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Oleg Aristov

Publications and source records attributed to Oleg Aristov.

At least 19 recordsLinked to original sources
arXiv2026

Algebras of analytic functionals and homological epimorphisms

It has been proved by the author [arXiv: 2404.19433] that the Arens-Michael envelope of a solvable Lie algebra is a homological epimorphism. We show here that for algebras of analytic functionals on a connected complex Lie group the analogous statement is satisfied without the assumption of solvability, and furthermore the completion homomorphisms of a more general form are also homological epimorphisms, including the envelope with respect to the class of Banach PI-algebras.

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arXiv2026

Homological dimensions of algebras of analytic functionals and their completions

We show that the main homological dimensions of the algebra of analytic functionals on a connected complex Lie group, as well as some of its completions, coincide with the dimension of the simply connected solvable factor in the canonical decomposition of the linearization of this group. Thus, the possible nontriviality of a linearly complex reductive factor does not affect the homological properties of the algebras under consideration.

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arXiv2025

On two versions of holomorphic quantum plane

We find power series descriptions of two versions of holomorphic quantum plane, the Arens--Michael envelope and the envelope with respect to the class of Banach PI algebras, in the case of non-unitary parameter.

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arXiv2025

Finitely $C^\infty$-generated associative and Hopf algebras

We introduce finitely $C^\infty$-generated algebras, which can be treated as `algebras of functions' on non-commutative $C^\infty$-differentiable spaces. Our approach uses the category of projective limits of real Banach algebras of polynomial growth. We prove the existence of some universal constructions in this and some similar categories. By analogy with holomorphically finitely generated algebras of Pirkovskii, a finitely $C^\infty$-generated algebra is defined as a quotient of a finite-rank algebra of `free $C^\infty$-functions'. The latter notion was introduced by the author in a previous article, where a structure theorem for algebras of `free $C^\infty$-functions' was announced and proved in dimension at most $2$. Here this theorem is proved in full generality. The central result asserts that the projective tensor product of a finite tuple of finitely $C^\infty$-generated algebras is finitely $C^\infty$-generated. In particular, this makes it natural to consider finitely $C^\infty$-generated topological Hopf algebras. Furthermore, a construction called `envelope' provides a functor from the category of affine real Hopf algebras to the category of finitely $C^\infty$-generated Hopf algebras.

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arXiv2024

Holomorphically finitely generated Hopf algebras and quantum Lie groups

We study topological Hopf algebras that are holomorphically finitely generated (HFG) as Fréchet Arens--Micheal algebras in the sense of Pirkovskii. Some of them, but not all, can be obtained from affine Hopf algebras by applying the analytization functor. We show that a commutative HFG Hopf algebra is always an algebra of holomorphic functions on a complex Lie group (actually a Stein group), and prove that the corresponding categories are equivalent. With a compactly generated complex Lie group~$G$, Akbarov associated a cocommutative topological Hopf algebra, the algebra ${\mathscr A}_{exp}(G)$ of exponential analytic functionals. We show that it is HFG but not every cocommutative HFG Hopf algebra is of this form. In the case when $G$ is connected, using previous results of the author we establish a theorem on the analytic structure of ${\mathscr A}_{exp}(G)$. It depends on the large-scale geometry of $G$. We also consider some interesting examples including complex-analytic analogues of classical $\hbar$-adic quantum groups.

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arXiv2024

Decomposition of the algebra of analytic functionals on a connected complex Lie group and its completions into iterated analytic smash products

We show that a decomposition of a complex Lie group $G$ into a semidirect product generates that of the algebra of analytic functional, ${\mathscr A}(G)$, into an analytic smash product in the sense of Pirkovskii. Also we find sufficient conditions for a semidirect product to generate similar decompositions of certain Arens-Michael completions of ${\mathscr A}(G)$. The main result: if $G$ is connected, then its linearization admits a decomposition into an iterated semidirect product (with the composition series consisting of abelian factors and a semisimple factor) that induces a decomposition of algebras in a class of completions of ${\mathscr A}(G)$ into iterated analytic smash products. Considering the extreme cases, the envelope of ${\mathscr A}(G)$ in the class of all Banach algebras (aka the Arens-Michael envelope) and the envelope in the class Banach PI-algebras (a new concept that is introduced in this article), we decompose, in particular, these envelopes into iterated analytic smash products.

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arXiv2023

Length functions exponentially distorted on subgroups of complex Lie groups

We introduce a notion of a length function exponentially distorted on a (compactly generated) subgroup of a locally compact group. We prove that for a connected linear complex Lie group there is a maximum equivalence class of length functions exponentially distorted on a normal integral subgroup lying between the exponential and nilpotent radicals. Moreover, a function in this class admits an asymptotic decomposition similar to that previously found by the author for word length functions, i.e., in the case of exponential radical [J. Lie Theory 29:4, 1045--1070, 2019]. In the general case we use auxiliary length functions constructed via holomorphic homomorphisms to Banach PI-algebras.

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arXiv2023

The structure of the linearizer of a connected complex Lie group

The Morimoto theorem states that each connected abelian complex Lie group $A$ can be decomposed into the direct product of a group on which all holomorphic functions are constant, finitely many copies of $\mathbb{C}^\times$ and a vector group. We prove that if $A$ is the complex linearizer of a connected complex Lie group then the last factor of the product is trivial.

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arXiv2023

Holomorphic reflexivity for locally finite and profinite groups: the abelian and general cases

Akbarov's theory of holomorphic reflexivity for topological Hopf algebras has been developed in two directions, namely, by the complication of definitions when expanding the scope and by their simplification when restricting. In the framework of the latter approach, we establish the holomorphic reflexivity for topological Hopf algebras associated with locally finite countable groups and second-countable profinite groups. In the Abelian case, the reflexivity is described in terms close to the classical ones.

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arXiv2023

Sheaves of non-commutative smooth and holomorphic functions associated with the non-abelian two-dimensional Lie algebra

Dosi and, quite recently, the author showed that, on the character space of a nilpotent Lie algebra, there exists a sheaf of Fréchet--Arens--Michael algebras (of noncommutative holomorphic functions in the complex case and of noncommutative smooth functions in the real case). We construct similar sheaves (both versions, holomorphic and smooth) on a special space of representations for the Lie algebra of the group of affine transformations of the real line (which is the simplest nonnilpotent solvable Lie algebra).

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arXiv2023

The relation "commutator equals function'' in Banach algebras

The relation $xy-yx=h(y)$, where $h$ is a holomorphic function, occurs naturally in the definitions of some quantum groups. To attach a rigorous meaning to the right-hand side of this equality, we assume that $x$ and $y$ are elements of a Banach algebra (or of an Arens--Michael algebra). We prove that the universal algebra generated by a commutation relation of this kind can be represented explicitly as an analytic Ore extension. An analysis of the structure of the algebra shows that the set of holomorphic functions of $y$ degenerates, but at each zero of $h$, some local algebra of power series remains. Moreover, this local algebra depends only on the order of the zero. As an application, we prove a result about closed subalgebras of holomorphically finitely generated algebras.

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arXiv2023

Banach space representations of Drinfeld-Jimbo algebras and their complex-analytic forms

We prove that every non-degenerate Banach space representation of the Drinfeld-Jimbo algebra $U_q(\mathfrak{g})$ of a semisimple complex Lie algebra $\mathfrak{g}$ is finite dimensional when $|q|\ne 1$. As a corollary, we find an explicit form of the Arens-Michael envelope of $U_q(\mathfrak{g})$, which is similar to that of $U(\mathfrak{g})$ obtained by Joseph Taylor in 70s. In the case when $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_2$, we also consider the representation theory of the corresponding analytic form $\widetilde U(\mathfrak{s}\mathfrak{l}_2)_\hbar$ (with $e^\hbar=q$) and show that it is simpler than for $U_q(\mathfrak{s}\mathfrak{l}_2)$. For example, all irreducible continuous representations of $\widetilde U(\mathfrak{s}\mathfrak{l}_2)_\hbar$ are finite dimensional for every admissible value of the complex parameter $\hbar$, while $U_q(\mathfrak{s}\mathfrak{l}_2)$ has a topologically irreducible infinite-dimensional representation when $|q|= 1$ and $q$ is not a root of unity.

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arXiv2023

Functions of class $C^\infty$ in non-commuting variables in the context of triangular Lie algebras

We construct a certain completion $C^\infty_\mathfrak{g}$ of the universal enveloping algebra of a triangular real Lie algebra $\mathfrak{g}$. It is a Fréchet-Arens-Michael algebra that consists of elements of polynomial growth and satisfies to the following universal property: every Lie algebra homomorphism from $\mathfrak{g}$ to a real Banach algebra all of whose elements are of polynomial growth has an extension to a continuous homomorphism with domain~$C^\infty_\mathfrak{g}$. Elements of this algebra can be called functions of class $C^\infty$ in non-commuting variables. The proof is based on representation theory and employs an ordered $C^\infty$-functional calculus. Beyond the general case, we analyze two simple examples. As an auxiliary material, the basics of the general theory of algebras of polynomial growth are developed. We also consider local variants of the completion and obtain a sheaf of non-commutative functions on the Gelfand spectrum of $C^\infty_\mathfrak{g}$ in the case when $\mathfrak{g}$ is nilpotent. In addition, we discuss the theory of holomorphic functions in non-commuting variables introduced by Dosi and apply our methods to prove theorems strengthening some his results.

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arXiv2022

Holomorphic functions of exponential type on connected complex Lie groups

Holomorphic functions of exponential type on a complex Lie group $G$ (introduced by Akbarov) form a locally convex algebra, which is denoted by $\cO_{exp}(G)$. Our aim is to describe the structure of $\cO_{exp}(G)$ in the case when $G$ is connected. The following topics are auxiliary for the claimed purpose but of independent interest: (1) a characterization of linear complex Lie group (a~result similar to that of Luminet and Valette for real Lie groups); (2) properties of the exponential radical when $G$ is linear; (3) an asymptotic decomposition of a word length function into a sum of three summands (again for linear groups). The main result presents $\cO_{exp}(G)$ as a complete projective tensor of three factors, corresponding to the length function decomposition. As an application, it is shown that if $G$ is linear then the Arens-Michael envelope of $\cO_{exp}(G)$ is just the algebra of all holomorphic functions.

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arXiv2022

When a completion of the universal enveloping algebra is a Banach PI-algebra?

We prove that a Banach algebra $B$ that is a completion of the universal enveloping algebra of a finite-dimensional complex Lie algebra $\mathfrak{g}$ satisfies a polynomial identity if and only if the nilpotent radical $\mathfrak{n}$ of $\mathfrak{g}$ is associatively nilpotent in $B$. Furthermore, this holds if and only if a certain polynomial growth condition is satisfied on $\mathfrak{n}$.

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arXiv2022

An analytic criterion for the local finiteness of a countable semigroup

We prove that a countable semigroup $S$ is locally finite if and only if the Arens-Michael envelope of its semigroup algebra is a $(DF)$-space. This is a counterpart to a recent result of the author, which asserts that $S$ is finitely generated if and only if the Arens-Michael envelope is a Fréchet space.

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