denote by I − (V )
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Definition 2.1.7 Let A be a K-algeb
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• We want to encode this informat
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ι g : g → T (g) ↠ T (g)/I(g) =
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We introduce some more language: De
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(a) The functor F is essentially su
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3. A coalgebra map is a linear map
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It is easy to check that a subspace
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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 and 36: Proposition 2.5.5. Let H be a Hopf
Page 37 and 38: Finally, apply ɛ to the equality
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: 2. If S 2 = id H , then by proposit
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
Page 87 and 88: 3. Hence, in braided tensor categor
Page 89 and 90: This shows that the vectors (b i )
Page 91 and 92: The structure of three-dimensional
Page 93 and 94: - Objects are oriented triangulated
Page 95 and 96: If the braided category is not stri
Page 97 and 98: • Conversely, suppose that the ca
Page 99 and 100: ✷ 4.3 Interlude: Yang-Baxter equa
Page 101 and 102: A vertex model with parameters λ i
Page 103 and 104: Proof. • We first show that By eq
Page 105 and 106: 1. The invertible element Q := R 21
Page 107 and 108: We will see below that any factoriz
Page 109 and 110: and trivial coaction K → A ⊗ K
Page 111 and 112: • Define an associative multiplic
Page 113 and 114: We leave the proof of the converse
Page 115 and 116: while for the right hand side, we f
Page 117 and 118: Proposition 4.5.22. This defines a
Page 119 and 120: 2. A pivotal Hopf algebra is called
Page 121 and 122: Proof. Identifying I ∼ = I ⊗ I,
Page 123 and 124: ✷ Definition 5.1.11 A spherical c
Page 125 and 126: Proposition 5.1.17. 1. Let (H, R,
Page 127 and 128: 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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