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## Quantum polydisk, quantum ball, and a q-analog of Poincaré's theorem

The classical Poincaré theorem (1907) asserts that the polydisk D^n and the ball B^n in C^n are not biholomorphically equivalent for *n* ≥ 2. Equivalently, this means that the Fréchet algebras O(D^n) and O(B^n) of holomorphic functions are not topologically isomorphic. Our goal is to prove a noncommutative version of the above result. Given a nonzero complex number *q,* we define two noncommutative power series algebras O_q(D^n) and O_q(B^n) which can be viewed as q-analogs of O(D^n) and O(B^n), respectively. Both O_q(D^n) and O_q(B^n) are the completions of the algebraic quantum affine space w.r.t. certain families of seminorms. In the case where 0 < *q* < 1, the algebra O_q(B^n) admits an equivalent definition related to L. L. Vaksman's algebra *C_q(B^n)* of continuous functions on the closed quantum ball. We show that both O_q(D^n) and O_q(B^n) can be interpreted as Fréchet algebra deformations (in a suitable sense) of O(D^n) and O(B^n), respectively. Our main result is that O_q(D^n) and O_q(B^n) are not isomorphic if *n* ≥ 2 and |*q*| = 1, but are isomorphic if |*q*| ≠ 1.

We introduce and study holomorphically finitely generated (HFG) Fr\'echet algebras, which are analytic counterparts of affine (i.e., finitely generated) C-algebras. Using a theorem of O. Forsterфn. We also show that the class of HFG algebras is stable under some standard constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions on the quantum affine space and on the quantum torus), the algebras of holomorphic functions on the free polydisk, on the quantum polydisk, and on the quantum ball. We further concentrate on the algebras of holomorphic functions on the quantum polydisk and on the quantum ball and show that they are isomorphic, in contrast to the classical case. Finally, we interpret our algebras as Fr\'echet algebra deformations of the classical algebras of holomorphic functions on the polydisk and on the ball in C^n.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

Let X be a Stein manifold, and let Y be a closed complex submanifold of X. Denote by O(X) the algebra of holomorphic functions on X. We show that the weak (i.e., flat) homological dimension of O(Y) as a Fr'echet O(X)-module equals the codimension of Y in X. In the case where X and Y are of Liouville type, the same formula is proved for the projective homological dimension of O(Y) over O(X). On the other hand, we show that if X is of Liouville type and Y is hyperconvex, then the projective homological dimension of O(Y) over O(X) equals the dimension of X.

We introduce and study noncommutative (or "quantized") versions of the algebras of holomorphic functions on the polydisk and on the ball in **C**^n. Specifically, for each nonzero complex number q we construct Fréchet algebras *O*_q(D^n) and *O*_q(B^n) such that for q=1 they are isomorphic to the algebras of holomorphic functions on the open polydisk D^n and on the open ball B^n, respectively. In the case where 0<q<1, we establish a relation between our holomorphic quantum ball algebra *O*_q(B^n) and L.L.Vaksman's algebra C_q(B^n) of continuous functions on the closed quantum ball. Finally, we show that *O*_q(D^n) and *O*_q(B^n) are not isomorphic provided that |q|=1 and n>1. This result can be interpreted as a q-analog of Poincaré's theorem, which asserts that D^n and B^n are not biholomorphically equivalent unless n=1.

We introduce and study holomorphically finitely generated (HFG) Fr\'echet algebras, which are analytic counterparts of affine (i.e., finitely generated) complex algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of HFG algebras is stable under some natural constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions on the quantum affine space and on the quantum torus), the algebras of holomorphic functions on the free polydisk, on the quantum polydisk, and on the quantum polyannulus.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.