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Register a new userSecond cohomology groups of the Hopf$^*$-algebras associated to universal unitary quantum groups
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We compute the second (and the first) cohomology groups of $^*$-algebras associated to the universal quantum unitary groups of not neccesarily Kac type, extending our earlier results for the free unitary group $U_d^+$. The extended setup forces us to use infinite-dimensional representations to construct the cocycles.
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A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of the super line. As applications, solutions to the YBE are given and quandle cocycles are constructed from groupoid cocycles.
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 give a new sufficient condition on a spectral triple to ensure that the quantum group of orientation and volume preserving isometries defined in cite{qorient} has a $C^*$-action on the underlying $C^*$ algebra.
We determine the Kac quotient and the RFD (residually finite dimensional) quotient for the Hopf *-algebras associated to universal orthogonal quantum groups.
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.
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