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The lemma implies that the u-algebra of a non-trivial restricted Lie algebra cannot be isomorphic, as a Hopf algebra, to a group algebra, since it contains no non-trivial group-like elements. Over fields of characteristic zero, we can recover a Lie algebra from the primitive elements in its universal enveloping algebra: Proposition 2.6.14. Let g be a Lie algebra over a field K of characteristic zero with an ordered basis and ι g : g → U(g) its universal enveloping algebra. Then P (U(g)) = ι g (g) . If char(K) = p, then the primitive elements of U(g) is the span of all x pk k ≥ 0. It is a restricted Lie algebra. with x ∈ g and Proof. Define and consider the direct sum: Since x ∈ g is primitive in U(g), we find U n (g) := span K {x n |x ∈ g} Δ(x n ) = U(g) ⊃ n∑ k=0 ∞⊕ U n (g) . n=0 ( n k) x k ⊗ x n−k . Thus the direct sum is a subcoalgebra of U(g) and the coproduct Δ : U(g) → U(g) ⊗ U(g) preserves the degree. The right hand side is thereby provided with the total degree. One checks inductively using the Poincaré-Birkhoff-Witt theorem, that the direct sum is closed under multiplication as well (the multiplication is not homogeneous, though). We can restrict to homogenous elements to investigate the coproduct: Δ(x) = x = l∑ j=1 l∑ λ j (x j ) n ∈ U n (g) j=1 λ j n∑ k=0 ( n k) (x j ) k ⊗ (x j ) n−k Then x is primitive, if and only if all components with bigrade (k, n − k) and 1 ≤ k ≤ n − 1 vanish. Applying multiplication, we find ( ) l∑ n λ j (x j ) n = 0 for all k = 1, . . . , n − 1 . k j=1 Over a field of characteristic zero, this implies x = 0. ✷ 52
3 Finite-dimensional Hopf algebras 3.1 Hopf modules and integrals The goal of this subsection is to introduce the notion of an integral on a Hopf algebra that is fundamental for representation theory and some applications to topological field theory. Hopf modules are an essential tool to this end. Definition 3.1.1 1. Let H be a K-Hopf algebra. A K-vector space V is called a right Hopf module, if • It has the structure of a right (unital) H-module. • It has the structure of a right (counital) H-comodule with right coaction Δ V : V → V ⊗ H. • Δ V is a morphism of right H-modules. 2. If V and W are Hopf modules, a K-linear map f : V → W is a map of Hopf modules, if it is both a module and a comodule map. 3. We denote by M H H the category of right Hopf modules. The categories HM H , H M H and M are defined analogously. H H Remarks 3.1.2. 1. We have in Sweedler notation with Δ V (v) = v (0) ⊗ v (1) where v (0) ∈ V and v (1) ∈ H Δ V (v.x) = v (0) .x (1) ⊗ v (1) ∙ x (2) for all x ∈ H, v ∈ V . 2. Any Hopf algebra H is a Hopf module over itself with action given by multiplication and coaction given by the coproduct. 3. More generally, let K ⊂ H be a Hopf subalgebra. We may consider the restriction of the right action to K, but the full coaction of H to get the category of right (H, K)-Hopf modules M H K . 4. Given any H-module M, the tensor product M ⊗ H is a right H-module by using the product on H. Using Δ M⊗H := id M ⊗ Δ as a coaction, one checks that it becomes a Hopf module. 5. In particular, let M be a trivial H-module, i.e. a K-vector space M with H-action defined by h.m = ɛ(h) ∙ m for all h ∈ H and m ∈ M. In this case, the right action is (m ⊗ k).h = m ⊗ k ∙ h. Such a module is called a trivial Hopf module. We also need the notion of invariants and coinvariants: Definition 3.1.3 Let H be a Hopf algebra. 1. Let M be a left H-module. The invariants of H on M are defined as the K-vector space M H := {m ∈ M | h.m = ɛ(h)m for all h ∈ H} . This defines a functor H−mod → vect(K). For invariants of left modules, the notation H M would be more logical, but is not common. Similarly, invariants are defined for right H-modules. 53
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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: Proof. 1. The equation x = (ɛ ⊗
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
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1. The invertible element Q := R 21
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We will see below that any factoriz
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and trivial coaction K → A ⊗ K
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• Define an associative multiplic
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We leave the proof of the converse
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while for the right hand side, we f
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Proposition 4.5.22. This defines a
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2. A pivotal Hopf algebra is called
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Proof. Identifying I ∼ = I ⊗ I,
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✷ Definition 5.1.11 A spherical c
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Proposition 5.1.17. 1. Let (H, R,
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a fibre functor in the category of
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Remark 5.3.3. 1. If one projects a
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In the third identity, we used the
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The composition is concatenation of
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Proof. One can show that any tangle
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the boundary points, for example gl
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• 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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