arXiv subjects

Pawel Kasprzak

Publications and source records attributed to Pawel Kasprzak.

4 recordsLinked to original sources

arXiv2019

Integrals in left coideal subalgebras and group-like projections

We develop a theory of right group-like projections in Hopf algebras linking them with the theory of left coideal subalgebras with two sided counital integrals. Every right group-like projection is associated with a left coideal subalgebra, maximal among the ones containing the given group-like projection as an integral, and we show that that subalgebra is finite dimensional. We observe that in a semisimple Hopf algebra $H$ every left coideal subalgebra has an integral and we prove a 1-1 correspondence between right group-like projections and left coideal subalgebras of $H$. We provide a number of equivalent conditions for a right group-like projections to be left group-like projection and prove a 1-1 correspondence between semisimple left coideal subalgebras preserved by the squared antipode and two sided group-like projections. We also classify left coideal subalgebras in Taft Hopf algebras $H_{n^2}$ over a field $\mathbb{k}$, showing that the automorphism group splits them into - a class of cardinality $|\mathbb{k}|-1$ of semisimple ones which correspond to right group-like projections which are not two sided; - finitely many semisimple singletons, each corresponding to two sided group-like projection; the number of those singletons for $H_{n^2}$ is equal to the number of divisors of $n$; - finitely many singletons, each non-semisimple and admitting no right group-like projection; the number of those singletons for $H_{n^2}$ is equal to the number of divisors of $n$. In particular we answer the question of Landstad and Van Daele showing that there do exist right group-like projections which are not left group-like projections.

math.QA↗

arXiv2016

Coideals, quantum subgroups and idempotent states

We establish a one to one correspondence between idempotent states on a locally compact quantum group G and integrable coideals in the von Neumann algebra of bounded measurable functions on G that are preserved by the scaling group. In particular we show that there is a 1-1 correspondence between idempotent states on G and psi-expected left invariant von Neumann subalgebras of bounded measurable functions on G. We characterize idempotent states of Haar type as those corresponding to integrable normal coideals preserved by the scaling group. We also establish a one to one correspondence between open subgroups of G and convolutionally central idempotent states on the dual of G. Finally we characterize coideals corresponding to open quantum subgroups of G as those that are normal and admit an atom. As a byproduct of this study we get a number of universal lifting results for Podles condition, normality and regularity and we generalize number of results known before to hold under the coamenability assumption.

math.OA↗

arXiv2014

On a certain approach to quantum homogeneous spaces

We propose a definition of a quantum homogeneous space of a locally compact quantum group. We show that classically it reduces to the notion of a homogeneous spaces. On the quantum level our definition goes beyond the quotient case. It provides a framework which, besides the Vaes' quotient of a locally compact quantum group by its closed quantum subgroup (our main motivation) is also compatible with, generically non-quotient, quantum homogeneous spaces of a compact quantum group studied by P. Podles as well as the Rieffel deformation of G-homogeneous spaces. Finally, our definition rules out the paradoxical examples of the non-compact quantum homogeneous spaces of a compact quantum group.

math.OA↗

arXiv2014

Rieffel deformation of tensor functor and braided quantum groups

In this paper we apply Rieffel deformation to C*- tensor product viewed as a functor on the category of C*-algebras with an abelian group action. In the case of the Rieffel deformation of a quantum group with the action by automorphisms the deformed tensor product enables us to view the deformed object as a braided quantum group. We construct a bicharacter for a braided quantum group and the dual braided quantum group. We employ our method to get a braided quantum Minkowski space. Its description in terms of the deformed space-time coordinates is provided.

math.OA↗