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It follows that Z(S 1 ) is given as the composition of the linear maps K ≃ Z(∅) Z(S1 − ) −→ Z(±1) Z(S1 + ) −→ Z(∅) ≃ K . These maps were described by (c) and (d) above. We thus get a map K ∼ = Z(∅) → Z(±1) ∼ = V ⊗ V ∨ → Z(∅) ∼ = K λ ↦→ λ ∑ v i ⊗ v i ↦→ λ ∑ i vi (v i ) = λ ∙ dim V where we have chosen a basis (v i ) i∈I of V and a dual basis (v i ) i∈I of V ∗ . Consequently, Z(S 1 ) is given by the dimension of V . In physical language, we have a quantum mechanical system which has only ground states and thus trivial Hamiltonian. Then the only invariant of the system is the degeneracy dim V of the space of ground states. Example 4.1.12 (Topological field theories in dimension 2). • A two-dimensional topological field theory assigns a vector space Z(M) to every closed, oriented 1-manifold M. Such a manifold is diffeomorphic to a disjoint union of circles, M ∼ = (S 1 ) ∐ n for some n ≥ 0. Since Z is monoidal, Z(M) ∼ = A ⊗n with A := Z(S 1 ) a finite-dimensional vector space. • One can show (see e.g. [Kock, Proposition 1.4.13]) that the monoidal category Cob(2) is generated under composition and disjoint union by six cobordisms: cap or disc, trinion, also called pair of pants, the cylinder, the trinion with two outoing circles, a disc with one ingoing circle and two exchanging cylinders. Applying the functor Z to these cobordisms, we get the following linear maps: cap trinion cylinder opposite trinion opposite cap exchanging cylinder η : K → A μ : A ⊗ A → A I A : A → A Δ : A → A ⊗ A ɛ : A → K τ : A ⊗ A → A ⊗ A • One can also classify all relations between the generators, see [Kock, 1.4.24-1.4.28]. The relations can be summarized that category Cob(2) is the free symmetric monoidal category on a commutative Frobenius object (Chapter 3). The relations imply that A has the structure of a commutative (Δ, ɛ)-Frobenius algebra. • In fact, the converse is true as well: given a commutative Frobenius algebra A, one can construct a 2-dimensional topological field theory Z such that A = Z(S 1 ) and the multiplication and Frobenius form on A are given by evaluating Z on a pair of pants and a disk, respectively. • In a categorical language, we thus arrive at the following classification result for twodimensional topological field theories: the topological field theories are described by the category [Cob(2), vect(K)] symm. monoidal of symmetric monoidal functors. This category is equivalent to the category of commutative K-Frobenius algebras. • As a general reference for this example, we refer to the book [Kock]. 88
The structure of three-dimensional topological field theories is much more involved. Here, the notion of a Hopf algebra will enter, as we will see later. Before turning to this, we consider a generalization and a specific construction: Example 4.1.13 (Open/closed topological field Theories in dimension 2). • We define a larger category Cob(2) o/cl of open-closed cobordisms: – Objects are compact oriented 1-manifolds which are allowed to have boundaries. These are disjoint unions of oriented intervals and oriented circles. – As a bordism B : M → N, we consider a smooth oriented two-dimensional manifold B, together with an orientation preserving smooth map φ B : M ∐ N → ∂B which is a diffeomorphism to its image. The map is not required to be surjective. In particular, we have parametrized and unparametrized intervals on the boundary circles of M. The unparametrized intervals are also called free boundaries and constitute physical boundaries. The other boundaries are cut-and-paste boundaries and implement (aspects of) locality of the topological field theory. Two bordisms B, B ′ give the same morphism, if there is an orientation-preserving diffeomorphism φ : B → B ′ such that the following diagram commutes: B φ B ′ φ B M ∐ φ ′ B N Thus the diffeomorphism respects parametrizations of intervals on boundary circles and parametrizations of whole boundary circles. – For any object M, the identity morphism id M is represented by the cylinder over M. – Composition is again by gluing. • Again, disjoint union endows Cob(2) o/cl with the structure of a symmetric monodial category with the empty set as the tensor unit. • An open-closed TFT is defined as a symmetric monoidal functor Z : Cob(2) o/cl → vect(K) . Again C := Z(S 1 ) is a commutative Frobenius algebra. One can again write generators and relations for the cobordism category and finds that the image O := Z(I) of the interval I carries the structure of a Frobenius algebra. C is called the bulk Frobenius algebra, O the boundary Frobenius algebra. • The Frobenius algebra O is not necessarily commutative: given three disjoint intervals on the boundary of a disk, two of them cannot be exchanged by a diffeomorphism of the disc. This situation is thus rather different from three boundary circles in a sphere, where two of them can be continuously commuted. For this reason, the bulk Frobenius algebra C is commutative. Still, the boundary Frobenius algebra is symmetric, i.e. the bilinear form κ 0 : O ⊗ O → K is symmetric: κ O (a, b) = κ O (b, a), as is shown graphically by cyclically exchanging two parametrized intervals on the boundary of a disc. 89
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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
Page 39 and 40: 2. Use again a convolution product
Page 41 and 42: 2. A monoidal category is called ri
Page 43 and 44: In the last line, we used the defin
Page 45 and 46: We discuss a final example. Example
Page 47 and 48: 3. Note that a pair of adjoint func
Page 49 and 50: We then conclude, since for the Taf
Page 51 and 52: 1. An element h ∈ H \ {0} of a Ho
Page 53 and 54: Proof. 1. The equation x = (ɛ ⊗
Page 55 and 56: 3 Finite-dimensional Hopf algebras
Page 57 and 58: We discuss a first simple applicati
Page 59 and 60: The transpose is a map m ∗ x : H
Page 61 and 62: 1. Consider H ∗ with the Hopf mod
Page 63 and 64: 2. Note that, unlike in the case of
Page 65 and 66: Proof. From the associativity and b
Page 67 and 68: 1. For every diagram with A-modules
Page 69 and 70: 2.⇒ 1. Trivial, since 1. is a spe
Page 71 and 72: splits. Thus H = ker ɛ ⊕ I with
Page 73 and 74: obeys and for all k, l = 1, . . . n
Page 75 and 76: It follows that the components Λ i
Page 77 and 78: 2. If S 2 = id H , then by proposit
Page 79 and 80: Lemma 3.3.10. Let a ∈ G(H) be the
Page 81 and 82: 2. By corollary 3.1.16, the Hopf al
Page 83 and 84: Since S 2 χ H = χ H and since S 2
Page 85 and 86: It defines a tensor product: given
Page 87 and 88: 3. Hence, in braided tensor categor
Page 89: This shows that the vectors (b i )
Page 93 and 94: - Objects are oriented triangulated
Page 95 and 96: If the braided category is not stri
Page 97 and 98: • Conversely, suppose that the ca
Page 99 and 100: ✷ 4.3 Interlude: Yang-Baxter equa
Page 101 and 102: A vertex model with parameters λ i
Page 103 and 104: Proof. • We first show that By eq
Page 105 and 106: 1. The invertible element Q := R 21
Page 107 and 108: We will see below that any factoriz
Page 109 and 110: 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,
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
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5.5 Quantum codes and Hopf algebras
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5.5.2 Classical gates To process in
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• Codes, i.e. interesting subspac
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5.5.6 Topological quantum computing
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Proof. 1. The operators A − (v,
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• The map is an isomorphism of K-
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5.7 Topological field theories of R
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A Facts from linear algebra A.1 Fre
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Remarks A.2.3. 1. This reduces the
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6. For any pair of K-vector spaces
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English quantum circuit quantum cod
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[S] Hans-Jürgen Schneider: Some pr
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invariant, 53 isomorphism, 9 isotop
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