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• The pentagon axiom: for all quadruples of objects U, V, W, X ∈ Obj(C) the following diagram commutes (U ⊗ V ) ⊗ (W ⊗ X) a U⊗V,W,X a U,V,W ⊗X ((U ⊗ V ) ⊗ W ) ⊗ X U ⊗ (V ⊗ (W ⊗ X)) a U,V,W ⊗id X id U ⊗a V,W,X (U ⊗ (V ⊗ W )) ⊗ X a U,V ⊗W,X U ⊗ ((V ⊗ W ) ⊗ X) • The triangle axiom: for all pairs of objects V, W ∈ Obj(C) the following diagram commutes a V,I,W (V ⊗ I) ⊗ W V ⊗ (I ⊗ W ) r V ⊗id W id V ⊗l W V ⊗ W Remarks 2.4.3. 1. A monoidal category can be considered as a higher analogue of an associative, unital monoid, hence the name. The associator a is, however, a structure, not a property. A property is imposed at the level of natural transformations in the form of the pentagon axiom. For a given category C and a given tensor product ⊗, inequivalent associators can exist. Any associator a gives for any triple U, V, W of objects an isomorphism such that all diagrams of the form a U,V,W : (U ⊗ V ) ⊗ W → U ⊗ (V ⊗ W ) (U ⊗ V ) ⊗ W a U,V,W U ⊗ (V ⊗ W ) (f⊗g)⊗h (U ′ ⊗ V ′ ) ⊗ W ′ a U ′ ,V ′ ,W ′ f⊗(g⊗h) U ⊗ (V ⊗ W ) commute. 2. The pentagon axiom can be shown to guarantee that one can change the bracketing of multiple tensor products in a unique way. This is known as Mac Lane’s coherence theorem. We refer to [Kassel, XI.5] for details. 3. A tensor category is called strict , if the natural transformations a, l and r are the identity. One can show that any tensor category can be replaced by an equivalent strict tensor category. Examples 2.4.4. 1. The category of vector spaces over a fixed field K is a tensor category which is not strict. (See the appendix for information about this tensor category.) Tacitly, it is frequently replaced by an equivalent strict tensor category. 26
2. Let G be a group and vect G (K) be the category of G-graded K-vector spaces, i.e. of K- vector spaces with a direct sum decomposition V = ⊕ g∈G V g . Then the tensor product V ⊗ W is bigraded, V ⊗ W = ⊕ g,h∈G V g ⊗ W h and becomes G-graded by V ⊗ W = ⊕ g∈G (⊕ h∈G V h ⊗ W h −1 g) . Together with the associativity constraints inherited from vect(K) and with K e , i.e. the ground field K in homogeneous degree e ∈ G, as the tensor unit, this is a monoidal category. For these considertations, inverses in G are not needed and we could consider vector spaces graded by any unital associative monoid. 3. Let C be a small category. The endofunctors F : C → C are the objects of a tensor category End(C). The morphisms in this category are natural transformations, the tensor product is composition of functors. 4. Let (G n ) n∈N0 be a family of groups such that G 0 = {1}. Define a category G whose objects are the natural numbers and whose morphisms are defined by { ∅ m ≠ n Hom G (m, n) = G n m = n Composition is the product in the group, the identity is the neutral element, id n = e ∈ G n . Suppose that we are given as further data a group homomorphism for any pair (m, n) such that for all m, n, p ∈ N, we have We define a functor ρ m,n : G m × G n → G m+n ρ m+n,p ◦ (ρ m,n × id Gp ) = ρ m,n+p ◦ (id Gm × ρ n,p ) . ⊗ : G × G → G on objects by m ⊗ n = m + n and on morphisms by This turns G into a strict tensor category. G m × G n → G m,n (f, g) ↦→ f ⊗ g := ρ m,n (f, g) . Such a structure is provided in particular by the collection (S n ) n∈Z≥0 of symmetric groups and the collection (B n ) n∈N of braid groups. Define ρ m,n : B m × B n → B m+n (σ i , σ j ) ↦→ σ i σ j+m , as the juxtaposition of a braid from B m to a braid B n . Remarks 2.4.5. 27
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: Lemma 2.3.5. Let (A, μ, η, Δ, ɛ
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 and 78: 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
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Since S 2 χ H = χ H and since S 2
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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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