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Let now C be a ribbon category. We describe the category T C of C-coloured framed oriented tangles. Observation 5.3.13. 1. Tangles are framed and oriented. Each component of a tangle is labelled with an object of C. Isotopies preserve the orientation, framing and labelling. 2. The objects of T C are finite sequences of pairs including the empty sequence. (V 1 , ɛ 1 ) . . . (V n , ɛ n ) V i ∈ C ɛ i ∈ {±1} , 3. Morphisms are isotopy classes of framed oriented tangles. If the source object has label ɛ = +1, the tangle is upward directed and labelled with V . It has to end on either an object (V, +1) at t = 1 or at (V, −1) at t = 0, where t ∈ [0, 1] parametrizes the tangle. 4. The category T C is endowed with a ribbon structure in complete analogy to the ribbon structure on the category T of framed oriented tangles. Proposition 5.3.14. Let C be a ribbon category. Then there is a unique braided tensor functor such that F = F C : T C → C, 1. F acts on objects as F (V, +) = V and F (V, −) = V ∗ . 2. For all objects V, W of C, we have c V,W V W θ V V b V d V 132
Proof. One can show that any tangle can be decomposed into the building blocks listed above. One then has to show the compatibility of F with the Reidemeister moves. This follows from the axioms of a ribbon category. ✷ Remarks 5.3.15. • In particular, restricting to morphisms between the tensor unit, we get an invariant of C-coloured framed oriented links in S 3 : F (L) ∈ End(I) . Since F is a monoidal functor, this invariant is multiplicative, F (L ⊗ L ′ ) = F (L) ⊗ F (L ′ ). • For the special case of the unknot labelled by the object V ∈ dim V ∈ End C (I). C, we get the invariant The following generalization will be needed in the construction of topological field theories: Observation 5.3.16. 1. We define a functor C ⊠n → vect by (V 1 , . . . , V n ) ↦→ Hom C (I, V 1 ⊗ ∙ ∙ ∙ ⊗ V n ) (14) for any collection V 1 , . . . , V n of objects of C. We thus consider invariant tensors in the tensor product. The pivotal structure gives functorial isomorphisms z : Hom C (I, V 1 ⊗ ∙ ∙ ∙ ⊗ V n ) ∼ = Hom C (I, V n ⊗ V 1 ⊗ ∙ ∙ ∙ ⊗ V n−1 ) (15) such that z n = id; thus, up to a canonical isomorphism, the space Hom C (I, V 1 ⊗ ∙ ∙ ∙ ⊗ V n ) only depends on the cyclic order of the objects V 1 , . . . , V n . 2. For a spherical category, we can define invariants of tangles where no crossings are allowed in the diagrams. We can also do the more general construction: consider an oriented graph Γ embedded in the sphere S 2 , where each edge e is colored by an object V (e) ∈ C, and each vertex v is colored by a morphism ϕ v ∈ Hom C (I, V (e 1 ) ± ⊗ ∙ ∙ ∙ ⊗ V (e n ) ± ), where e 1 , . . . , e n are the edges adjacent to vertex v, taken in clockwise order, and V (e i ) ± = V (e i ) if e i is outgoing edge, and V ∗ (e i ) if e i is the incoming edge. By removing a point from S 2 and identifying S 2 \ pt ≃ R 2 , we can consider Γ as a planar graph to which our rules assign a number Z(Γ) ∈ K. One shows that this number is an invariant of coloured graphs on the sphere. 3. Note that up to this point, only graphs on S 2 without crossings were allowed. We generalize this setup by allowing finitely many non-intersecting edges of a different type, labelled by objects of the Drinfeld double Z(C). These edges are supposed to start and end at the vertices as well. We colour edges of such a graph ˆΓ with objects in C and Z(C) respectively and morphisms as before. We get an invariant Z(Γ) ∈ K for this type of graph as well. 133
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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
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The transpose is a map m ∗ x : H
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1. Consider H ∗ with the Hopf mod
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2. Note that, unlike in the case of
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Proof. From the associativity and b
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1. For every diagram with A-modules
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2.⇒ 1. Trivial, since 1. is a spe
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splits. Thus H = ker ɛ ⊕ I with
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obeys and for all k, l = 1, . . . n
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It follows that the components Λ i
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2. If S 2 = id H , then by proposit
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Lemma 3.3.10. Let a ∈ G(H) be the
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2. By corollary 3.1.16, the Hopf al
Page 83 and 84: Since S 2 χ H = χ H and since S 2
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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
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Page 113 and 114: We leave the proof of the converse
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Page 117 and 118: Proposition 4.5.22. This defines a
Page 119 and 120: 2. A pivotal Hopf algebra is called
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Page 123 and 124: ✷ Definition 5.1.11 A spherical c
Page 125 and 126: Proposition 5.1.17. 1. Let (H, R,
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Page 129 and 130: Remark 5.3.3. 1. If one projects a
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Page 141 and 142: 5.5 Quantum codes and Hopf algebras
Page 143 and 144: 5.5.2 Classical gates To process in
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Page 147 and 148: 5.5.6 Topological quantum computing
Page 149 and 150: Proof. 1. The operators A − (v,
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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
Page 161 and 162: English quantum circuit quantum cod
Page 163 and 164: [S] Hans-Jürgen Schneider: Some pr
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