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Register a new userNew products and $mathbb{Z}_2$-extensions of compact matrix quantum groups
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There are two very natural products of compact matrix quantum groups: the tensor product $Gtimes H$ and the free product $G*H$. We define a number of further products interpolating these two. We focus more in detail to the case where $G$ is an easy quantum group and $H=hat{mathbb{Z}}_2$, the dual of the cyclic group of order two. We study subgroups of $G*hat{mathbb{Z}}_2$ using categories of partitions with extra singletons. Closely related are many examples of non-easy bistochastic quantum groups.
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We study glued tensor and free products of compact matrix quantum groups with cyclic groups -- so-called tensor and free complexifications. We characterize them by studying their representation categories and algebraic relations. In addition, we generalize the concepts of global colourization and alternating colourings from easy quantum groups to arbitrary compact matrix quantum groups. Those concepts are closely related to tensor and free complexification procedures. Finally, we also study a more general procedure of gluing and ungluing.
We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of approximating the identity component as well as the maximal normal (in the sense of Wang) connected subgroup by introducing canonical, but possibly transfinite, sequences of subgroups. These sequences have a trivial behaviour in the classical case. We give examples, arising as free products, where the identity component is not normal and the associated sequence has length 1. We give necessary and sufficient conditions for normality of the identity component and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending chain condition on the representation ring, called Lie property, which characterizes Lie groups in the commutative case and reduces to group Noetherianity of the dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that A_u(F) is not of Lie type. We discuss an example arising from the compact real form of U_q(sl_2) for q<0.
Let $G$ be one of the classical compact, simple, centre-less, connected Lie groups or rank $n$ with a maximal torus $T$, the Lie algebra $clg$ and let ${ E_i, F_i, H_i, i=1, ldots, n }$ be the standard set of generators corresponding to a basis of the root system. Consider the adjoint-orbit space $M={ {rm Ad}_g(H_1),~g in G }$, identified with the homogeneous space $G/L$ where $L={ g in G:~{rm Ad}_g(H_1)=H_1}$. We prove that the `coordinate functions ${ f_i, i=1, ldots, n }$, (where $f_i(g):=lambda_i({rm Ad}_g(H_1))$, ${ lambda_1, ldots, lambda_n}$ is basis of $clg^prime$) are `quadratically independent in the sense that they do not satisfy any nontrivial homogeneous quadratic relations among them. Using this, it is proved that there is no genuine compact quantum group which can act faithtully on $C(M)$ such that the action leaves invariant the linear span of the above cordinate functions. As a corollary, it is also shown that any compact quantum group having a faithful action on the noncommutative manifold obtained by Rieffel deformation of $M$ satisfying a similar `linearity condition must be a Rieffel-Wang type deformation of some compact group.
We prove that a compact quantum group with faithful Haar state which has a faithful action on a compact space must be a Kac algebra, with bounded antipode and the square of the antipode being identity. The main tool in proving this is the theory of ergodic quantum group action on $C^*$ algebras. Using the above fact, we also formulate a definition of isometric action of a compact quantum group on a compact metric space, generalizing the definition given by Banica for finite metric spaces, and prove for certain special class of metric spaces the existence of the universal object in the category of those compact quantum groups which act isometrically and are `bigger than the classical isometry group.
The aim of this paper is to give all quasitriangular structures on a class of semisimple Hopf algebras constructed through abelian extensions of $Bbbkmathbb{Z}_{2}$ by $Bbbk^G$ for an abelian group $G$. We first introduce the concept of symmetry of quasitriangular structures of Hopf algebras and obtain some related propositions which can be used to simplify our calculations of quasitriangular structures. Secondly, we find that quasitriangular structures of these semisimple Hopf algebras can do division-like operations. Using such operations we transform the problem of solving the quasitriangular structures into solving general solutions and giving a special solution. Then we give all general solutions and get a necessary and sufficient condition for the existence of a special solution.
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