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Index A-linear map, 9 abelian Lie algebra, 13 absolutely simple object, 149 additive tensor category, 148 adjoint functor, 43 algebra, 4 alternating algebra, 7 antialgebra map, 32 anticoalgebra map, 32 antipode, 31 associator, 25, 26 augmentation ideal, 24 autonomous category, 39 bialgebra, 22 bialgebra map, 23 biideal, 24 bilinear map, 154 Boltzmann weights, 98 boundary condition, 90 braid, 4 braid group, 2 braided tensor category, 83 braided tensor functor, 85 braided vector space, 1 braiding, 83 Cardy relation, 90 Casimir element, 72 category, 9 central form, 103 character, 49, 79 character algebra, 103 class function, 103 coaction, 21 coalgebra, 18 coassociativity, 18 cobordism, 41 cocommutative coalgebra, 18 code, 139 coevaluation, 38 coideal, 20 cointegral, 55 coinvariant, 54 commutative algebra, 5 commutativity constraint, 83 convolution product, 31 coopposed coalgebra, 18 counitality, 18 derivation, 12 dimension, 119 distinguished group-like element, 60 dominating family, 149 Drinfeld center, 115 Drinfeld double, 109 Drinfeld element, 101 Drinfeld map, 103 dual bases for a Frobenius form, 72 endomorphism, 9 enriched category, 10 enveloping algebra, 69 equivalence of categories, 16 error correcting code, 140 essentially small category, 124 evaluation, 38 extended topological field theory, 92, 134 exterior algebra, 7 factorizable Hopf algebra, 103 fibre functor, 124 forgetful functor, 153 framed link, 126 free vector space on a set, 153 Frobenius algebra, 62 Frobenius map, 58 functor, 15 fundamental groupoid, 10 fusion category, 134 gate, 141 gauge transformation, 113 global dimension, 134 group-like element, 49 groupoid, 10 Haar integral, 147 Hemming distance, 139 hexagon axioms, 83 Hopf algebra, 31 Hopf ideal, 32 Hopf module, 53 integrable vertex model, 99 162
invariant, 53 isomorphism, 9 isotopy, 4, 126 Jacobi identity, 11 Jones polynomial, 130 Kauffman relations, 127 knot, 126 knowledgeable Frobenius algebra, 90 left adjoint functor, 43 left dual object, 39 left integral, 55 left module, 7 Leibniz rule, 12 length of a code, 139 Lie algebra, 11 linear code, 139 link, 126 linking number, 127 modular category, 149 modular element, 60 monodromy element, 103 monoidal category, 25 monoidal functor, 28 monoidal natural transformation, 29 morphism, 9 Nakayama automorphism, 76 natural isomorphism, 16 natural transformation, 16 object, 9 open-closed TFT, 89 opposite algebra, 5 pentagon axiom, 26 pivotal category, 117 pivotal Hopf algebra, 116 Poincaré-Birkhoff-Witt theorem, 13 primitive element, 49 projective module, 66 pullback of a representation, 11 quantum circuit, 143 quantum code, 144 quantum gate, 143 quantum group, 94 quasi-cocommutative bialgebra, 93 quasi-triangular bialgebra, 93 qubits, 142 Reidemeister moves, 127 representation, 8, 14 restricted Lie algebra, 47 restriction functor, 15 restriction of scalars, 11 ribbon category, 121 ribbon element, 122 ribbon Hopf algebra, 122 right adjoint functor, 43 right comodule, 21 right dual object, 38 right integral, 55 rigid category, 39 semisimple algebra, 64 semisimple module, 64 separability idempotent, 70 separable algebra, 70 simple module, 64 site, 145 skein category, 131 skein functors, 131 skein module, 128 skew antipode, 36 spherical category, 121 spherical Hopf algebra, 117 straightening formula, 109 strict tensor category, 26 strict tensor functor, 29 super vector spaces, 84 surgery for a 3-manifold, 151 Sweedler notation, 21 symmetric algebra, 6, 89 symmetric Frobenius algebra, 76 symmetric tensor category, 83 Taft Hopf algebra, 45 tangle, 130 tensor algebra, 6 tensor category, 25 tensor functor, 28 tensor product, 25, 154 tensor unit, 25 tensors, 155 topological field theory, 85 topological quantum computing, 144 trace, 119, 121 transfer matrix, 98 163
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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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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
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
Page 161 and 162: English quantum circuit quantum cod
Page 163: [S] Hans-Jürgen Schneider: Some pr
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