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The ground state or protected subspace is the zero eigenspace of H: It is a quantum code. K(Σ, Δ) := {v ∈ H ⊗n : Hv = 0} Remarks 5.5.20. 1. One shows that x ∈ K(Σ, Δ), if and only if A v x = x and B p x = x for all vertices v and all plaquettes p. 2. Up to canonical isomorphism, the ground space does not depend on the choice of cell decomposition Δ of Σ. 3. In the case of a group algebra of a finite group, H = C[G], we use the distinguished basis of H consisting of group elements of G. (Kitaev’s toric code uses the cyclic group Z 2 .) A basis of V (Σ, Δ) is given by assigning to any edge of Δ a group element g. We interpret the group elements g as the holonomy of a connection along the edge. • The projection by the operator A v implements gauge invariance at the vertex v by averaging with respect to the Haar measure. • The projection by the operator B p implements that locally on the face p the field strength vanishes, i.e. that the connection is locally flat. Indeed, integrals project to invariants and thus for the holonomy around a plaquette, we have ɛ(g 1 ∙g 2 ∙. . .∙g n ) = 1 which amounts to the flatness condition g 1 ∙ g 2 ∙ . . . ∙ g n = e. 4. One can then easily modify at single sites the projection condition: instead of requiring invariance under the action of the double associated to the site, one only keeps states transforming in a specific representation of the double. In this way, again the category of D(H−mod) appears in the description of degrees of freedoms at insertions. One can now show: Theorem 5.5.21 (Balsam-Kirillov). Let H be a finite-dimensional semisimple Hopf algebra. The vector spaces of ground states constructed from the Hopf algebra H are canonically isomorphic to the vector spaces of the Turaev-Viro topological field theory based on H. For the proof, we refer to [BK2]. 5.6 Modular tensor categories We want to discuss yet another construction of three-dimensional extended topological field theories, this time starting from a class of braided monoidal categories. Again it provides in particular invariants of links and three-manifolds. Definition 5.6.1 Let K be a field. 1. A tensor category C is called additive with ground ring K if the following conditions hold: • All Hom sets Hom C (U, V ) are endowed withe the structure of a K-vector space. • Composition and tensor product are K-bilinear. 148
• The map is an isomorphism of K-vector spaces. K → End C (I) λ ↦→ λ id I 2. More generally, an object V of an additive category C is called absolutely simple, if the map is an isomorphism of K-vector spaces. K → End C (V ) λ ↦→ λ id V 3. Let {V i } i∈I be a family of objects in a K-linear category C. An object V in this category is said to be dominated by the family {V i }, if the image of the map additively generates End C (V ). Hom C (V, V i ) ⊗ Hom C (V i , V ) → End C (V ) (f, g) → g ◦ f Definition 5.6.2 A modular category is pair (C, {V i } i∈I ) consisting of • an additive ribbon category C with ground field K • a finite family {V i } i∈I of simple objects such that (A1) There exists an index 0 ∈ I with V 0 = I. (A2) For any index i ∈ I there is an index i ∗ ∈ I such that V i ∗ ∼ = V ∗ i . (A3) The family {V i } dominates all objects of C. (A4) The |I| × |I|-matrix with entries S ij = Tr c Vj ,V i ◦ c Vi ,V j ∈ End (I) ∼ = K is invertible over K. Remarks 5.6.3. 1. The symmetry of the trace implies that the matrix S is symmetric, S ij = S ji . 2. The matrix element S ij equals the invariant of the Hopf link with the two components coloured by the objects V i and V j . Observation 5.6.4. 149
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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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Page 101 and 102: A vertex model with parameters λ i
Page 103 and 104: Proof. • We first show that By eq
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Page 107 and 108: 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
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Page 117 and 118: Proposition 4.5.22. This defines a
Page 119 and 120: 2. A pivotal Hopf algebra is called
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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
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Page 135 and 136: Proof. One can show that any tangle
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Page 141 and 142: 5.5 Quantum codes and Hopf algebras
Page 143 and 144: 5.5.2 Classical gates To process in
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Page 147 and 148: 5.5.6 Topological quantum computing
Page 149: Proof. 1. The operators A − (v,
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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 and 164: [S] Hans-Jürgen Schneider: Some pr
Page 165 and 166: invariant, 53 isomorphism, 9 isotop
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