Faculty of Physics University of Warsaw > Events > Seminars > Operator Algebras and Their Applications in Physics

Operator Algebras and Their Applications in Physics

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room 2.23, Pasteura 5 at 13:15

Paweł Józiak (IMPAN)

Hopf images in locally compact quantum goups

In the theory of classical groups, given a (closed) subset of a group one may ask about the (closed) subgroup it generates. The quantum version of this generation procedure is what was called Hopf Image by Banica and Bichon in the context of Hopf algebras. The aim of this talk is to present a uniform method for constructing a closed quantum subgroup generated by a "quantum subset" in the context of locally compact quantum groups. The method we will present is (essentially) consistent with previous constructions: the one for CQG algebras due to Banica and Bichon, the one for compact quantum groups due to Skalski and Sołtan, and with the notion of (finite) generating set of a discrete quantum group due to Vergnioux. We will sketch some of the properties and applications of this construction and, hopefully, present some examples.

room 2.23, Pasteura 5 at 13:15

Tatiana Shulman (IMPAN)

Mathematical aspects of zero-error quantum information theory

In quantum information theory for mathematical description ofquantum channels one uses trace-preserving completely positive maps on matrix (or operator) spaces. I will start with that and then I will focus on some mathematical problems arising in zero-error quantum information theory, namely I will talk on various zero-error capacitiesof quantum channels and superactivation effect. This is a joint work with M. Shirokov.

room 2.23, Pasteura 5 at 13:15

Colin Mrozinski (IMPAN)

A Borel-Bott-Weil theorem for the Quantum Grassmannians (II)

As a continuation of Réamon's talk, we introduce the Quantum Grassmannians Gr_q(k,n) and study them in the light of results on the Quantum Projective space. In particular, we will outline how the coquasitriangular structure on U_q(n) allows us to construct a Hopf Galois extension of the non-universal calculus over Gr_q(k,n). Finally, we will move toward a Borel-Bott-Weil theorem for the Quantum Grassmannians.

room 2.23, Pasteura 5 at 13:15

Paweł Kasprzak (KMMF)

Coideals, quantum subgroups and idempotent states

We characterise coideals of the algebra of functions on a locally compact quantum group G which are assigned to open subgroups by taking the quotient. We shall also extend (beyond coamenable case) Salmi's characterisation of coideals that are assigned to compact subgroups of G . Finally we relate open subgroups of G with a certain class of idempotent states on the dual of G. This is work in progress with F. Khosravi.

room 2.23, Pasteura 5 at 13:15

Piotr M. Sołtan (KMMF)

Rieffel induced representations (III)

room 2.23, Pasteura 5 at 13:15

Réamonn Ó Buachalla (IMPAN)

A Borel-Bott-Weil theorem for the Quantum Grassmannians (I)

We begin by recalling the classical Borel-Bott-Weil theorem for complex projective space. Brzezinski and Majid's theory of quantum principal bundles is then recalled, and quantum projective space presented as an example. A connection form is introduced and the induced covariant derivative for the line bundles examined. The notion of a holomorphic structure for a quantum vector bundle, and its associated space of holomorphic sections, is then presented. Finally, the space of holomorphic of the line bundles of quantum projective space is calculated and shown to satisfy a direct q-generalisation of Borel-Bott-Weil.

room 2.23, Pasteura 5 at 13:15

Alan Czuroń (IMPAN)

On the isomorphisms of Fourier algebras of finite abelian groups

We prove that if G_1 and G_2 are two infinite Vielenkin groups of bounded exponents such that G_1 is not a subgroup of G_2 then there are finite dimensional invariant convolution subalgebras of L^1(G_1) distant from any invariant convolution subalgebras of L^1(G_2). In fact, the norm of any algebraic isomorphims between them grows to infinity with the dimension.

room 2.23, Pasteura 5 at 13:15

Fatemeh Khosravi

Homomorphisms of quantum groups satisfying the integrability condition

room 2.23, Pasteura 5 at 13:15

Piotr Sołtan (KMMF)

Convolutions of functionals and characterization of compact quantum subgroups

room 2.23, Pasteura 5 at 13:15

Paweł Kasprzak (KMMF)

The Hopf (co)center of a Hopf algebra (II)

The notion of the Hopf center and Hopf cocenter of a Hopf algebra isinvestigated using the extension theory of Hopf algebras. We prove thateach of them yields an exact sequence of Hopf algebras. Moreover, the exact sequences are shown to satisfy the faithful (co)flatness condition.The Hopf center and cocenter are computed for the quantized universalenveloping algebra of a simple complex Lie algebra and for the Hopf algebra of the Drinfeld-Jimbo quantization of a compact semisimple simply connected Lie group. Joint work with Alexandru Chirvasitu.In this second part of the talk I will be focused on the concept of a Hopf cocenter.

room 2.23, Pasteura 5 at 13:15

Paweł Kasprzak (KMMF)

The Hopf (co)center of a Hopf algebra

The notion of the Hopf center and Hopf cocenter of a Hopf algebra isinvestigated using the extension theory of Hopf algebras. We prove thateach of them yields an exact sequence of Hopf algebras. Moreover, the exact sequences are shown to satisfy the faithful (co)flatness condition.The Hopf center and cocenter are computed for the quantized universalenveloping algebra of a simple complex Lie algebra and for the Hopf algebra of the Drinfeld-Jimbo quantization of a compact semisimple simply connected Lie group. Joint work with Alexandru Chirvasitu.For those who attended my talk at IMPAN - this will be much more detailed version of what I presented at Noncommutative Geometry Seminar

room 2.23, Pasteura 5 at 13:15

Piotr Sołtan (KMMF)

Rieffel induced representations (II)

room 2.23, Pasteura 5 at 12:30

Piotr Sołtan (KMMF)

Rieffel Induced representations

room 2.23, Pasteura 5 at 13:15

Paweł Kasprzak (KMMF)

Open quantum subgroups of locally compact quantum groups (II)

The notion of an open quantum subgroup of a locally compact quantum group is introduced and given several equivalent characterisations in terms of group-like projections, inclusions of quantum group -algebras and properties of respective quantum homogenous spaces. Open quantum subgroups are shown to be closed in the sense of Vaes and normal open quantum subgroups are proved to be in 1-1 correspondence to compact quantum subgroups of the dual quantum group.(Joint work with Mehrdad Kalantar and Adam Skalski)

room 2.23, Pasteura 5 at 13:15

Paweł Kasprzak (KMMF)

Open quantum subgroups of locally compact quantum groups

room 2.23, Pasteura 5 at 13:15

Adam Skalski (IMPAN)

Fixed points of completely contractive maps, ternary rings of operators and convolution operators on classical and quantum groups

The Choi-Effros product, granting the fixed point space of a unital completely positive map a unique von Neumann algebra structure is the key tool in the construction of the abstract Poisson boundary, generalising the classical concept of a probabilistic-type boundary for a random walk. In this talk we will discuss how replacing the completely positive map by a completely contractive one leads instead to a construction of a (weak*-closed) ternary ring of operators and present some applications to the study of fixed point spaces of contractive convolution operators on classical and quantum locally compact groups.(Mainly based on joint work with Pekka Salmi, Matthias Neufang and Nico Spronk)