arXiv subjects
Publications and source records attributed to S. Sinel'shchikov.
A necessary condition is established for a function to be in the image of a quantum Poisson integral operator associated to the Shilov boundary of the quantum matrix ball. A quantum analogue of the Hua equations is introduced.
This paper is devoted to study of differential calculi over quadratic algebras, which arise in the theory of quantum bounded symmetric domains. We prove that in the quantum case dimensions of the homogeneous components of the graded vector spaces of k-forms are the same as in the classical case. This result is well-known for quantum matrices. The quadratic algebras, which we consider in the present paper, are q-analogues of the polynomial algebras on prehomogeneous vector spaces of commutative parabolic type. This enables us to prove that the de Rham complex is isomorphic to the dual of a quantum analogue of the generalized Bernstein-Gelfand-Gelfand resolution.
We consider Knapp-Vogan Hecke algebras in the quantum group setting. This allows us to produce a quantum analogue of the Bernstein functor as a first step towards the cohomological induction for quantum groups.
In this paper we study the Fock representation of a certain $*$-algebra which appears naturally in the framework of quantum group theory. It is also a generalization of the twisted CCR-algebra introduced by W. Pusz and S.~Woronowicz. We prove that the Fock representation is a faithful irreducible representation of the algebra by bounded operators in a Hilbert space, and, moreover, it is the only (up to unitary equivalence) representation possessing these properties. Keywords and phrases: Fock representation, quantum groups, bounded symmetric domain, non-compact Hermitian symmetric spaces
Recent results of the authors on quantum bounded symmetric domains and quantum Harish-Chandra modules are expounded.
The maximum principle for holomirphic functions in the quantum ball is formulated. A proof can be found in [8] (see the bibliography).
We proceed with studying the q-analogues of Cartan domains introduced in q-alg/9703005 and turn to the case of a ball in the space of complex matrices. An explicit expression for a positive $U_q\frak{su}_{nm}$-invariant integral (see math.QA/9803110) is used to define q-analogues for weighted Bergman spaces. The orthogonal projections onto these spaces are integral operators; this work presents an explicit formula for their kernels.
In our preprint q-alg/9703005 q-analogues of bounded symmetric domains were defined to be homogeneous spaces of the associated quantum groups. The investigation of a simplest among those domains, the quantum matrix ball, was started in math.QA/9803110. This work presents a construction of q-analogues for Hardy-Bergman spaces of 'functions in those balls', together with an explicit form of the Bergman kernel. Besides that, two auxiliary results are also established: a boundedness of matrix balls is proved, and de Rham complexes of differential forms with finite coefficients in those balls are constructed.
The present work considers one of the simplest homogeneous spaces of the quantum group SU(1,1), the q-analogue of the unit disc in ${\Bbb C}$. We state without proofs q-analogues of Cauchy-Green formulae, integral representations of eigenfunctions of the Laplace-Beltrami operator, Green functions for Poisson equation and an inversion formula for Fourier transform. It is also demonstrated that the two-parameter quantization of the disc introduced before by S. Klimec and A. Lesniewski, can be derived via an application of the method of F. Berezin.
This work presents proofs of the main results of (math.QA/9808015), except those on q-Berezin transform to appear in a subsequent work. The notation and the results of (math.QA/9808037) and (math.QA/9808047) are used.
We have introduced q-analogues of bounded symmetric domains in our work q-alg/9703005. Given the simplest ones among those, the works q-alg/9603012 and math.QA/9803110 announce the relations describing the algebras of functions, differential and integral calculi. This paper presents the proofs of these results.
We use Berezin's quantization procedure to obtain a formal $U_q su_{1,1}$-invariant deformation of the quantum disc. Explicit formulae for the associated q-bidifferential operators are produced.
A very well known result by Harish-Chandra claims that any Hermitian symmetric space of non-compact type admits a canonical embedding into a complex vector space $V$. The image of this embedding is a bounded symmetric domain in $V$. This work provides a construction of q-analogues of a polynomial algebra on $V$ and the differential algebra of exterior forms on $V$. A way of producing a q-analogue of the bounded function algebra in a bounded symmetric domain is described. All the constructions are illustrated by detailed calculations in the case of the simplest Hermitian symmetric space $SU(1,1)/U(1)$. The development of these ideas can be found in math.QA/9803110 and math.QA/9809038 .
This work contains a proof of theorem 7.3 from math.QA/9808015. This theorem demonstrates the Berezin method to be applicable for producing a well known one-parameter deformation of the quantum disc.
In our earlier work math.QA/9808015 some results on integral representations of functions in quantum disc were announced. It was then shown in math.QA/9808037 that the validity of those results is related to the invariance of kernels of some integral operators. We introduce here a method which allows us to prove the invariance of the above kernels.
Our previous work (math.QA/9808015) introduces the basic notions and announces some results on function theory in the quantum disc. The present paper establishes a relationship between those results and the quantum groups theory.
This work considers the algebras of functions in the quantum matrix ball. An explicit formula for a positive invariant integral is presented.
This work considers a formal deformation of the quantum disc (it is developed via an application of the Berezin method) and presents an explicit formula for this deformation.