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For the special case of the energy, we have ∑ ɛ λ (ϕ)e −βɛ λ(ϕ) E β,λ [ɛ] = states Z(β, λ) = − ∂ ln Z(β, λ) ∂β It is thus an important goal to compute the partition function. To this end, we introduce Boltzmann weights We then get R kl ij (β, λ) = e −βɛkl ij (λ) . e −βɛ λ(ϕ) = ∏ Consider the contribution of the first line to the partition function where we temporarily allow different values for the leftmost and rightmost bond: atoms R kl ij . It equals i 1 k 1 k 2 k 3 k N−1 k N . . . r 1 r 2 r 3 r N−1 i ′ 1 l . . . 1 l 2 l 3 l N−1 l N T i′ 1 l 1...l N i 1 k 1 ...k N = ∑ R r 1l 1 i 1 k 1 R r 2l 2 r 1 ...r N−1 r 1 k 2 . . . R i′ 1 l N r N−1 k N (12) To eliminate indices, we introduce a complex vector space V freely generated on the set {1, . . . , n} with basis {v 1 , . . . v n } and a family of endomorphisms R = R(β, λ) : V ⊗ V → V ⊗ V v i ⊗ v j ↦→ ∑ k,l Rkl ij v k ⊗ v l . Definition (12) leads to the definition of the endomorphism T ∈ End (V ⊗ V N ) with T = R 01 R 02 R 03 . . . R 0n . Here we understand that the endomorphism R ij acts on the i-th and j-th copy of V in the tensor product V ⊗ V N . Periodic boundary conditions imply for the sum over the first line Tr V (T ) l 1...l N k 1 ...k N . This endomorphism is called the row-to-row transfer matrix. To sum over all M lines, we take the matrix product and then the trace so that we find: Z = tr V ⊗N (tr V (T )) M . This raises the problem of understanding the eigenvalues of the endomorphism Tr V (T ) ∈ End(V ⊗N ): in the thermodynamic limit, we take M → ∞ so that Z ∼ κ M N with κ N the eigenvalue with the largest modulus. As usual in eigenvalue problems, we try to find as many endomorphisms of V ⊗N as possible commuting with Tr V (T ) which allows us to solve the eigenproblem separately on eigenspaces of these operators. Definition 4.3.1 98
A vertex model with parameters λ is called integrable, if for any pair μ, ν of values for the parameters there is a value λ such that the equation R 12 (λ)R 13 (μ)R 23 (ν) = R 23 (ν)R 13 (μ)R 12 (λ) (QYBE) holds in End (V ⊗ V ⊗ V ). A specific case is the quantum Yang-Baxter equation with spectral parameters: R 12 (λ − μ)R 13 (λ − ν)R 23 (μ − ν) = R 23 (μ − ν)R 13 (λ − ν)R 12 (λ − μ) Bialgebras are not enough to describe such a structure; Etingof and Varchenko have instead proposed algebroids. Lemma 4.3.2. Consider the tensor product V ⊗ V ⊗ V N and denote the index for the first copy of V by 0 and the index for the second copy of V by 0. Then the following equation holds in End (V ⊗V ⊗V N ) R 00 (λ)T 0 (μ)T 0 (ν) = T 0 (ν)T 0 (μ)R 00 (λ) . Proof. We suppress spectral parameters and calculate: R 00 T 0 T 0 = R 00 R 01 R 02 . . . R 0N R 01 . . . R 0N = R 00 R 01 R 01 R 02 . . . R 0N R 02 . . . R 0N = (QYBE) R 01 R 01 R 00 R 02 . . . R 0N R 02 . . . R 0N Here, we first used that the endomorphisms R 0j and R 01 for j ≥ 2 act on different factors of the tensor product V ⊗ V ⊗ V N and thus commute. Then we applied the integrability equation (QYBE) on the indices 0, 0, 1. Repeating this N-times, we get = R 01 R 02 . . . R 0N R 01 . . . R 0N R 00 = T 0 T 0 R 00 ✷ Proposition 4.3.3. Suppose that for the integrable lattice model, the endomorphism R(λ) is invertible for all values λ of the parameters. Then the endomorphism commutes with C(μ) for all values λ, μ. C(λ) := Tr V T (λ) ∈ End (V ⊗N ) We thus have a set of commuting endomorphisms which make the eigenproblem for any operator C(λ) more tractable, hence the name integrable. Proof. 99
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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
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 and 90: This shows that the vectors (b i )
Page 91 and 92: The structure of three-dimensional
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: ✷ 4.3 Interlude: Yang-Baxter equa
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
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,
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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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