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Yury A. Neretin

Publications and source records attributed to Yury A. Neretin.

At least 19 recordsLinked to original sources
arXiv2026

Spherical harmonics, operators of multiplication by coordinates, and infinitesimal conformal transformations

Consider the space of $C^\infty$-functions on the two-dimensional sphere $S^2$ and its decomposition $\oplus\mathcal H_n$ into a direct sum of minimal rotation-invariant spaces. We consider elements of $\oplus\mathcal H_n$ as functions of two variables, a nonnegative integer variable $n$ and a complex variable $u$ (a restriction of such function to the set $n=k$ is a polynomial in $u$ of degree $\le 2k$). For operators of multiplication by $x_1$, $x_2$, $x_3$ in $C^\infty(S^2)$ we obtain the corresponding operators in $\oplus\mathcal H_n$, they are differential-difference operators in the variables $u$, $n$ (including second derivatives in $u$ and shifts $n\mapsto n\pm1$). We obtain the similar correspondence for operators of differentiation along conformal vector fields on $S^2$.

math.CA↗
arXiv2026

Riesz products and spectral decompositions for rank 1 measure preserving transformations

We consider rank one measure preserving transformations $g$ and the corresponding Koopman unitary operators $U(g)$. It is known that a generic (in the sense of Baire category) measure preserving transformation has rank one, spectral measure of $U(g)$ is purely singular and is given by a Riesz product. For such transformations we write explicitly spectral decompositions of operators $U(g)$ in generalized eigenfunctions.

math.DS↗
arXiv2026

On a super-Virasoro group, a semigroup of annuli, and Gauss--Berezin integral operators

Denote by ${\mathcal A}$ the Grassmann algebra with a countable number of generators, by ${\mathcal A}_{\overline 0}$, ${\mathcal A}_{\overline 1}$ its even and odd parts. We consider supergroups as groups over $\mathcal A$. Consider the Neveu--Schwarz Lie superalgebra $\mathfrak{ns}= \mathfrak{ns}_{\overline 0}\oplus \mathfrak{ns}_{\overline 1}$. Consider its Grassmannization $\mathfrak{ns}(\mathcal{A}):= (\mathfrak{ns}_{\overline 0}\otimes {\mathcal A}_{\overline 0}) \oplus (\mathfrak{ns}_{\overline 1}\otimes {\mathcal A}_{\overline 1})$ and the corresponding supergroup $\mathrm{NS}(\mathcal {A})$. We describe this group explicitly in different ways. Next, we define a complexification $Γ({\mathcal A})$ of $\mathrm{NS}({\mathcal A})$. It is a semigroup, whose elements are superannuli of dimension $1|1$ equipped with contact structures; the multiplication is gluing of such superannuli. Under some inequalities for parameters, we show that unitary representations of $\mathrm{NS}({\mathcal A})$ admit extensions to representations of $Γ(\mathcal {A})$. We do not assume that the reader has prior knowledge of Lie superalgebras and supergroups.

math.RT↗
arXiv2026

The double gamma function and Vladimir Alekseevsky

This paper is about a forgotten function and a forgotten mathematician. The double gamma function is now an important special function, which appears for different reasons in many branches of mathematics and in mathematical physics, as it satisfies a large and amusing collection of identities parallel to the classical gamma function. It was discovered and investigated in detail by Vladimir Alekseevsky in 1888-89. We outline his starting point here: considering the Weierstrass product for the entire periodic function $\sin πx$ and taking half of the factors corresponding to non-positive roots of the sine, we obtain the function $1/Γ(x)$. Considering the Weierstrass product for the doubly quasiperiodic Jacobi theta function $\vartheta_1$ and taking the half of factors we come to the $q$-gamma function (which appears, with a slight change in notation, in Eduard Heine's work). Considering the quarter of the factors (corresponding to the positive quadrant in the lattice $n_1ω+n_2ω_2$ of periods), Alekseevsky arrived at his double gamma function. The work in this direction was continued by Ernest Barnes, Jean Beaupin, Godfrey Hardy, and Vladimir Steklov in 1899--1907. After 1907, publications on this topic stopped, and the double gamma appeared again 70 years later due to Takuro Shintani. In this paper we discuss Alekseevsky's seminal work and its genesis, the history of the double gamma, and Alekseevsky's biography (1858-1916).

math.HO↗
arXiv2025

Algebra of double cosets of a symmetric group by a smaller symmetric group

Fix a natural $α$. Let $n\ge α$ be an integer. Consider the symmetric group $S_{α+n}$ and its subgroup $S_n$. We consider the group algebra of $S_{α+n}$ and its subalgebra $\mathbb{O}[α;n]$ consisting of $S_n$-biinvariant functions, i.e., functions, which are constant on double cosets of $S_{α+n}$ with respect to $S_n$. We obtain two simple descriptions of $\mathbb{O}[α;n]$. First, we write explicitly formulas for multiplication in a natural basis (structure constants are Pochhammer symbols). Secondly, we describe this algebra in terms of generators and relations. We also construct an interpolating family of algebras $\mathbb{O}[α;ν]$ depending on a complex parameter $ν$.

math.RT↗
arXiv2025

One algebra of double cosets for a general linear group over a finite field

Let $\mathbb {F}_q$ be finite field with $q$ elements. Let $α\leqslant n$ be positive integers. Consider the general linear group $\mathrm{GL}(α+n, \mathbb {F}_q) $ and its subgroup $H(n)$, which fixes the first $α$ basis elements in $\mathbb {F}_q^{α+n}$. Denote $\mathcal{A}_n$ by the convolution algebra of $H(n)$-biinvariant functions on $\mathrm{GL}(α+n, \mathbb {F}_q) $. We describe algebras $\mathcal{A}_n$ in terms of generators and relations and show that the family $\mathcal{A}_n$ admits a natural interpolation to arbitrary complex $n$ (the field $\mathbb {F}_q$ and $α$ are fixed).

math.RT↗
arXiv2025

Oligomorphic groups, categories of partial bijections, and ultrahomogeneous cubic spaces over finite fields

We show that for certain class of oligomorphic groups there is a version of multiplication of double cosets in the Ismagilov--Olshanski sense. Categories of (reduced) double cosets are realized as certain categories of partial bijections. As an example, we consider the ultrahomogeneous linear space $\mathcal{Z}$ of countable dimension over afinite field equipped with a cubic form. The group of automorphisms of $\mathcal{Z}$ is an oligomorphic group, we describe its open subgroups. According Tsankov, this gives a classification of its unitary representations. The category of reduced double cosets in this case is the category of partial linear bijections of finite-dimensional cubic spaces preserving cubic forms.

math.RT↗
arXiv2025

A Lie group corresponding to the free Lie algebra and its universality

Consider the real free Lie algebra $\mathfrak{fr}_n$ with generators $ω_1$, \dots, $ω_n$. Since it is positively graded, it has a completion $\overline{\mathfrak{fr}}_n$ consisting of formal series. By the Campbell--Hausdorff formula, we have a corresponding Lie group $\overline{\mathrm{Fr}}_n$. It is the set $\exp\bigl(\overline{\mathfrak{fr}}_n\bigr)$ in the completed universal enveloping algebra of $\mathfrak{fr}_n$. Also, the group $\overline{\mathrm{Fr}}_n$ is a 'submanifold' in the algebra of formal associative noncommutative series in $ω_1$, \dots, $ω_n$, the 'submanifold' is determined by a certain system of quadratic equations. We consider a certain dense subgroup $\mathrm{Fr}_n^\infty\subset \overline{\mathrm{Fr}}_n$ with a stronger (Polish) topology and show that any homomorphism $π$ from $\mathfrak{fr}_n$ to a real finite-dimensional Lie algebra $\mathfrak{g}$ can be integrated in a unique way to a homomorphism $Π$ from $\mathrm{Fr}_n^\infty$ to the corresponding simply connected Lie group $G$. If $π$ is surjective, then $Π$ also is surjective. Note that Pestov (1993) constructed a separable Banach--Lie group such that any separable Banach--Lie group is its quotient.

math.GR↗
arXiv2025

Some remarks on the group of formal diffeomorphisms of the line

Consider a strictly positively graded finitely generated infinite-dimensional real Lie algebra $\mathfrak{g}$. It has a well-defined Lie group $\overline{\mathbf{G}}$, which is an inverse limit of finite-dimensional nilpotent Lie groups (a pro-unipotent group). Generally, representations (even finite-dimensional representations) of $\mathfrak{g}$ and actions of $\mathfrak{g}$ on manifolds do not admit liftings to $\overline{\mathbf{G}}$. There is a canonically defined dense subgroup $\mathbf{G}^\circ\subset \overline{\mathbf{G}}$ with a stronger (Polish) topology, which admits lifting of representations of $\mathfrak{g}$ in finite-dimensional spaces (and, more generally, of representations of $\mathfrak{g}$ by bounded operators in Banach spaces). We describe this completion for the group $\overline{\mathbf{Diff}}$ of formal diffeomorphisms of the line, i.e., substitutions of the form $x\mapsto x+ p(x)$, where $p(x)=a_2 x^2+\dots$ are formal series, and show that the group $\mathbf{Diff}^\circ$ consists of series with subfactorial growth of coefficients.

math.RT↗
arXiv2024

Algebras of conjugacy classes in symmetric groups

In 1999 V. Ivanov and S. Kerov observed that structure constants of algebras of conjugacy classes of symmetric groups $S_n$ admit a stabilization (in a non-obvious sense) as $n\to \infty$. We extend their construction to a class of pairs of groups $G\supset K$ and algebras of conjugacy classes of $G$ with respect to $K$. In our basic example $G$ is a product of symmetric groups, $G=S_n \times S_n$, $K$ is the diagonal subgroup $S_n$.

math.GR↗
arXiv2024

Non-singular actions of infinite-dimensional groups and polymorphisms

Let $Z$ be a probabilistic measure space with a measure $ζ$, $\mathbb{R}^\times$ be the multiplicative group of positive reals, let $t$ be the coordinate on $\mathbb{R}^\times$. A polymorphism of $Z$ is a measure $π$ on $Z\times Z\times \mathbb{R}^\times$ such that for any measurable $A$, $B\subset Z$ we have $π(A\times Z\times \mathbb{R}^\times)=ζ(A)$ and the integral $\int t\,dπ(z,u,t)$ over $Z\times B\times \mathbb{R}^\times$ is $ζ(B)$. The set of all polymorphisms has a natural semigroup structure, the group of all nonsingular transformations is dense in this semigroup. We discuss a problem of closure in polymorphisms for certain types of infinite dimensional ('large') groups and show that a non-singular action of an infinite-dimensional group generates a representation of its train (category of double cosets) by polymorphisms.

math.DS↗
arXiv2023

On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions

The Selberg integral has a twin (`the Dotsenko--Fateev integral') of the following form. We replace real variables $x_k$ in the integrand $\prod |x_k|^{σ-1}\,|1-x_k|^{τ-1} \prod|x_k-x_l|^{2θ}$ of the Selberg integral by complex variables $z_k$, integration over a cube we replace by an integration over the whole complex space $\mathbb{C}^n$. According to Dotsenko, Fateev, and Aomoto, such integral is a product of Gamma functions. We define and evaluate a family of beta integrals over spaces $\mathbb{C}^m\times \mathbb{C}^{m+1}\times \dots \times \mathbb{C}^n$, which for $m=n$ gives the complex twin of the Selberg integral mentioned above (with three additional integer parameters)

math.CA↗
arXiv2023

On Derkachov--Manashov $R$-matrices for the principal series of unitary representations

In 2001--2013 Derkachov and Manashov with coauthors obtained simple and natural expressions of $R$-matrices for the principal series of representations of the groups $\mathrm{SL}(2,\mathbb{C})$, $\mathrm{SL}(2,\mathbb{R})$, $\mathrm{SL}(n,\mathbb{C})$, $\mathrm{SO}(1,n)$. The Yang--Baxter identities for these intertwining operators are kinds of multivariate hypergeometric transformations. Derivations of the identities are based on calculations 'of physical level of rigor' with divergent integrals. Our purpose is a formal mathematical justification of these results.

math.RT↗
arXiv2023

On the group of homeomorphisms of the Basilica

We show that the group of all homeomorphisms of the Basilica fractal coincides with a group of transformations of a certain non-locally finite ribbon tree. Also, we show that Basilica Thompson group defined by Belk and Forrest is dense in the group of all orientation preserving homeomorphisms of the Basilica.

math.GR↗
arXiv2023

On algebras of double cosets of symmetric groups with respect to Young subgroups

We consider the subalgebra $Δ$ in the group algebra of the symmetric group $G=S_{n_1+\dots+n_ν}$ consisting of all functions invariant with respect to left and right shifts by elements of the Young subgroup $H:=S_{n_1}\times \dots \times S_{n_ν}$. We discuss structure constants of the algebra $Δ$ and construct an algebra with continuous parameters $n_1$ extrapolating algebras $Δ$, it can be also can be rewritten as an asymptotic algebra as $n_j\to\infty$ (for fixed $ν$). We show that there is a natural map from the Lie algebra of the group of pure braids to $Δ$ (and therefore this Lie algebra acts in spaces of multiplicities of the quasiregular representation of the group $G$ in functions on $G/H$).

math.GR↗
arXiv2022

Groups of isometries of ultrametric Urysohn spaces and their unitary representations

We consider groups $\mathbb{I}$ of isometries of ultrametric Urysohn spaces $\mathbb{U}$. Such spaces $\mathbb{U}$ admit transparent realizations as boundaries of certain $R$-trees and the groups $\mathbb{I}$ are groups of automorphisms of these $R$-trees. Denote by $\mathbb{I}[X]\subset \mathbb{I}$ stabilizers of finite subspaces $X\subset \mathbb{U}$. Double cosets $\mathbb{I}[X]\cdot g\cdot \mathbb{I}[Y]$, where $g\in \mathbb{I}$, are enumerated by ultrametrics on union of spaces $X\cup Y$. We construct natural associative multiplications on double coset spaces $\mathbb{I}[X]\backslash \mathbb{I}/\mathbb{I}[X]$ and, more generally, multiplications $\mathbb{I}[X]\backslash \mathbb{I}/\mathbb{I}[Y]\,\times\, \mathbb{I}[Y]\backslash \mathbb{I}/\mathbb{I}[Z]\to \mathbb{I}[X]\backslash \mathbb{I}/\mathbb{I}[Z]$. These operations are a kind of canonical amalgamations of ultrametric spaces. On the other hand, this product can be interpreted in terms of partial isomorphisms of certain $R$-trees (in particular, we come to an inverse category). This allows us to classify all unitary representations of the groups $\mathbb{I}$ and to prove that groups $\mathbb{I}$ have type $I$. We also describe a universal semigroup compactification of $\mathbb{I}$ whose image in any unitary representation of $\mathbb{I}$ is compact.

math.RT↗