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Register a new userSome remarks on the action of Quantum Isometry Groups
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Debashish Goswami
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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.
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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.
Given a spectral triple of compact type with a real structure in the sense of [Dabrowski L., J. Geom. Phys. 56 (2006), 86-107] (which is a modification of Connes original definition to accommodate examples coming from quantum group theory) and references therein, we prove that there is always a universal object in the category of compact quantum group acting by orientation preserving isometries (in the sense of [Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572]) and also preserving the real structure of the spectral triple. This gives a natural definition of quantum isometry group in the context of real spectral triples without fixing a choice of volume form as in [Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572].
In this follow-up of the article: Quantum Group of Isometries in Classical and Noncommutative Geometry(arXiv:0704.0041) by Goswami, where quantum isometry group of a noncommutative manifold has been defined, we explicitly compute such quantum groups for a number of classical as well as noncommutative manifolds including the spheres and the tori. It is also proved that the quantum isometry group of an isospectral deformation of a (classical or noncommutative) manifold is a suitable deformation of the quantum isometry group of the original (undeformed) manifold.
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.
The twin group $T_n$ is a right angled Coxeter group generated by $n- 1$ involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this note, we study some properties of twin groups whose analogues are well-known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups $T_n$ have $R_infty$-property and are not co-Hopfian for $n ge 3$.
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