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English derivation distinguished group-like element dominating family Drinfeld center Drinfeld double enriched category enveloping algebra error correcting code essentially small category evaluation exterior algebra factorizable Hopf algebra fibre functor forgetful functor free vector space fundamental groupoid fusion category gate gauge transformation group-like element groupoid Hamming distance Haar integral hexagon axioms Hopf algebra isotopy knot left adjoint functor left integral left module link linking number modular category modular element monodromy element opposite algebra pentagon axiom pivotal category primitive element projective module pullback of a representation German Derivation ausgezeichnetes gruppenartiges Element dominierende Familie Drinfeld Zentrum Drinfeld-Doppel angereicherte Kategorie einhüllende Algebra fehlerkorrigierender Code wesentlich kleine Kategorie Evaluation äußere Algebra faktorisierbare Hopf-Algebra Faserfunktor Vergissfunktor freier Vektorraum Fundamentalgruppoid Fusionskategorie Gatter Eichtransformation gruppenartiges Element Gruppoid Hamming-Abstand Haarsches Maß Hexagon-Axiome Hopf-Algebra Isotopie Knoten linksadjungierter Funktor Linksintegral Linksmodul Verschlingung Verschlingungszahl modulare Kategorie modulares Element Monodromie-Element opponierte Algebra Pentagon-Axiom pivotale Kategorie primitives Element projektiver Modul Pullback einer Darstellung 158
English quantum circuit quantum code quantum gate quantum group quasi-triangular bialgebra Reidemeister move representation right adjoint functor right integral semisimple algebra semisimple module separable algebra simple module skew antipode spherical category surgery tensor unit trace trefoil knot triangle axiom trivial module unimodular Hopf algebra universal R-matrix universal enveloping algebra uversal quantum gaute unknot Yang-Baxter equation Yetter-Drinfeld module German Quantenschaltkreis Quantenkode Quantengatter Quantengruppe quasitrianguläre Bialgebra Reidemeister-Bewegung Darstellung rechtsadjungierter Funktor Rechtsintegral halbeinfache Algebra halbeinfacher Modul (der) separable Algebra einfacher Modul (der) Schiefantipode sphärische Kategorie Chirurgie Tensoreins Spur Kleeblattknoten Dreiecksaxiom trivialer Modul (der) unimodulare Hopf-Algebra universelle R-Matrix universelle einhüllende Algebra universelles Quantengatter trivialer Knoten, Unknoten Yang-Baxter- Gleichung Yetter-Drinfeld-Modul (der) 159
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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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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 and 158: Remarks A.2.3. 1. This reduces the
Page 159: 6. For any pair of K-vector spaces
Page 163 and 164: [S] Hans-Jürgen Schneider: Some pr
Page 165 and 166: invariant, 53 isomorphism, 9 isotop
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