Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra

Abstract

Banica and Vergnioux have shown that the dual discrete quantum group of a compact simply connected Lie group has polynomial growth of order the real manifold dimension. We extend this result to a general compact group and its topological dimension, by connecting it with the Gelfand-Kirillov dimension of an algebra. Furthermore, we show that polynomial growth for a compact quantum group G of Kac type implies * -regularity of the Fourier algebra A(G), that is every closed ideal of C(G) has a dense intersection with A(G). In particular, A(G) has a unique C * -norm.

Gelfand-Kirillov dimension of the quantized algebra of regular functions on homogeneous spaces

“…In order to give a precise estimate for the growth exponent, Banica and Vergnioux [1] proved that for a connected simply connected compact real Lie group G, GKdim of the Hopf-algebra O(G) generated by matrix co-efficients of all finite dimensional unitary representations of G is same as manifold dimension of G. In the same article, they mentioned that they do not have any other example of Hopf algebra having polynomial growth. Later DÁndrea, Pinzari and Rossi ( [4]) extended their result to compact Lie groups (see Theorem 3.1, [4]). But apart from these commuatative examples, not much is known about the growth of other Hopf algebras.…”

Quasi-flat representations of uniform groups and quantum groups

“…We have then Irr( Γ) = Γ, and the length function is the usual one on Γ, with respect to S. Thus, the above numbers v n are the volumes of the corresponding balls, and f is their generating series. See [17].…”

Gelfand-Kirillov dimension of cosemisimple Hopf algebras

“…The result in Remark 3.3 was then extended by different methods, close in spirit to what we achieve here, to representative functions on arbitrary compact groups (i.e. regular functions on classical reductive groups) in work of D'Andrea-Pinzari-Rossi[8, Corollary 3.5].…”

“…We introduce necessary terminology and verify the results mentioned above in Section 2. Then, we discuss in Section 3 how our main theorem compares to previous results on the growth of cosemisimple Hopf algebras by Goodearl-Zhang [12], by Banica-Vergnioux [3], and by D'Andrea-Pinzari-Rossi [8]. We also compute in Section 3 the GK-dimensions of Dubois-Violette and Launer's and Mrozinski's universal quantum groups [11,17] using Theorem 2.9.…”