Some advances about the existence of compact involutions in semisimple Hopf algebras

Abstract

In this paper we show that all complex semisimple Hopf algebras of dimension less than 24 are compact quantum groups. To do this, we survey all the above algebras and show explicitly that they can be described by bicrossed products of group algebras and its duals. We also study the behaviour under twisting of compact quantum groups. Using this we show that certain families of triangular semisimple Hopf algebras are compact quantum groups.

Quasitriangular Hopf algebras of dimension pq^2

“…Here by definition R 12 = R (1) ⊗ R (2) ⊗ 1 and similarly for R 13 and R 23 . The element R is called a universal R-matrix of H or a quasitriangular structure on H. Dually, the definition of coquasitriangular Hopf algebra can be given as follows.…”

“…we have e h g i , e k g j e r g m = (e h g i ) (1) , e r g m (e h g i ) (2) , e k g j . Finally, we will show the following equation (e h g i ) (1) , (e k g j ) (1) (e h g i ) (2) (e k g j ) (2) = (e k g j ) (1) (e h g i ) (1) (e h g i ) (2) , (e k g j ) (2) . (4.1) Since (e h g i ) (1) , (e k g j ) (1) (e h g i ) (2) (e k g j ) (2) = rs=h cd=k e r g i , e c g j (e s g i )(e and R∆(a) = r,s∈G w(r, s)e r (s ⊲ a) ⊗ e s a.…”

“…Finally, we will show the following equation (e h g i ) (1) , (e k g j ) (1) (e h g i ) (2) (e k g j ) (2) = (e k g j ) (1) (e h g i ) (1) (e h g i ) (2) , (e k g j ) (2) . (4.1) Since (e h g i ) (1) , (e k g j ) (1) (e h g i ) (2) (e k g j ) (2) = rs=h cd=k e r g i , e c g j (e s g i )(e and R∆(a) = r,s∈G w(r, s)e r (s ⊲ a) ⊗ e s a. Since e g a ⊗ e g (g ⊲ a) appear in ∆ cop (a)R while not in R∆(a), we know ∆ cop (a)R = R∆(a).…”

On the quasitriangular structures of abelian extensions of $Z_2$

“…Take an odd number n and let t be a primitive nth root of 1 in k, then the Hopf algebras A 2n 2 ,t were defined in [8,Definition 1.2]. By definition, they belong to k G # σ,τ kZ 2 and the data (G, ⊳, σ, τ ) of A 2n 2 ,t can be described as follows (see [1,Section 2.3.4]…”

Quasitriangular structures on abelian extensions of $\mathbb{Z}_{2}$

“…1 ⊳x,t 2 ⊳x) τ (t 2 ,t 1 ) w(t 1 , t 2 ) for t 1 , t 2 ∈ T . Proposition 4.14 above simplifies the test for the condition (v) in Definition 4.5, so it will be used frequently in next sections.…”