pdf file

More documents

Recommendations

Info

Proof. We have already seen in corollary 3.3.4.2 that 3. implies 1. and 2. One can show that 2 implies 1, see Larson and Radford. Here we only show that 1. and 2. together imply 3. Suppose that H and H ∗ are both semisimple and thus, by corollary 3.2.15, unimodular. By corollary 3.3.13.3, then S 4 = (S 2 ) 2 = id. Hence the eigenvalues of S 2 on H and of S 2 | χH H ∗ are all ±1. Call them (μ j) 1≤j≤n with n := dim H and (η i ) 1≤i≤m with m := dim χ H H ∗ . Thus Tr H ∗(S 2 ) = n∑ m μ j and Tr(S 2 | χH H ∗) = ∑ η i , j=1 i=1 By lemma 3.3.18, This implies n ∙ | n∑ μ j = n j=1 m∑ η i | ≤ i=1 m∑ η i . (8) i=1 n∑ |μ j | = n . For a semisimple Hopf algebra, we have seen in corollary 3.3.4 that 0 ≠ TrS 2 = ∑ n j=1 μ j and thus by the equality (8) we find ∑ m i=1 η i ≠ 0. This implies ∑ m i=1 η i = ±1 and, as a further consequence of equation (8) we have ∑ n j=1 μ j = ±n. Since S 2 (1 H ) = 1 H , we have at least one eigenvalue +1. Thus all eigenvalues of S 2 on H have to be +1 which amounts to S 2 = id H . ✷ There are some important results we do not cover in these lectures. The following theorem is proven in [Schneider]: Theorem 3.3.20 (Nichols-Zoeller, 1989). Let H be a finite-dimensional Hopf algebra, and let R ⊂ H be a Hopf subalgebra. Then H is a free R-module. Corollary 3.3.21 (“Langrange’s theorem for Hopf algebras”). If R ⊂ H are finite-dimensional Hopf algebras, then the order of R divides the order of H. We finally refer to chapter 4 of Schneider’s lecture notes [Schneider] for a character theory for finite-dimensional semisimple Hopf algebras that closely parallels the character theory for finite groups. j=1 4 Quasi-triangular Hopf algebras and braided categories 4.1 Interlude: topological field theory In this subsection, we introduce the notion of a topological field theory and investigate lowdimensional topological field theories. To this end, we need more structure on monoidal categories. Let (C, ⊗, a, l, r) be a tensor category. From the tensor product ⊗ : C × C → C, we can get the functor ⊗ opp = ⊗ ◦ τ with V ⊗ opp W := W ⊗ V and f ⊗ opp g := g ⊗ f . 82
It defines a tensor product: given an associator a for ⊗, one verifies that a opp U,V,W := a−1 W,V,U is an associator for the tensor product ⊗ opp . Similarly, one obtains left and right unit constraints. Definition 4.1.1 1. A commutativity constraint for a tensor category (C, ⊗) is a natural isomorphism c : ⊗ → ⊗ opp of functors C × C → C. Explicitly, we have for any pair (V, W ) of objects of C an isomorphism c V,W : V ⊗ W → W ⊗ V such that for all morphisms V commute. f −→ V ′ and W V ⊗ W cV,W W ⊗ V f⊗g g −→ W ′ the diagrams g⊗f V ′ ⊗ W ′ c V ′ ,W ′ W ′ ⊗ V ′ 2. Let C be, for simplicity, a strict tensor category. A braiding is a commutatitivity constraint such that for all objects U, V, W the compatibility relations with the tensor product hold. c U⊗V,W = (c U,W ⊗ id V ) ◦ (id U ⊗ c V,W ) c U,V ⊗W = (id V ⊗ c U,W ) ◦ (c U,V ⊗ id W ) If the category is not strict, the following two hexagon axioms have to hold: c U,V ⊗W U ⊗ (V ⊗ W ) a U,V,W (U ⊗ V ) ⊗ W c U,V ⊗id W (V ⊗ U) ⊗ W a V,U,W (V ⊗ W ) ⊗ U a V,W,U V ⊗ (W ⊗ U) id V ⊗c U,W V ⊗ (U ⊗ W ) and (U ⊗ V ) ⊗ W cU⊗V,W W ⊗ (U ⊗ V ) a −1 a −1 U,V,W W,U,V U ⊗ (V ⊗ W ) (W ⊗ U) ⊗ V id U ⊗c V,W a −1 c U,W ⊗id V U,W,V U ⊗ (W ⊗ V ) (U ⊗ W ) ⊗ V 3. A braided tensor category is a tensor category together with the structure of a braiding. 4. With c UV , also c −1 V U is a braiding. In case, the identity c U,V = c −1 V,U tensor category is called symmetric. 83 holds, the braided
Page 1 and 2:
Hopf algebras, quantum groups and t
Page 3 and 4:
1 Introduction 1.1 Braided vector s
Page 5 and 6:
Proposition 1.2.3. Let (V, c) with
Page 7 and 8:
In the first identity, the identifi
Page 9 and 10:
3. Similarly, denote by I − (V )
Page 11 and 12:
Definition 2.1.7 Let A be a K-algeb
Page 13 and 14:
• We want to encode this informat
Page 15 and 16:
ι g : g → T (g) ↠ T (g)/I(g) =
Page 17 and 18:
We introduce some more language: De
Page 19 and 20:
(a) The functor F is essentially su
Page 21 and 22:
3. A coalgebra map is a linear map
Page 23 and 24:
It is easy to check that a subspace
Page 25 and 26:
in pictures = or in Sweedler notati
Page 27 and 28:
Lemma 2.3.5. Let (A, μ, η, Δ, ɛ
Page 29 and 30:
2. Let G be a group and vect G (K)
Page 31 and 32:
• Compatibility with the right un
Page 33 and 34: Remark 2.4.8. Let (A, μ, Δ) again
Page 35 and 36: Proposition 2.5.5. Let H be a Hopf
Page 37 and 38: 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: Since S 2 χ H = χ H and since S 2
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 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
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 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
show all

pdf file

Create successful ePaper yourself

Delete template?

Save as template?