pdf file

More documents

Recommendations

Info

Remark 5.7.3. • Given a modular tensor category C, the invariants can be explicitly computed. We find, e.g. τ(S 1 × S 2 ) = 1 τ(S 3 ) = D −1 • One can consider invariants of a C-coloured link L in an arbitrary three-manifold M: They are topological as well. τ(M, L) ∈ K . In the special case M = S 3 , we find the link invariant τ(S 3 , L) = D −1 F C (L) . If one takes a modular category constructed from representations of a deformation of the universal enveloping algebra of the Lie algebra sl(2) and labels the link with the fundamental representation, one obtains the Jones polynomial from definition 5.3.8. This is shown by checking the Kauffman relations. • We can even construct an extended three-dimensional topological field theory. For modular tensor category extracted from deformed universal enveloping algebras U q (g) with g a semisimple complex Lie algebra, the theory obtained is closely related to threedimensional Chern-Simons gauge theories. This is, unfortunately, beyond the scope of these lectures. The following theorem relates the Turaev-Viro and the Reshetikhin-Turaev invariant: Theorem 5.7.4 (Balsam-Kirillov, Turaev-Virelizier). As extended three-dimensional topological field theories, the Turaev-Viro TFT based on a semisimple ribbon category C and the Reshetikhin-Turaev TFT based on its Drinfeld center Z(C) are isomorphic. Two independent proofs can be found in [BK1] and [TV]. 152
A Facts from linear algebra A.1 Free vector spaces Let K be a fixed field. To any K-vector space, we can associate the underlying set. Any K-linear map is in particular a map of sets. We thus have a so-called forgetful functor U : vect(K) → Set. We also need a functor from sets to K-vector spaces. Definition A.1.1 Let S be a set and K be a field. A K-vector space V (S), together with a map of sets ι S : S → V (S), is called the free vector space on the set X, if for any K-vector space W and any map f : X → W of sets, there is a unique K-linear map ˜f : V (S) → W such that ˜f ◦ ι S = f. As a commuting diagram: ι S S V (S) ∃! f ˜f W Remarks A.1.2. 1. A free vector space, if it exists, is unique. Suppose that (V ′ , ι ′ ) is another free vector space on the same set S. Consider the commutative diagram V (S) ι S φ S ι′ V ′ ι S φ ′ V (S) By the defining property of V (S), applied to the map ι ′ of sets, we find a (unique) K-linear map φ : V (S) → V ′ such that the upper triangle commutes. By the defining property of V ′ , applied to the map ι S of sets, we find a (unique) K-linear map φ ′ : V (S) → V ′ such that the lower triangle commutes. Thus the outer triangle commutes, ι S = φ ′ ◦ φ ◦ ι S . On the other hand, ι S = id V (S) ◦ ι S , and by the defining property of V (S), such a map is unique. Thus φ ′ ◦ φ = id V (S) . Exchanging the roles of V (S) and V ′ , we find φ ◦ φ ′ = id V . Thus V (S) and V are isomorphic. 2. The free vector space exists: take V (S) the set of maps S → K which take value zero almost everywhere. Adding the values of two maps (f + g)(s) := f(s) + g(s) and taking scalar multiplication on the values (λf)(s) := λ ∙ f(s) endows V (S) with the structure of a K-vector space. To define the map ι S , let ι S (s) for s ∈ S be the map which takes value 1 on s and zero on all other elements, ι S (s)(s) = 1 ι S (s)(t) = 0 for t ≠ s . Then the set ι S (S) ⊂ V (S) is a K-basis. The condition ˜f ◦ ι S (s) = f(s) then fixes ˜f on a basis which is possible and unique. This shows that we have constructed the free vector space on the set S. 153
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 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 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: 5.7 Topological field theories of R
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?