The Drinfel’d Double for Group-cograded Multiplier Hopf Algebras

Abstract

Let G be any

Group-cograded Multiplier Hopf ${\left( { * {\text{ - }}} \right)}$ algebras

“…We will see some of the consequences in the rest of the paper (see Sections 3 and 4). We also refer to our work on the quantum double in this context [3].…”

“…Similarly as in a), we have products of functionals with at least one factor in A * and anyway, in both sides of the equation, we have a result with functionals in A * . Using a) the right hand side of the formula in b) can be written as i, (a) f p,i ( · S(a (2) ))( f ( · S(a (1) )) ⊗ e p,i a (3) . Now we use b) of Lemma 4.6 to see that this expression gives i, j,(a) f p, j (S(a (2) ))( f p,i ( f ( · S(a (1) )))) ⊗ e p,i e p, j a (3) = i,(a) f p,i ( f ( · S(a (1) ))) ⊗ e p,i S(a (2) )a (3) = i f p,i ( f ( · S(a))) ⊗ e p,i .…”

“…In a separate paper [3], we follow these ideas to study the quantum double for Hopf group-coalgebras. We generalize the setting, considered in the paper by Zunino [16], and construct a family of associated multiplier Hopf algebras.…”

A Lot of Quasitriangular Group-cograded Multiplier Hopf Algebras

“…We denote the group of multiplier Hopf automorphism by Aut(A). By [6,Lemma 3.3] we get ε f (a) = ε(a) and S A • f = f • S A . Let B = p∈G B p be a G-cograded multiplier Hopf algebra.…”

“…Assume that there is a family of (non-trivial) subalgebras (A p ) p∈G of A so that (i)A = p∈G A p with A p A q = 0 whenever p, q ∈ G and p = q and (ii) (A pq )(1 ⊗ A q ) = A p ⊗ A q and (A p ⊗ 1) (A pq ) = A p ⊗ A q for all p, q ∈ G. Then (A, ) is called a G-cograded multiplier Hopf algebra. The theory of groupcograded multiplier Hopf algebras was further developed in [4,6,7,13] and [14]. In particular in [7], the authors studied quasitriangular group-cograded multiplier Hopf algebras in the following sense: a G-cograded multiplier Hopf algebra with a crossing action ξ is called quasitriangular if there is a multiplier R = α,β∈G R α,β with R α,β ∈ M(A α ⊗ A β ) so that (GQT0): 1 ⊗ b )), for all a ∈ A and b ∈ A q .…”