Translations in quantum groups

Abstract

Let H be the Hopf C * -algebra of continuous functions on a (locally) compact quantum group of either reduced or full type. We show that endomorphisms of H that respect its right regular comodule structure are translations by elements of the largest classical subgroup of G.Furthermore, we show that for compact G such an endomorphism is automatically an automorphism regardless of the quantum group norm on the C * -algebra H. Introduction.The starting point for the present note is the observation that for a compact group G the map sending g ∈ G to the translationinduces an isomorphism between the group G and the monoid of C * -algebra endomorphisms α : C(G) → C(G) that respect the right comodule structure of C(G) in the sense that

Braided Hopf algebras and gauge transformations

“…These two constructions are one the inverse of the other. This equivalence holds also for locally compact quantum groups [13]. For example, the gauge group of the exention C ⊆ O q (SL(2)), q 2 = 1, is just the multiplicative group C × .…”

Gauge groups and bialgebroids

“…3.1.4]. When H is the Hopf algebra of a compact quantum group G this isomorphism is the result of [10] that equivariant endomorphisms of H are automorphisms and are 'translations' by elements of the largest classical subgroup of G.…”

Gauge groups and bialgebroids

“…3.1.4]. For H the Hopf algebra of a compact quantum group G, this isomorphism is the result of [10] that equivariant endomorphisms of H are automorphisms and are 'translations' by elements of the largest classical subgroup of G.…”