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1. We comment on the results in a language using bases. Let A be a Frobenius algebra. It is finite-dimensional and let (l i ) i=1,...N be a K-basis of A. Since the Frobenius form κ is non-degenerate, we can find another basis (r i ) i=1,...,N such that κ(l i , r i ) = δ ij . Such a pair of bases is called a pair (r i , l i ) of dual bases for the Frobenius form κ. 2. Since (l i ) i=1,...N is a basis, we can write any x ∈ A as a linear combination, x = ∑ N i=1 x il i . Now N∑ κ(x, r j ) = x i κ(l i , r j ) = x j and thus similarly, x = x = i=1 N∑ κ(x, r i )l i for all x ∈ A ; (6) i=1 N∑ κ(l i , x)r i for all x ∈ A . (7) i=1 3. Conversely, given a pair of bases (r i , l i ) such that equation (6) holds for all x ∈ A, we find with x = l j that κ(l i , r i ) = δ ij and conclude that (7) holds for all x ∈ A. 4. Consider for any dual bases the element C := N∑ r i ⊗ l i ∈ A ⊗ A . We claim that it is a Casimir element, i.e. xC = Cx for all x ∈ A. Indeed, implies Similarly, we find Cx = xC = i=1 l i x = N∑ κ(l i x, r i )l i i=1 N∑ r i ⊗ l i x = i=1 N∑ xr i ⊗ l i = i=1 N∑ κ(l i x, r j )r i ⊗ l i . i,j=1 N∑ κ(l i , xr j )r i ⊗ l i . i,j=1 The invariance of the Frobenius form κ now implies xC = Cx. Remark 3.2.21. We can give explicitly a separability idempotent of a finite-dimensional semisimple Hopf algebra. 1. Let λ ∈ H ∗ be a non-zero left integral and let Λ ∈ H be a right integral that λ(Λ) = 1, c.f. proposition 3.1.25. Then we have S(Λ (1) )〈λ, Λ (2) x〉 = S(Λ (1) )Λ (2) x (1) 〈λ, Λ (3) x (2) 〉 [λ cointegral] = x (1) 〈λ, Λx (2) 〉 [S antipode] = x〈λ, Λ〉 = x [Λ right integral, normalization] 72
It follows that the components Λ i (1) of any representation of Δ(Λ) Δ(Λ) = ∑ i Λ i (1) ⊗ Λ i (2) ∈ H ⊗ H form a generating system of H. We can thus find a form for Δ(Λ) such that the components (Λ i (1) ) form a basis. Thus (S(Λ (1)), Λ (2) ) form a dual basis for the standard Frobenius structure on on the Hopf algebra H given by λ. 2. Assume now that H is semisimple. By Maschke’s theorem κ := ɛ(Λ) ≠ 0. Then is a separability idempotent. Indeed, e := κ −1 ∙ S(Λ (1) ) ⊗ Λ (2) ∈ H ⊗ H μ(e) := κ −1 S(Λ (1) ) ∙ Λ (2) = κ −1 1 H ɛ(Λ) = 1 H by the defining property of the antipode. The Casimir property of a separability idempotent follows directly from observation 3.2.20.4, since it is built from a pair of dual bases. 3.3 Powers of the antipode Observation 3.3.1. Let V be a finite-dimensional K-vector space. Under the canonical identification V ∗ ⊗ V → End K (V ) β ⊗ v ↦→ (w ↦→ β(w)v) the trace becomes Tr(β ⊗ v) = β(v). Indeed, with dual bases (e i ) i∈I of V and (e i ) i∈I of V ∗ , we find β = ∑ i β ie i and v = ∑ i vi e i . The corresponding linear map is ∑ i,j β iv j e j ⊗ e i which has trace ∑ i β iv i = β(v). Lemma 3.3.2. Let H be a finite-dimensional Hopf algebra with λ ∈ I l (H ∗ ) and a right integral Λ ∈ H such that λ(Λ) = 1. Let F be a linear endomorphism of H. Then Tr(F ) = 〈λ, F (Λ (2) )S(Λ (1) )〉 . Proof. We know by remark 3.2.21.1 that for all x ∈ H, we have F (x) = 〈λ, F (x)S(Λ (1) )〉Λ (2) . Thus under the identification H ∗ ⊗ H ∼ = End(H), the endomorphism F corresponds to 〈λ, F (−)S(Λ (1) )〉 ⊗ Λ (2) ; thus Tr(F ) = 〈λ, F (Λ (2) )S(Λ (1) )〉 . ✷ 73
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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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Page 33 and 34: Remark 2.4.8. Let (A, μ, Δ) again
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Page 37 and 38: Finally, apply ɛ to the equality
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Page 45 and 46: We discuss a final example. Example
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Page 55 and 56: 3 Finite-dimensional Hopf algebras
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Page 79 and 80: Lemma 3.3.10. Let a ∈ G(H) be the
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Page 83 and 84: Since S 2 χ H = χ H and since S 2
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Page 99 and 100: ✷ 4.3 Interlude: Yang-Baxter equa
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Page 117 and 118: Proposition 4.5.22. This defines a
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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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