Noncommutative transforms and free pluriharmonic functions

In this paper, we study free pluriharmonic functions on noncommutative balls γ[Bn(H)], γ>0, and their boundary behavior. These functions have the form

     

Noncommutative Berezin transforms and multivariable operator model theory

In this paper, we initiate the study of a class of noncommutative domains of n-tuples of bounded linear operators on a Hilbert space H, where m?2, n?2, and p is a positive regular polynomial in n noncommutative indeterminates. These domains are defined by certain positivity conditions on p, i.e.,

     

Poisson transforms adapted to BGG-complexes

We present a new construction of Poisson transforms between vector bundle valued differential forms on homogeneous parabolic geometries and vector bundle valued differential forms on the corresponding Riemannian symmetric space, which can be described in terms of finite dimensional representations of reductive Lie groups. In particular, we use these operators to relate the BGG-sequences on the domain to twisted deRham sequences on the target space. Finally, we explicitly design a family of Poisson transforms between standard tractor valued differential forms for the real hyperbolic space and its boundary which are compatible with the BGG-complex.     

Poisson transforms on vector bundles

Let be a connected real semisimple Lie group with finite center, and a maximal compact subgroup of . Let be an irreducible unitary representation of , and the associated vector bundle. In the algebra of invariant differential operators on the center of the universal enveloping algebra of induces a certain commutative subalgebra . We are able to determine the characters of . Given such a character we define a Poisson transform from certain principal series representations to the corresponding space of joint eigensections. We prove that for most of the characters this map is a bijection, generalizing a famous conjecture by Helgason which corresponds to the trivial representation.

     

Noncommutative,quasi-conformal integral transforms over quaternions and octonions

This paper is devoted to the study of holomorphic and meromorphic functions of quaternion and octonion variables. New classes of quasi-conformal and quasi-meromorphic mappings are defined and studied. Different properties of these functions such as residues and the argument principle are studied. We prove that the family of all quasi-conformal diffeomorphisms of a domain is a topological group relative to composition of mappings. In particular, cases where it is a finite-dimensional Lie group over R are studied. Relations between quasi-conformal functions and integral transformations of functions over quaternions and octonions are investigated. Noncommutative analogs of the Mellin transformations are studied and used. In addition, examples of such functions are given. Finally, applications to problems of complex analysis are discussed. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 52, Functional Analysis, 2008.     

Zeta distributions and boundary values of Poisson transforms

Let G be the conformal group of a non-Euclidean Jordan algebra and let P be the maximal parabolic subgroup canonically associated to G. Standard intertwining operators between spherical degenerate principal series induced from P determine Zeta distributions. In this article, we obtain a functional equations for Zeta distributions by considering boundary values of Poisson transforms. We relate the constant occurring in the Zeta functional equation to that occurring in the functional equation of Wallach's Generalized Jacquet functionals.     

Poisson integrals and Riesz transforms for Hermite function expansions with weights

We consider expansions with respect to the multi-dimensional Hermite functions which are eigenfunctions of the harmonic oscillator L=−Δ+|x|². For the heat-diffusion and Poisson semigroups corresponding to a self-adjoint extension of L we investigate their boundary behaviour and mapping properties. All this is done for functions from L^p(w), 1?p<∞, w∈A_p. Then Riesz transforms and conjugate Poisson integrals are considered. The Riesz transforms occur to be Calderón-Zygmund operators hence their mapping properties follow by using results from a general theory.     

Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures

Let and denote the Gaussian and Poisson measures on , respectively. We show that there exists a unique measure on such that under the Segal-Bargmann transform the space is isomorphic to the space of analytic -functions on with respect to . We also introduce the Segal-Bargmann transform for the Poisson measure and prove the corresponding result. As a consequence, when and have the same variance, and are isomorphic to the same space under the - and -transforms, respectively. However, we show that the multiplication operators by on and on act quite differently on .

     

On the Poisson envelope of a Lie algebra. “Noncommutative” moment space

Moscow State University; e-mail: theodore@mech.math.msu.su, theodore@nw.math.msu.su. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 3, pp. 61–64, July–September, 1995.     

Noncommutative Constrained KP Hierarchy and Multi-component Noncommutative Constrained KP Hierarchy

In this paper, the authors define the noncommutative constrained Kadomtsev-Petviashvili (KP) hierarchy and multi-component noncommutative constrained KP hierarchy. Then they give the recursion operators for the noncommutative constrained KP (NcKP) hierarchy and multi-component noncommutative constrained KP (NmcKP) hierarchy. The authors hope these studies might be useful in the study of D-brane dynamics whose noncommutative coordinates emerge from limits of the M theory and string theory.     

Noncommutative structures

We propose a method for constructing noncommutative analogs of objects from classical calculus, differential geometry, topology, dynamical systems, etc. The standard (commutative) objects can be obtained from noncommutative ones by natural projections (a set of canonical homomorphisms). The approach is ideologically close to the noncommutative geometry of A. Connes but differs from it in technical details.     

Noncommutative schemes

The main purpose of this work is to introduce noncommutative relative schemes and establish some of basic properties of schemes and scheme morphisms. In particular, we prove an analogue of the canonical bijection: ((X, O), Spec(A)) Hom (A, (X, O)). We define a noncommutative version of the ech cohomology of an affine cover and show that the ech cohomology can be used to compute higher direct images. This fact is applied here to compute cohomology of invertible sheaves on skew projective spaces and in [LR3] to study D-modules on quantum flag varieties.     

Noncommutative unitons

By Uhlenbeck’s results, every harmonic map from the Riemann sphere S² to the unitary group U(n) decomposes into a product of so-called unitons: special maps from S² to the Grassmannians Gr _k(ℂⁿ) ⊂ U(n) satisfying certain systems of first-order differential equations. We construct a noncommutative analogue of this factorization, applicable to those solutions of the noncommutative unitary sigma model that are finite-dimensional perturbations of zero-energy solutions. In particular, we prove that the energy of each such solution is an integer multiple of 8π, give examples of solutions that are not equivalent to Grassmannian solutions, and study the realization of non-Grassmannian zero modes of the Hessian of the energy functional by directions tangent to the moduli space of solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 220–239, February, 2008.     

Noncommutative structures

We propose a method for constructing noncommutative analogs of objects from classical calculus, differential geometry, topology, dynamical systems, etc. The standard (commutative) objects can be obtained from noncommutative ones by natural projections (a set of canonical homomorphisms). The approach is ideologically close to the noncommutative geometry of A. Connes but differs from it in technical details. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 203–242.     

Noncommutative topological dynamics

We study noncommutative dynamical systems associated to unimodal and bimodal maps of the interval. To these maps we associate subshifts and the correspondent AF-algebras and Cuntz–Krieger algebras. As an example we consider systems having equal topological entropy log(1 + ϕ), where ϕ is the golden number, but distinct chaotic behavior and we show how a new numerical invariant allows to distinguish that complexity. Finally, we give a statistical interpretation to the topological numerical invariants associated to bimodal maps.