Coactions of Hopf algebras on Cuntz algebras and their fixed point algebras

Abstract

Abstract. We study coactions of Hopf algebras coming from compact quantum groups on the Cuntz algebra. These coactions are the natural generalization to the coalgebra setting of the canonical representation of the unitary matrix group U (d) as automorphisms of the Cuntz algebra O d .In particular we study the fixed point subalgebra under the coaction of the quantum compact groups Uq(d) on the Cuntz algebra O d by extending to any dimension d < ∞ a result of Konishi (1992).Furthermore we give a description of the fixed point subalgebra under the coaction of SUq(d) on O d in terms of generators.

Fixed Points of Compact Quantum Groups Actions on Cuntz Algebras

“…Remark 6.7. This proposition complements the explicit description by generators of O α provided for this case by Paolucci (see[Pao97], Lemma 7).…”

“…If (e i ) is an orthonormal basis of H , we define the associated action of on H as the map δ α : H → H ⊗ C( ) given by δ α (e i ) = j e j ⊗ α ji . Note that our conventions differ from those of [Pao97,Ban99].…”

“…It is readily checked [Pao97] that the gauge action γ on O d "commutes" with the coaction of on O d , therefore O α also carries a gauge action and we set F α := O α ∩ F . We use the notation F α,ℓ := O α ∩ H ℓ (H * ) ℓ .…”

“…He gave an explicit description of the fixed point algebra by means of generators and relations. The general case of the natural irreducible representation of = SU q (d) on d was treated by Paolucci in her article [Pao97]. She gave an explicit family of generators based on the infinite braid group B ∞ .…”

Quantum Group Actions on the Cuntz Algebra

“…This paper is motivated by earlier work of one of us [12] on co-actions of (Woronowicz' [15]) quantum groups on Cuntz algebras [2] and by studies by Doplicher and Roberts [4] of group actions on Cuntz algebras.…”