A note on the order of the antipode of a pointed Hopf algebra

Let k be a field and let H denote a pointed Hopf k-algebra with antipode S. We are interested in determining the order of S. Building on the work done by Taft and Wilson in [7], we define an invariant for H, denoted mH, and prove that the value of this invariant is connected to the order of S. In the case where char k = 0, it is shown that if S has finite order then it is either the identity or has order 2 mH. If in addition H is assumed to be coradically graded, it is shown that the order of S is finite if and only if mH is finite. We also consider the case where char k = p > 0, generalizing the results of [7] to the infinite-dimensional setting

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

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.Comment: to appear in Annales de l'Institut Fourie

Quantum homogeneous spaces of connected Hopf algebras

Let H be a connected Hopf k-algebra of finite Gel'fand-Kirillov dimension over an algebraically closed field k of characteristic 0. The objects of study in this paper are the left or right coideal subalgebras T of H. They are shown to be deformations of commutative polynomial k-algebras. A number of well-known homological and other properties follow immediately from this fact. Further properties are described, examples are considered, invariants are constructed and a number of open questions are listed.Comment: 26 pages; comments welcom

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

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