Pairing and Quantum Double of Finite Hopf C*-Algebras

Abstract

This paper defines a pairing of two finite Hopf C*-algebras A and B, and investigates the interactions between them. If the pairing is non-degenerate, then the quantum double construction is given. This construction yields a new finite Hopf C*algebra D (A, B). The canonical embedding maps of A and B into the double are both isometric.

The Crossed Product of Finite Hopf C*-Algebra and C*-Algebra

“…Define the map θ(a, b) := ϕ A (a)ϕ B (b), where ϕ A , ϕ B are Haar functionals on A and B, respectively. Then θ is a faithful positive linear functional on D(A, B).Lemma 3.7 in[8] plays an important role in the above assertion. However, the proof of Lemma 3.7 is incomplete.…”

“…Moreover, the quantum symmetry in Hopf spin models is implemented by the quantum double of finite Hopf C * -algebras. In particular, Liu et al [8] investigated the relations between the pairing of two finite Hopf C * -algebras A and B, and then proved that nondegenerate pairing leads to the quantum double D(A, B), which is a new finite Hopf C * -algebra. This result is illustrated by their Lemma 3.7.…”

“…What is more, to reveal the quantum symmetry in quantum spin models, finite Hopf C * -algebra actions are used to construct the quantum double. In particular, Liu et al [8] established a faithful positive linear functional, and then showed that the quantum double of the pairing of two finite Hopf C * -algebras is a C * -algebra. However, the proof of the main result in their work is insufficient.…”

“…However, the proof of Lemma 3.7 is incomplete. Notice that in order to establish the positivity of θ, one needs to verify θ((n ∑ i=1 (a i , b i ))( n ∑ i=1 (a i , b i )) * ) ≥ 0 instead of just θ((a, b)(a, b) * ) ≥ 0.In the same way, θ is faithful means that the condition θ((n ∑ i=1 (a i , b i ))( n ∑ i=1 (a i , b i )) * ) = 0 implies n ∑ i=1 (a i , b i ) = 0.We now give a few words to sketch the proof for Lemma 3.7 in[8] as follows. Forn ∑ i=1 (a i , b i ) ∈ D(A, B), one has θ ( a * j (2) , (b i b * j ) (2) ) a * j (1) , S −1 (b i b * j ) (3) a * j (3) , (b i b * j ) j )(b i b * j ) ϕ A (a i a * j (2) )ϕ B ((b i b * j ) (2) ) a * j (1) , S −1 (b i b * j ) (3) a * j (3) , (b i b * j ) (1)…”