Weighted Bergman spaces¶associated with causal symmetric spaces

Let H\G be a causal symmetric space sitting inside its complexification H _ℂ\G _ℂ. Then there exist certain G-invariant Stein subdomains Ξ of H _ℂ\G _ℂ. The Haar measure on H _ℂ\G _ℂ gives rise to a G-invariant measure on Ξ. With respect to this measure one can define the Bergman space B ²(Ξ) of square integrable holomorphic functions on Ξ. The group G acts unitarily on the Hilbert space B ²(Ξ) by left translations in the arguments. The main result of this paper is the Plancherel Theorem for B ²(Ξ), i.e., the disintegration formula for the left regular representation into irreducibles. Received: Received: 23 November 1998     

Analytic sets extending the graphs of holomorphic mappings

Let D, D′ ⊂ ℂⁿ be bounded domains with smooth real analytic boundaries and ƒ: D → D′ be a proper holomorphic map. Our main result implies that if the graph of ƒ extends as an analytic set to a neighborhood of a poìnt (a, a′) ∈ ∂D × 3D′ with a′ ∈ cl_ƒ(a), then ƒ extends holomorphically to a neighborhood of a.     

One-parameter families of operators in ℂ

We introduce classes of one-parameter families (OPF) of operators on C _c ^t8 (ℂ) which characterize the behavior of operators associated to the problem in the weighted space L² (ℂ, e^−2p) where p is a subharmonic, nonharmonic polynomial. We prove that an order 0 OPF operator extends to a bounded operator from L^q (ℂ) to itself, 1 < q < ∞, with a bound that depends on q and the degree of p but not on the parameter τ or the coefficients of p. Last, we show that there is a one-to-one correspondence given by the partial Fourier transform in τ between OPF operators of order m ≤ 2 and nonisotropic smoothing (NIS) operators of order m ≤ 2 on polynomial models in ℂ².     

Classification of ternary forms with zero Hessian

In this paper we study the differential invariant algebra of the GL₃(ℂ)-action on the space of ternary forms with zero Hessian and classify GL₃(ℂ)-orbits of such forms.     

Non-Autonomous basins of attraction and their boundaries

We study whether the basin of attraction of a sequence of automorphisms of ℂ ^{k is biholomorphic to} ℂ ^{k. In particular, we show that given any sequence of automorphisms with the same attracting fixed point, the basin is biholomorphic to} ℂ ^{k, if every map is iterated sufficiently many times. We also construct Fatou-Bieberbach domains in} ℂ² whose boundaries are four-dimensional.     

Complex<Emphasis Type="BoldItalic">n</Emphasis>-dimensional manifolds with a real<Emphasis Type="BoldItalic">n</Emphasis><Superscript>2</Superscript>-dimensional automorphism group

The main theorem of this article is a characterization of non compact simply connected complete Kobayashi hyperbolic complex manifold of dimension n≽ 2 with real n ²-dimensional holomorphic automorphism group. Together with the earlier work [11, 12] and [13] of Isaev and Krantz, this yields a complete classification of the simply-connected, complete Kobayashi hyperbolic manifolds with dim_ℝ Aut (M) ≽ (dim_ℂ M)².     

On <Emphasis Type="Italic">T</Emphasis> -spaces and relations in relatively free,lie nilpotent,associative algebras

The purpose of this work is to obtain the commutator relations and Frobenius relations in a relatively free algebra F ^(l) specified by the identity [x ₁ , . . . , x _l ] = 0 over a field of characteristic p > 0. These relations for l > 3 are analogous to the relations in the algebra F ⁽³⁾ and are applied to the T-spaces in the algebra F ^(l). In order to study the relations in F ^(l) in more detail, we construct a model algebra analogous to the Grassmann algebra.     

Realizing enveloping algebras via varieties of modules

By using the Ringel-Hall algebra approach, we investigate the structure of the Lie algebra L（A） generated by indecomposable constructible sets in the varieties of modules for any finite- dimensional C-algebra A. We obtain a geometric realization of the universal enveloping algebra R（A） of L（A）, this generalizes the main result of Riedtmann. We also obtain Green＇s formula in a geometric form for any finite-dimensional C-algebra A and use it to give the comultiplication formula in R（A）.     

Sharp and Meager Elements in Orthocomplete Homogeneous Effect Algebras

We prove that every orthocomplete homogeneous effect algebra is sharply dominating. Let us denote the greatest sharp element below x by x ^↓. For every element x of an orthocomplete homogeneous effect algebra and for every block B with x ∈ B, the interval [x ^↓,x] is a subset of B. For every meager element (that means, an element x with x ^↓ = 0), the interval [0,x] is a complete MV-effect algebra. As a consequence, the set of all meager elements of an orthocomplete homogeneous effect algebra forms a commutative BCK-algebra with the relative cancellation property. We prove that a complete lattice ordered effect algebra E is completely determined by the complete orthomodular lattice S(E) of sharp elements, the BCK-algebra M(E) of meager elements and a mapping h:S(E)→2 ^M(E) given by h(a) = [0,a] ∩ M(E).     

Characterizations of G-tube domains

Let G be a real connected Lie group for which the universal complexification G ^ℂ has a polar decomposition G ^ℂ≅G exp(i?), where ? denotes the Lie algebra of G. The present paper is concerned with Riemann G-domains over the complex group G ^ℂ viewed as a G-manifold via the left multiplication. Such a Riemann domain X is said to be of Reinhardt type if G contains a discrete cocompact subgroup $\Gamma$ for whichG/Γ is a Stein manifold. Here the following is proved: Every Riemann G-domain of Reinhardt type is schlicht, hence a G-tube domain, i.e., a G-invariant subdomain of G ^ℂ. As an application one obtains conditions for a holomorphically separable G-manifold to be a G-tube domain. Received: 22 October 1998     

Growth of transcendental entire functions on algebraic varieties

LetX be a complete intersection algebraic variety of codimensionm>1 in ℂ ^m+n . We define the notion of (p,q)-order and (p,q)-K-type for transcendental entire functionsfεO(ℂ ^m+n ) whereK is a non-pluripolar compact subset of ℂ ^m+n . Further, we consider the analogues of (p,q)-order and (p,q)-K-type inO(X). We discuss the series expansions of the functions inO(X) in terms of an orthogonal basis in a Hilbert spaceL ²(X, μ), where μ is a capacitary extremal measure onK. Author is grateful to the NSA for partial support during the period of this research.     

Systems of <Emphasis Type="Italic">sl</Emphasis>(2, ℂ) tops as two-particle systems

We show that Euler-Arnold tops on the algebra sl(2, ℂ) are equivalent to a two-particle system of Calogero type. We show that an arbitrary quadratic Hamiltonian of an sl(2, ℂ) top can be reduced to one of the three canonical Hamiltonians using the automorphism group of the algebra. For each canonical Hamiltonian, we obtain the corresponding two-particle system and write the bosonization formulas for the coadjoint orbits explicitly. We discuss the relation of the obtained formulas to nondynamical Antonov-Zabrodin-Hasegawa R-matrices for Calogero-Sutherland systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 8–21, October, 2008.     

Quantum Spaces Associated to Multipermutation Solutions of Level Two

We study finite set-theoretic solutions (X,r) of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over ℂ with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra A(\mathbbC,X,r)\mathcal{A}(\mathbb{C},X,r) having a q-commutation form of relations determined by complex phase factors. These complex factors are roots of unity and all roots of a prescribed form appear as determined by the representation theory of the finite abelian group G\mathcal{G} of left actions on X. We study the structure of A(\mathbbC,X,r)\mathcal{A}(\mathbb{C},X,r) and show that they have a ∙-product form ‘quantizing’ the commutative algebra of polynomials in |X| variables. We obtain the ∙-product both as a Drinfeld cotwist for a certain canonical 2-cocycle and as a braided-opposite product for a certain crossed G\mathcal{G}-module (over any field k). We provide first steps in the noncommutative differential geometry of A(k,X,r)\mathcal{A}(k,X,r) arising from these results. As a byproduct of our work we find that every such level 2 solution (X,r) factorises as r = f ∘ τ ∘ f ^− 1 where τ is the flip map and (X,f) is another solution coming from X as a crossed G\mathcal{G}-set.     

Weighted Composition Operators from H ∞ to the Bloch Space of a Bounded Homogeneous Domain

Let D be a bounded homogeneous domain in ℂ ⁿ . In this paper, we study the bounded and the compact weighted composition operators mapping the Hardy space H ^∞(D) into the Bloch space of D. We characterize the bounded weighted composition operators, provide operator norm estimates, and give sufficient conditions for compactness. We prove that these conditions are necessary in the case of the unit ball and the polydisk. We then show that if D is a bounded symmetric domain, the bounded multiplication operators from H ^∞(D) to the Bloch space of D are the operators whose symbol is bounded.     

A Remark on a Theorem by Kodama and Shimizu

We prove a characterization theorem for the unit polydisc Δ ⁿ ⊂ℂ ⁿ in the spirit of a recent result due to Kodama and Shimizu. We show that if M is a connected n-dimensional complex manifold such that (i) the group Aut (M) of holomorphic automorphisms of M acts on M with compact isotropy subgroups, and (ii) Aut (M) and Aut (Δ ⁿ ) are isomorphic as topological groups equipped with the compact-open topology, then M is holomorphically equivalent to Δ ⁿ .      

Piecewise Hereditary Nakayama Algebras

Let k be an algebraically closed field. Let Λ be the path algebra over k of the linearly oriented quiver \mathbb A_n\mathbb A_n for n ≥ 3. For r ≥ 2 and n > r we consider the finite dimensional k −algebra Λ(n,r) which is defined as the quotient algebra of Λ by the two sided ideal generated by all paths of length r. We will determine for which pairs (n,r) the algebra Λ(n,r) is piecewise hereditary, so the bounded derived category D ^b (Λ(n,r)) is equivalent to the bounded derived category of a hereditary abelian category H\mathcal H as triangulated category.