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1. In any monoidal category, we have a notion of an associative unital algebra (A, μ, η): this is a triple, consisting of an object A ∈ C, multiplication μ : A ⊗ A → A and a unit morphism η : I → A such that associativity identity μ ◦ (μ ⊗ id A ) = μ ◦ (id A ⊗ μ) ◦ a A,A,A for the morphism A ⊗ A ⊗ A → A and the unit property μ ◦ (id A ⊗ η) = id A ◦ r A and μ ◦ (η ⊗ id A ) = id A ◦ l A hold. Note that the associator enters. For a general monoidal category, we do not have a notion of a commutative algebra. 2. Similarly, we can define the notion of a coalgebra (C, Δ, ɛ) in any monoidal category. For a general monoidal category, we do not have a notion of a cocommutative coalgebra. 3. Similarly, one can define modules and comodules in any monoidal category. 4. A coalgebra in C gives an algebra in C opp and vice versa. Tensor categories are categories with some additional structure. It should not come as a surprise that we need also a class of functors and natural transformations that is adapted to this extra structure. Definition 2.4.6 1. Let (C, ⊗ C , I C , a C , l C , r C ) and (D, ⊗ D , I D , a D , l D , r D ) be tensor categories. (We will sometimes suppress indices indicating the category to which the data belong.) A tensor functor or monoidal functor from C to D is a triple (F, ϕ 0 , ϕ 2 ) consisting of a functor F : C → D an isomorphism ϕ 0 : I D → F (I C ) in the category D a natural isomorphism ϕ 2 : ⊗ D ◦ (F × F ) → F ◦ ⊗ C of functors C × C → D. This includes in particular an isomorphism for any pair of objects U, V ∈ C ϕ 2 (U, V ) : F (U) ⊗ D F (V ) −→ ∼ F (U ⊗ C V ) . These data have to obey a series of constraints expressed by commuting diagrams: • Compatibility with the associativity constraint: (F (U) ⊗ F (V )) ⊗ F (W ) ϕ 2 (U,V )⊗id F (W ) F (U ⊗ V ) ⊗ F (W ) ϕ 2 (U⊗V,W ) F ((U ⊗ V ) ⊗ W ) a F (U),F (V ),F (W ) F (a U,V,W ) F (U) ⊗ (F (V ) ⊗ F (W )) id F (U) ⊗ϕ 2 (V,W ) F (U) ⊗ F (V ⊗ W ) ϕ 2 (U,V ⊗W ) F (U ⊗ (V ⊗ W )) • Compatibility with the left unit constraint: I D ⊗ F (U) ϕ 0 ⊗id F (U) l F (U) F (U) F (l U ) F (I C ) ⊗ F (U) ϕ2 (I C ,U) F (I C ⊗ U) 28
• Compatibility with the right unit constraint: r F (U) F (U) ⊗ I D F (U) id F (U) ⊗ϕ 0 F (r U ) F (U) ⊗ F (I C ) ϕ 2 (U,I C ) F (U ⊗ I C ) 2. A tensor functor is called strict, , if ϕ 0 and ϕ 2 are identities in D. In general, the isomorphism and the natural isomorphism is additional structure. 3. A monoidal natural transformation η : (F, ϕ 0 , ϕ 2 ) → (F ′ , ϕ ′ 0, ϕ ′ 2) between tensor functors is a natural transformation η : F → F ′ such that the diagram involving the tensor unit F (I C ) ϕ 0 I η D I and for all pairs (U, V ) of objects the diagram ϕ ′ 0 F ′ (I C ) F (U) ⊗ F (V ) ϕ 2 (U,V ) F (U ⊗ V ) η U ⊗η V F ′ (U) ⊗ F ′ (V ) η U⊗V ϕ ′ 2 (U,V ) F ′ (U ⊗ V ) commute. 4. One then defines monoidal natural isomorphisms as invertible monoidal natural transformations. An equivalence of tensor categories C,D is given by a pair of tensor functors F : C → D and F : D → C and natural monoidal isomorphisms η : id D → F G and θ : GF → id C . We can now characterize algebras whose representation categories are monoidal categories. Proposition 2.4.7. Let (A, μ) be a unital associative algebra. Suppose we are given unital algebra maps Δ : A → A ⊗ A and ɛ : A → K . Use the pullback along ɛ to endow the ground field K with the structure of an A-module (K, ɛ), i.e. a.λ := ɛ(a) ∙ λ for a ∈ A and λ ∈ K. Let ⊗ : A−mod × A−mod → A−mod 29
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: 2. Let G be a group and vect G (K)
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
Page 79 and 80: 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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