arXiv subjects

Jacek Krajczok

Publications and source records attributed to Jacek Krajczok.

22 records · Page 2Linked to original sources

arXiv2018

Symmetry of eigenvalues of operators associated with representations of compact quantum groups

We ask the question whether for a given unitary representation $U$ the associated operator $ρ_{U}\in\operatorname{Mor}(U,U^{c\, c})$ has spectrum invariant under inversion -- in this case we say that $ρ_{U}$ has symmetric eigenvalues. This does not have to be the case. However, we give affirmative answer whenever a certain condition on the growth of dimensions of irreducible subrepresentations of tensor powers of $U$ is imposed. This condition is satisfied whenever $\widehat{\mathbb{G}}$ is of subexponential growth.

math.QA↗

arXiv2018

Center of the algebra of functions on the quantum group $SU_q(2)$ and related topics

The center of the algebra of continuous functions on the quantum group $SU_q(2)$ is determined as well as centers of other related algebras. Several other results concerning this quantum group are given with direct proofs based on concrete realization of these algebras as algebras of operators on a Hilbert space.

math.OA↗

arXiv2017

Compact quantum groups with representations of bounded degree

We show that a compact quantum group all whose irreducible representations have dimension bounded by a fixed constant must be of Kac type, in other words, its Haar measure is a trace. The proof is based on establishing several facts concerning operators related to modular properties of the Haar measure. In particular we study spectrum of these operators and the dimension of some of their eigenspaces in relation to the quantum dimension of the corresponding irreducible representation.

math.QA↗

arXiv2015

Non-signalling theories and generalized probability

We provide mathematicaly rigorous justification of using term "probability" in connection to the so called non-signalling theories,known also as Popescu's and Rohrlich's box worlds. No only do we prove correctness of these models (in the sense that they describe composite system of two independent subsystems) but we obtain new properties of non-signalling boxes and expose new tools for further investigation. Moreover, it allows strightforward generalization to more complicated systems.

quant-ph↗