arXiv subjects

Paweł Kasprzak

Publications and source records attributed to Paweł Kasprzak.

25 records · Page 2Linked to original sources

arXiv2016

On the Hopf (co)center of a Hopf algebra

The notion of Hopf center and Hopf cocenter of a Hopf algebra is investigated by the extension theory of Hopf algebras. We prove that each of them yields an exact sequence of Hopf algebras. Moreover the exact sequences are shown to satisfy the faithful (co)flatness condition. Hopf center and cocenter are computed for $\mathsf{U}_q(\mathfrak{g})$ and the Hopf algebra $\textrm{Pol}(\mathbb{G}_q)$, where $\mathbb{G}_q$ is the Drinfeld-Jimbo quantization of a compact semisimple simply connected Lie group $\mathbb{G}$ and $\mathfrak{g}$ is a simple complex Lie algebra.

math.QA↗

arXiv2016

Integrable actions and quantum subgroups

We study homomorphisms of locally compact quantum groups from the point of view of integrability of the associated action. For a given homomorphism of quantum groups $Π\colon\mathbb{H}\to\mathbb{G}$ we introduce quantum groups $\mathbb{H}/\!\kerΠ$ and $\overline{\mathrm{im}\,Π}$ corresponding to the classical quotient by kernel and closure of image. We show that if the action of $\mathbb{H}$ on $\mathbb{G}$ associated to $Π$ is integrable then $\mathbb{H}/\!\kerΠ\cong\overline{\mathrm{im}\,Π}$ and characterize such $Π$. As a particular case we consider an injective continuous homomorphism $Π\colon{H}\to{G}$ between locally compact groups $H$ and $G$. Then $Π$ yields an integrable action of $H$ on $L^\infty\;\!\!(G)$ if and only if its image is closed and $Π$ is a homeomorphism of $H$ onto $\mathrm{im}\,Π$. We also give characterizations of open quantum subgroups and of compact quantum subgroups in terms of integrability and show that a closed quantum subgroup always gives rise to an integrable action. Moreover we prove that quantum subgroups closed in the sense of Woronowicz whose associated homomorphism of quantum groups yields an integrable action are closed in the sense of Vaes.

math.OA↗

arXiv2015

Braided quantum SU(2) groups

We construct a family of q-deformations of SU(2) for complex parameters q not equal to 0. For real q, the deformation coincides with Woronowicz' compact quantum SU_q(2) group. For q not real, SU_q(2) is only a braided compact quantum group with respect to a certain tensor product functor for C*-algebras with an action of the circle group.

math.OA↗

arXiv2014

Quantum automorphism groups of finite quantum groups are classical

In a recent paper of Bhowmick, Skalski and Sołtan the notion of a quantum group of automorphisms of a finite quantum group was introduced and, for a given finite quantum group G, existence of the universal quantum group acting on G by automorphisms was proved. We show that this universal quantum group is in fact a classical group. The key ingredient of the proof is the use of multiplicative unitary operators, and we include a thorough discussion of this notion in the context of finite quantum groups.

math.OA↗

arXiv2014

Quantum groups with projection on von Neumann algebra level

We introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study properties. Our approach is slightly different from the one introduced in the thesis of S.~Roy and, unlike the investigations of Roy, we work within the von Neumann algebraic picture. This allows the use of powerful techniques related to crossed products by actions of locally compact quantum groups. In particular we show existence of a "braided comultiplication" on the algebra spanned by slices of "braided multiplicative unitary".

math.OA↗

arXiv2013

Embeddable Quantum Homogeneous Spaces

We discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups. On the von Neumann algebra level we find an interesting duality for such objects. A definition of a quantum homogeneous space is proposed along the lines of the pioneering work of Vaes on induction and imprimitivity for locally compact quantum groups. The concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces. Among various examples we single out the quantum analog of the quotient of the Cartesian product of a quantum group with itself by the diagonal subgroup, analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles.

math.OA↗

arXiv2012

Closed quantum subgroups of locally compact quantum groups

We investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group. Two definitions - one due to S.Vaes and one due to S.L.Woronowicz - are analyzed and relations between them discussed. Among many reformulations we prove that the former definition can be phrased in terms of quasi-equivalence of representations of quantum groups while the latter can be related to an old definition of Podleś from the theory of compact quantum groups. The cases of classical groups, duals of classical groups, compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context. A deep relationship with the quantum group generalization of Herz restriction theorem from classical harmonic analysis is also established, in particular, in the course of our analysis we give a new proof of Herz restriction theorem.

math.OA↗