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Proof. We show the following statements by graphical calculations: • The above definition indeed defines H-coactions on V ∗ . • The coaction defined with S −1 has the property that the right evaluation and right coevaluation are morphisms of H-comodules. The statement that the coaction defined with S on the left dual is compatible with left evaluation and coevaluation follows in complete analogy. • The left and right dual actions and coactions obey the Yetter-Drinfeld axiom. This raises the question whether for any given Hopf algebra H the category H HYD can be seen as the category of left modules over a quasi-triangular Hopf algebra D(H). Observation 4.5.6. 1. To investigate this in more detail, assume that the Hopf algebra H is finite-dimensional and recall from example 2.2.8.1 that a coaction of H then amounts to an action of H ∗ . Thus the quasi-triangular Hopf algebra D(H) should account for an action of H and H ∗ . If the two actions would commute, H ∗ ⊗ H with the product structure for algebra and coalgebra would be an obvious candidate. This is not the case, and we have to encode the Yetter-Drinfeld condition in the product on D(H). 2. It will be convenient to consider a slightly different category H YD H of Yetter-Drinfeld modules: these are triples (V, ρ V , Δ V ) such that • (V, ρ V ) is a unital left A-module. • (V, Δ V ) is a counital right A-comodule: Δ V : V → V ⊗ H v ↦→ v (V ) ⊗ v (H) • The Yetter-Drinfeld condition holds in the form h (1) .v (V ) ⊗ h (2) ∙ v (H) = (h (2) .v) (V ) ⊗ (h (2) .v) (H) ∙ h (1) . Morphisms of Yetter-Drinfeld modules in H YD H are morphisms of left modules and right comodules. This observation leads to the following definition: Definition 4.5.7 Let H be a finite-dimensional Hopf algebra. Endow the vector space D(H) := H ∗ ⊗ H • with the structure of a counital coalgebra using the tensor product structure, i.e. for f ∈ H ∗ and a ∈ H, we have ɛ(f ⊗ a) := ɛ(a)f(1) Δ(f ⊗ a) := (f (1) ⊗ a (1) ) ⊗ (f (2) ⊗ a (2) ) This encodes the fact that the tensor product of Yetter-Drinfeld modules is the ordinary tensor product of modules and comodules. 108 ✷
• Define an associative multiplication for a, b ∈ H and f, g ∈ H ∗ by (f ⊗ a) ∙ (g ⊗ b) := fg(S −1 (a (3) )?a (1) ) ⊗ a (2) b . The unit for this multiplication is ɛ ⊗ 1 ∈ H ∗ ⊗ H. A tedious, but direct calculation (see [Kassel, Chapter IX]) shows: Proposition 4.5.8. This defines a finite-dimensional Hopf algebra. Moreover, if (e i ) is any basis of H with dual basis (e i ) of H ∗ , then the element R := ∑ i (1 ⊗ e i ) ⊗ (e i ⊗ 1) ∈ D(H) ⊗ D(H) , which is independent of the choice of basis, is a universal R-matrix for D(H). Definition 4.5.9 We call the quasi-triangular Hopf algebra (D(H), R) the Drinfeld double of the Hopf algebra H. Remarks 4.5.10. 1. The Drinfeld double D(H) contains H and H ∗ as Hopf subalgebras with embeddings i H : H → D(H) a ↦→ 1 ⊗ a and i H ∗ : H ∗ → D(H) f ↦→ f ⊗ 1 . 2. One checks that ι H ∗(f) ∙ ι H (a) = (f ⊗ 1) ∙ (1 ⊗ a) = fɛ(S −1 1 (3) ?1 (1) ) ⊗ 1 (2) ∙ a = f ⊗ a and therefore writes f ∙a instead of f ⊗a. The multiplication on D(H) is then determined by the straightening formula a ∙ f = f(S −1 (a (3) ?a (1) ) ∙ a (2) . 3. The Hopf algebra D(H) is quasi-triangular, even if the Hopf algebra H does not admit an R-matrix. If (H, R) is already quasi-triangular, then one can show that the linear map π R : D(H) → H fa ↦→ f(R 1 )R 2 ∙ a is a morphism of Hopf algebras. The multiplicative inverse R of R gives a second projection π R : D(H) → H. For more details, we refer to the article [S]. Proposition 4.5.11. The Drinfeld double D(H) of a finite-dimensional Hopf algebra H is factorizable. 109
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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
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Lemma 2.3.5. Let (A, μ, η, Δ, ɛ
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2. Let G be a group and vect G (K)
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• Compatibility with the right un
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Remark 2.4.8. Let (A, μ, Δ) again
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Proposition 2.5.5. Let H be a Hopf
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Finally, apply ɛ to the equality
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2. Use again a convolution product
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2. A monoidal category is called ri
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In the last line, we used the defin
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We discuss a final example. Example
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3. Note that a pair of adjoint func
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We then conclude, since for the Taf
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1. An element h ∈ H \ {0} of a Ho
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Proof. 1. The equation x = (ɛ ⊗
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3 Finite-dimensional Hopf algebras
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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
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 trivial coaction K → A ⊗ K
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
Page 129 and 130: Remark 5.3.3. 1. If one projects a
Page 131 and 132: In the third identity, we used the
Page 133 and 134: The composition is concatenation of
Page 135 and 136: Proof. One can show that any tangle
Page 137 and 138: the boundary points, for example gl
Page 139 and 140: • D is the dimension of the categ
Page 141 and 142: 5.5 Quantum codes and Hopf algebras
Page 143 and 144: 5.5.2 Classical gates To process in
Page 145 and 146: • Codes, i.e. interesting subspac
Page 147 and 148: 5.5.6 Topological quantum computing
Page 149 and 150: Proof. 1. The operators A − (v,
Page 151 and 152: • The map is an isomorphism of K-
Page 153 and 154: 5.7 Topological field theories of R
Page 155 and 156: A Facts from linear algebra A.1 Fre
Page 157 and 158: Remarks A.2.3. 1. This reduces the
Page 159 and 160: 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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