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References [AAITC] N. Andruskiewitsch, I. Angiono, A. Iglesias, B. Torrecillas, C. Vay: From Hopf algebras to tensor categories. Preprint arXiv:1204.5807 [math.QA] [BK1] B. Balsam and A.N. Kirillov, Turaev-Viro invariants as an extended TQFT. Preprint math.GT/1004.1533 [BK2] B. Balsam and A.N. Kirillov, Kitaev’s lattice model and Turaev-Viro TQFTS. Preprint math.QA/1206.2308 [BMCA] O. Buerschaper, J.M. Mombelli, M. Christandl and M. Aguado: A hierarchy of topological tensor network states. http://arxiv.org/abs/1007.5283 [D] P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift II 111-197, Birkhäuser Boston, 2007. [DNR] S. Dascalescu, C. Nastasescu, S. Raianu, Hopf Algebras. An Introduction. Monographs and Textbooks in Pure and Applied Mathematics 235, Marcel-Dekker, New-York, 2001. [ENO] P. Etingof, D. Nikshych, V. Ostrik, On fusion categories Ann. of Mathematics 162 (2005) 581-642, http://arxiv.org/abs/math/0203060 [FKLW] M.H. Freedman, A. Kitaev, M.J. Larsen and Z. Wang: Topological quantum computation Bull. Amer. Math. Soc. 40 (2003), 31-38 [Kassel] C. Kassel, Quantum Groups. Graduate Texts in Mathematics 155, Springer, Berlin, 1995. [KRT] C. Kassel, M. Rosso, Vl. Turaev: Quantum groups and knot invariants. Panoramas et Synthèses, Soc. Math. de France, Paris, 1993 [Kock] J. Kock: Frobenius algebras and 2D topological quantum field theories. Cambridge University Press, London Mathematical Society Student Texts Nr. 59, Cambridge University Press, 2003 [LP] A. Lauda and H. Pfeiffer: Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras. Topology and its Applications (155) 2008 623-666. http://arxiv.org/abs/math/0510664 [L] J. Lurie: On the classification of topological field theories. Current Developments in Mathematics 2008 (2009) 129-280. http://www.math.harvard.edu/ lurie/papers/cobordism.<strong>pdf</strong> [McL] S. Mac Lane: Categories for the working mathematician, Springer, New York, 1971. [Montgomery] S. Montgomery, Hopf algebras and their actions on rings, CMBS Reg. Conf. Ser. In Math. 82, Am. Math. Soc., Providence, 1993. [NS] Th. Nikolaus and C. Schweigert: Bicategories in field theories - an invitation preprint http://arxiv.org/abs/arXiv:1111.6896 [Schneider] H.J. Schneider, Lectures on Hopf algebras, Notes by Sonia Natale. Trabajos de Matemática 31/95, FaMAF, 1995. http://www.famaf.unc.edu.ar/ andrus/papers/Schn1.<strong>pdf</strong> 160
[S] Hans-Jürgen Schneider: Some properties of factorizable Hopf algebras. Proc. AMS 129 (2001) 1891-1898. www.mathematik.uni-muenchen.de/ hanssch/Publications/Fact.ps [TV] V.G. Turaev and A. Virelizier, On two approaches to 3-dimensional TQFTs. preprint math.GT/1006.3501 161
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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
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,
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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: English quantum circuit quantum cod
Page 165 and 166: invariant, 53 isomorphism, 9 isotop
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