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Remarks A.2.5. 1. The bilinearity of κ implies that the tensor product of linear maps is bilinear: (λ 1 ϕ 1 + λ 2 ϕ 2 ) ⊗ ψ = λ 1 ϕ 1 ⊗ ψ + λ 2 ϕ 2 ⊗ ψ ϕ ⊗ (λ 1 ψ 1 + λ 2 ψ 2 ) = ϕ ⊗ λ 1 ψ 1 + ϕ ⊗ λ 2 ψ 2 2. Similarly, one deduces the following compatibility with direct sums: and analogously in the second argument. 3. There are canonical isomorphisms (V 1 ⊕ V 2 ) ⊗ W ∼ = (V 1 ⊗ W ) ⊕ (V 2 ⊗ W ) , a U,V,W : (U ⊗ V ) ⊗ W −→ ∼ U ⊗ (V ⊗ W ) u ⊗ (v ⊗ w) ↦→ (u ⊗ v) ⊗ w which allow to identify the K-vector spaces U ⊗ (V ⊗ W ) and (U ⊗ V ) ⊗ W . With this identification, the tensor product is strictly associative. One verifies that the following diagram commutes: (U ⊗ V ) ⊗ (W ⊗ X) (U ⊗ V ) ⊗ (W ⊗ X) α U⊗V,W,X ((U ⊗ V ) ⊗ W ) ⊗ X α U,V,W ⊗X U ⊗ (V ⊗ (W ⊗ X)) α U,V,W ⊗id X id U ⊗α V,W,X (U ⊗ (V ⊗ W )) ⊗ X αU,V ⊗W,X U ⊗ ((V ⊗ W ) ⊗ X) One can show that, as a consequence, any bracketing of multiple tensor products gives canonically isomorphic vector spaces. 4. There are canonical isomorphisms ∼ K ⊗ V −→ V λ ⊗ v ↦→ λ ∙ v with inverse map v ↦→ 1 ⊗ v which allow to consider the ground field K as a unit for the tensor product. There is also a similar canonical isomorphism V ⊗ K ∼ = V . We will tacitly apply the identifications described in 3. and 4. 5. For any pair U, V of K-vector spaces, there are canonical isomorphisms c U,V : U ⊗ V → V ⊗ U u ⊗ v ↦→ v ⊗ u , which allow to permute the factors. One has c V,U ◦ c U,V = id U⊗V as well as the identity (c V,W ⊗ id U ) ◦ (id V ⊗ c U,W ) ◦ (c U,V ⊗ id W ) = (id U ⊗ c V,W ) ◦ (c U,W ⊗ id V ) ◦ (id W ⊗ c U,V ) Note that in this identify, we have tacitly used the identification (U ⊗ V ) ⊗ W ∼ = U ⊗ (V ⊗ W ) from 3. 156
6. For any pair of K-vector spaces vector spaces V, W , the canonical map V ∗ ⊗ W ∗ → (V ⊗ W ) ∗ α ⊗ β ↦→ (v ⊗ w ↦→ α(v) ∙ β(w)) is an injection. If both V and W are finite-dimensional, this is an isomorphism. (Give an example of a pair of infinite-dimensional vector spaces and an element that is not in the image!) 7. For any pair of K-vector spaces V, W , the canonical map V ∗ ⊗ W → Hom K (V, W ) α ⊗ w ↦→ (v ↦→ α(v)w) is an injection. If both V and W are finite-dimensional, this is an isomorphism. (Give again an example of a pair of infinite-dimensional vector spaces and an element that is not in the image!) B Glossary German-English For the benefit of German speaking students, we include a table with German versions of important notions. English German abelian Lie algebra abelsche Lie-Algebra absolutely simple object absolut einfaches Objekt additive tensor category additive Tensorkategorie adjoint functor adjungierter Funktor alternating algebra alternierende Algebra antipode Antipode associator Assoziator augmentation ideal Augmentationsideal autonomous category autonome Kategorie Boltzmann weights braid braid group braided tensor category braided tensor functor braided vector space braiding character class function coaction code coevaluation coinvariant commutativity constraint convolution product coopposed coalgebra counitality Boltzmann-Gewichte Zopf Zopfgruppe verzopfte Tensorkategorie verzopfter Tensorfunktor verzopfter Vektorraum Verzopfung Charakter Klassenfunktion Kowirkung Code Koevaluation Koinvariante Kommutatitivitätsisomorphismus Konvolutionsprodukt, Faltungsprodukt koopponierte Algebra Kounitarität 157
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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
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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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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
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: Remarks A.2.3. 1. This reduces the
Page 161 and 162: English quantum circuit quantum cod
Page 163 and 164: [S] Hans-Jürgen Schneider: Some pr
Page 165 and 166: invariant, 53 isomorphism, 9 isotop
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