Searcharxiv⌕ Search

arXiv subjects

Alexander Stolin

Publications and source records attributed to Alexander Stolin.

At least 19 recordsLinked to original sources
arXiv2024

Generalized classical Yang-Baxter equation and regular decompositions

The focus of the paper is on constructing new solutions of the generalized classical Yang-Baxter equation (GCYBE) that are not skew-symmetric. Using regular decompositions of finite-dimensional simple Lie algebras, we construct Lie algebra decompositions of $\mathfrak{g}(\!(x)\!) \times \mathfrak{g}[x]/x^m \mathfrak{g}[x]$. The latter decompositions are in bijection with the solutions to the GCYBE. Under appropriate regularity conditions, we obtain a partial classification of such solutions. The paper is concluded with the presentations of the Gaudin-type models associated to these solutions.

math.RA↗
arXiv2022

Topological Manin pairs and $(n,s)$-type series

Lie subalgebras of $ L = \mathfrak{g}(\!(x)\!) \times \mathfrak{g}[x]/x^n\mathfrak{g}[x] $, complementary to the diagonal embedding $\Delta$ of $ \mathfrak{g}[\![x]\!] $ and Lagrangian with respect to some particular form, are in bijection with formal classical $r$-matrices and topological Lie bialgebra structures on the Lie algebra of formal power series $ \mathfrak{g}[\![x]\!] $. In this work we consider arbitrary subspaces of $ L $ complementary to $\Delta$ and associate them with so-called series of type $ (n,s) $. We prove that Lagrangian subspaces are in bijection with skew-symmetric $ (n,s) $-type series and topological quasi-Lie bialgebra structures on $ \mathfrak{g}[\![x]\!] $. Using the classificaiton of Manin pairs we classify up to twisting and coordinate transformations all quasi-Lie bialgebra structures. Series of type $ (n,s) $, solving the generalized Yang-Baxter equation, correspond to subalgebras of $L$. We discuss their possible utility in the theory of integrable systems.

math.RA↗
arXiv2022

Topological Lie bialgebra structures and their classification over $ \mathfrak{g}[\![x]\!] $

This paper is devoted to a classification of topological Lie bialgebra structures on the Lie algebra $\mathfrak{g}[\![x]\!]$, where $ \mathfrak{g} $ is a finite-dimensional simple Lie algebra over an algebraically closed field $ F $ of characteristic $ 0 $. We introduce the notion of a topological Manin pair $(L, \mathfrak{g}[\![x]\!])$ and present their classification by relating them to trace extensions of \( F[\![x]\!] \). Then we recall the classification of topological doubles of Lie bialgebra structures on $\mathfrak{g}[\![x]\!]$ and view the latter as a special case of the classification of Manin pairs. The classification of topological doubles states that up to some notion of equivalence there are only three non-trivial doubles. It is proven that topological Lie bialgebra structures on $\mathfrak{g}[\![x]\!]$ are in bijection with certain Lagrangian Lie subalgebras of the corresponding doubles. We then attach algebro-geometric data to such Lagrangian subalgebras and, in this way, obtain a classification of all topological Lie bialgebra structures with non-trivial doubles. When $F = \mathbb{C}$ the classification becomes explicit. Furthermore, this result enables us to classify formal solutions of the classical Yang-Baxter equation.

math.RA↗
arXiv2020

Vandiver's Conjecture via K-theory

Vandiver's conjecture states that any prime p does not divide the class number $h(R)$ of the maximal real subfield R of the p-th cyclotomic field. The aim of this paper is to prove Vandiver's conjecture, which has several consequences including the first case of Fermat's Great Theorem. The main idea lies in using relations of the algebraic K-theory and the Iwasawa theory, discovered by M.Kervaire and M.P.Murthy in 1977.

math.NT↗
arXiv2018

Classification of Quantum Groups via Galois cohomology

The first example of a quantum group was introduced by P.~Kulish and N.~Reshetikhin. In their paper "Quantum linear problem for the sine-Gordon equation and higher representations" published in Zap. Nauchn. Sem. LOMI, 1981, Volume 101 (English version: Journal of Soviet Mathematics, 1983, 23:4), they found a new algebra which was later called $U_q (sl(2))$. Their example was developed independently by V.~Drinfeld and M.~Jimbo, which resulted in the general notion of quantum group. Recently, the so-called Belavin-Drinfeld cohomologies (twisted and untwisted) have been introduced in the literature to study and classify certain families of quantum groups and Lie bialgebras. Later, the last two authors interpreted non-twisted Belavin-Drinfeld cohomologies in terms of non-abelian Galois cohomology $H^1(\mathbb{F}, \mathbf{H})$ for a suitable algebraic $\mathbb{F}$-group $\mathbf{H}$. Here $\mathbb{F}$ is an arbitrary field of zero characteristic. The untwisted case is thus fully understood in terms of Galois cohomology. The twisted case has only been studied using Galois cohomology for the so-called ("standard") Drinfeld-Jimbo structure. The aim of the present paper is to extend these results to all twisted Belavin-Drinfeld cohomologies and thus, to present classification of quantum groups in terms of Galois cohomologies and orders. Our results show that there exist yet unknown quantum groups for Lie algebras of the types $A_n, D_{2n+1}, E_6$.

math.QA↗
arXiv2016

Belavin-Drinfeld quantum groups and Lie bialgebras: Galois cohomology considerations

We relate the Belavin--Drinfeld cohomologies (twisted and untwisted) that have been introduced in the literature to study certain families of quantum groups and Lie bialgebras over a non algebraically closed field $\mathbb K$ of characteristic 0 to the standard non-abelian Galois cohomology $H^1(\mathbb K, \mathbf H)$ for a suitable algebraic $\mathbb K$-group $\mathbf H.$ The approach presented allows us to establish in full generality certain conjectures that were known to hold for the classical types of the split simple Lie algebras.

math.QA↗
arXiv2016

Lie Bialgebras, Fields of Cohomological Dimension at Most 2 and Hilbert's Seventeenth Problem

We investigate Lie bialgebra structures on simple Lie algebras of non-split type $A$. It turns out that there are several classes of such Lie bialgebra structures, and it is possible to classify some of them. The classification is obtained using Belavin--Drinfeld cohomology sets, which are introduced in the paper. Our description is particularly detailed over fields of cohomological dimension at most two, and is related to quaternion algebras and the Brauer group. We then extend the results to certain rational function fields over real closed fields via Pfister's theory of quadratic forms and his solution to Hilbert's Seventeenth Problem.

math.QA↗
arXiv2015

Classification of quantum groups and Belavin--Drinfeld cohomologies for orthogonal and symplectic Lie algebras

In this paper we continue to study Belavin-Drinfeld cohomology introduced in arXiv:1303.4046 [math.QA] and related to the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra. Here we compute Belavin-Drinfeld cohomology for all non-skewsymmetric $r$-matrices from the Belavin-Drinfeld list for simple Lie algebras of type $B$, $C$, and $D$.

math.QA↗
arXiv2014

Classification of quantum groups and Lie bialgebra structures on $sl(n,\mathbb{F})$. Relations with Brauer group

Given an arbitrary field $\mathbb{F}$ of characteristic 0, we study Lie bialgebra structures on $sl(n,\mathbb{F})$, based on the description of the corresponding classical double. For any Lie bialgebra structure $\delta$, the classical double $D(sl(n,\mathbb{F}),\delta)$ is isomorphic to $sl(n,\mathbb{F})\otimes_{\mathbb{F}} A$, where $A$ is either $\mathbb{F}[\varepsilon]$, with $\varepsilon^{2}=0$, or $\mathbb{F}\oplus \mathbb{F}$ or a quadratic field extension of $\mathbb{F}$. In the first case, the classification leads to quasi-Frobenius Lie subalgebras of $sl(n,\mathbb{F})$. In the second and third cases, a Belavin--Drinfeld cohomology can be introduced which enables one to classify Lie bialgebras on $sl(n,\mathbb{F})$, up to gauge equivalence. The Belavin--Drinfeld untwisted and twisted cohomology sets associated to an $r$-matrix are computed. For the Cremmer--Gervais $r$-matrix in $sl(3)$, we also construct a natural map of sets between the total Belavin--Drinfeld twisted cohomology set and the Brauer group of the field $\mathbb{F}$.

math.QA↗
arXiv2013

On classification of quantum groups and Belavin-Drinfeld twisted cohomologies

The present article is a continuation of QA/1303.4046, where we discussed the classification of quantum groups with quasi-classical limit $\mathfrak{g}$ and introduced a theory of Belavin-Drinfeld cohomology associated to any non-skewsymmetric $r$-matrix. Depending on the form of the corresponding double, there exists a one-to-one correspondence between gauge equivalence classes of Lie bialgebra structures on $\mathfrak{g}\otimes_{\mathbb{C}}\mathbb{K}$, where $\mathbb{K}=\mathbb{C}((\hbar))$, and untwisted or twisted cohomology classes. In the present paper we investigate twisted cohomologies for $sl(n)$ associated to generalized Cremmer-Gervais $r$-matrices, and twisted cohomologies for $o(n)$.

math.QA↗
arXiv2013

Classification of quantum groups and Belavin-Drinfeld cohomologies

In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra $\mathfrak{g}$. This problem reduces to the classification of all Lie bialgebra structures on $\mathfrak{g}(\mathbb{K})$, where $\mathbb{K}=\mathbb{C}((\hbar))$. The associated classical double is of the form $\mathfrak{g}(\mathbb{K})\otimes_{\mathbb{K}} A$, where $A$ is one of the following: $\mathbb{K}[\epsilon]$, where $\epsilon^{2}=0$, $\mathbb{K}\oplus \mathbb{K}$ or $\mathbb{K}[j]$ where $j^{2}=\hbar$. The first case relates to quasi-Frobenius Lie algebras. In the second and third cases we introduce a theory of Belavin-Drinfeld cohomology associated to any non-skewsymmetric $r$-matrix from the Belavin-Drinfeld list. We prove a one-to-one correspondence between gauge equivalence classes of Lie bialgebra structures on $\mathfrak{g}(\mathbb{K})$ and cohomology classes (in case II) and twisted cohomology classes (in case III) associated to any non-skewsymmetric $r$-matrix.

math.QA↗
arXiv2010

On Kervaire--Murthy conjecture, Bernoulli and Iwasawa numbers, and zeroes of $p$-adic $L$-function

The aim of the present paper is to establish relations between Iwasawa and Bernoulli numbers based on some results by M. Kervaire and M. P. Murthy about the structure of the $K_0$ groups of the integer group rings of cyclic groups of prime power order $p^n .$ In particular, we will prove that $\lambda_{i}\leq p-1$ under assumption that the generalized Bernoulli number $B_{1,\omega^{-i}}$ is not divisible by $p^2$. Here $\omega$ is the Teichm\"{u}ller character of $\mathbb{Z}/(p-1)\mathbb{Z}$. $\lambda_{i}=1$ if $B_{1,\omega^{-i}}$ is divisible by $p^2$. We will prove that $S_{n,i}\cong \mathbb{Z}/(p^{n+k_i})$, where $S_n$ is the Sylow $p$-subgroup of the class group of the field $\mathbb{Q}(\zeta_n)$. Here, $\zeta_n$ is a primitive $p^{n+1}$-root of unity, $\varepsilon_{i}$ are idempotents in the group ring ${\mathbb Z}_{p}[{\rm Gal}(\mathbb{Q} (\zeta_0) /\mathbb{Q})]$, $S_{n,i}=\varepsilon_i (S_n)$, and $k_i$ is the $p$-adic valuation of $B_{1,\omega^{-i}}$. At the end we will prove that $k_i \leq 1$ and also $v_p (L_p (0, \omega^j))\leq 1$ for even $j$ under certain conditions on zeroes of $L_p (0, \omega^j) .$ Throughout the paper we assume that $p$ satisfies Vandiver's conjecture.

math.NT↗
arXiv2009

Poisson structures compatible with the cluster algebra structure in Grassmannians

We describe all Poisson brackets compatible with the natural cluster algebra structure in the open Schubert cell of the Grassmannian $G_k(n)$ and show that any such bracket endows $G_k(n)$ with a structure of a Poisson homogeneous space with respect to the natural action of $SL_n$ equipped with an R-matrix Poisson-Lie structure. The corresponding R-matrices belong to the simplest class in the Belavin-Drinfeld classification. Moreover, every compatible Poisson structure can be obtained this way.

math.QA↗
arXiv2008

Classification of quasi-trigonometric solutions of the classical Yang-Baxter equation

It was proved by Montaner and Zelmanov that up to classical twisting Lie bialgebra structures on $\mathfrak{g}[u]$ fall into four classes. Here $\mathfrak{g}$ is a simple complex finite-dimensional Lie algebra. It turns out that classical twists within one of these four classes are in a one-to-one correspondence with the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. In this paper we give a complete list of the quasi-trigonometric solutions in terms of sub-diagrams of the certain Dynkin diagrams related to $\mathfrak{g}$. We also explain how to quantize the corresponding Lie bialgebra structures.

math.QA↗
arXiv2004

Dynamical Yang-Baxter equations, quasi-Poisson homogeneous spaces, and quantization

This paper is a continuation of [KS]. We develop the results of [KS] principally in two directions. First, we generalize the main result of [KS], the connection between the solutions of the classical dynamical Yang-Baxter equation and Poisson homogeneous spaces of Poisson Lie groups. We hope that now we present this result in its natural generality. Secondly, we propose a partial quantization of the results of [KS]. [KS] E. Karolinsky and A. Stolin, Classical dynamical r-matrices, Poisson homogeneous spaces, and Lagrangian subalgebras, Lett. Math. Phys., 60 (2002), p.257-274; e-print math.QA/0110319.

math.QA↗
arXiv2002

Fine Structure of Class Groups $\cl^{(p)}\Q(\z_n)$ and the Kervaire--Murthy Conjectures II

There is an Mayer-Vietoris exact sequence involving the Picard group of the integer group ring $\Z C_{p^n}$ where $C_{p^n}$ is the cyclic group of order $p^n$ and $ζ_{n-1}$ is a primitive $p^n$-th root of unity. The "unknown" part of the sequence is a group. $V_n$. $V_n$ splits as $V_n\cong V_n^+\oplus V_n^-$ and $V_n^-$ is explicitly known. $V_n^+$ is a quotient of an in some sense simpler group $\mathcal{V}_n$. In 1977 Kervaire and Murthy conjectured that for semi-regular primes $p$, $V_n^+ \cong \mathcal{V}_n^+ \cong \cl^{(p)}(\Q (ζ_{n-1}))\cong (\mathbb{Z}/p^n \mathbb{Z})^{r(p)}$, where $r(p)$ is the index of regularity of $p$. Under an extra condition on the prime $p$, Ullom calculated $V_n^+$ in 1978 in terms of the Iwasawa invariant $λ$ as $V_n^+ \cong (\mathbb{Z}/p^n \mathbb{Z})^{r(p)}\oplus (\mathbb{Z}/p^{n-1} \mathbb{Z})^{λ-r(p)}$. In the previous paper we proved that for all semi-regular primes, $\mathcal{V}_n^+\cong \cl^{(p)}(\Q (ζ_{n-1}))$ and that these groups are isomorphic to \[(\mathbb{Z}/p^n \mathbb{Z})^{r_0}\oplus (\mathbb{Z}/p^{n-1} \mathbb{Z})^{r_1-r_0} \oplus \hdots \oplus (\mathbb{Z}/p \mathbb{Z})^{r_{n-1}-r_{n-2}} \] for a certain sequence $\{r_k\}$ (where $r_0=r(p)$). Under Ulloms extra condition it was proved that \[V_n^+ \cong \mathcal{V}_n^+ \cong \cl^{(p)}(\Q(\z_{n-1})) \cong (\mathbb{Z}/p^n \mathbb{Z})^{r(p)}\oplus (\mathbb{Z}/p^{n-1}\mathbb{Z})^{λ-r(p)}.\] In the present paper we prove that Ullom's extra condition is valid for all semi-regular primes and it is hence shown that the above result holds for all semi-regular primes.

math.NT↗