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A characterization of coboundary Poisson Lie groups and Hopf algebras

Stanisław Zakrzewski (1997)

Banach Center Publications

We show that a Poisson Lie group (G,π) is coboundary if and only if the natural action of G×G on M=G is a Poisson action for an appropriate Poisson structure on M (the structure turns out to be the well known π + ). We analyze the same condition in the context of Hopf algebras. A quantum analogue of the π + structure on SU(N) is described in terms of generators and relations as an example.

Bargmann representation of q-commutation relations for q > 1 and associated measures

Ilona Królak (2007)

Banach Center Publications

The classical Bargmann representation is given by operators acting on the space of holomorphic functions with the scalar product z | z k q = δ n , k [ n ] q ! = F ( z z ̅ k ) . We consider the problem of representing the functional F as a measure for q > 1. We prove the existence of such a measure and investigate some of its properties like uniqueness and radiality. The above problem is closely related to the indeterminate Stieltjes moment problem.

Canonical commutation relations and interacting Fock spaces

Zied Ammari (2004)

Journées Équations aux dérivées partielles

We introduce by means of reproducing kernel theory and decomposition in orthogonal polynomials canonical correspondences between an interacting Fock space a reproducing kernel Hilbert space and a square integrable functions space w.r.t. a cylindrical measure. Using this correspondences we investigate the structure of the infinite dimensional canonical commutation relations. In particular we construct test functions spaces, distributions spaces and a quantization map which generalized the work of...

Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise

Luigi Accardi, Andreas Boukas (2010)

Banach Center Publications

In the first part of the paper we discuss possible definitions of Fock representation of the *-Lie algebra of the Renormalized Higher Powers of White Noise (RHPWN). We propose one definition that avoids the no-go theorems and we show that the vacuum distribution of the analogue of the field operator for the n-th renormalized power of WN defines a continuous binomial process. In the second part of the paper we present without proof our recent results on the central extensions of RHPWN, its subalgebras...

Factoriality of von Neumann algebras connected with general commutation relations-finite dimensional case

Ilona Królak (2006)

Banach Center Publications

We study a certain class of von Neumann algebras generated by selfadjoint elements ω i = a i + a i , where a i , a i satisfy the general commutation relations: a i a j = r , s t j i r s a r a s + δ i j I d . We assume that the operator T for which the constants t j i r s are matrix coefficients satisfies the braid relation. Such algebras were investigated in [BSp] and [K] where the positivity of the Fock representation and factoriality in the case of infinite dimensional underlying space were shown. In this paper we prove that under certain conditions on the number of generators...

Generalized q-deformed Gaussian random variables

Marek Bożejko, Hiroaki Yoshida (2006)

Banach Center Publications

We produce generalized q-Gaussian random variables which have two parameters of deformation. One of them is, of course, q as for the usual q-deformation. We also investigate the corresponding Wick formulas, which will be described by some joint statistics on pair partitions.

Measures connected with Bargmann's representation of the q-commutation relation for q > 1

Ilona Królak (1998)

Banach Center Publications

Classical Bargmann’s representation is given by operators acting on the space of holomorphic functions with scalar product z n , z k q = δ n , k [ n ] q ! = F ( z n z ¯ k ) . We consider the problem of representing the functional F as a measure. We prove the existence of such a measure for q > 1 and investigate some of its properties like uniqueness and radiality.

On path integration on noncommutative geometries

Achim Kempf (1997)

Banach Center Publications

We discuss a recent approach to quantum field theoretical path integration on noncommutative geometries which imply UV/IR regularising finite minimal uncertainties in positions and/or momenta. One class of such noncommutative geometries arise as `momentum spaces' over curved spaces, for which we can now give the full set of commutation relations in coordinate free form, based on the Synge world function.

On quantum weyl algebras and generalized quons

WŁadysŁaw Marcinek (1997)

Banach Center Publications

The model of generalized quons is described in an algebraic way as certain quasiparticle states with statistics determined by a commutation factor on an abelian group. Quantization is described in terms of quantum Weyl algebras. The corresponding commutation relations and scalar product are also given.

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