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Oleg K. Sheinman

Publications and source records attributed to Oleg K. Sheinman.

15 recordsLinked to original sources
arXiv2026

Inversion of the Abel--Prym map for real curves with involutions

Riemann vanishing theorem is a main ingredient of the conventional technique related to the Jacobi inversion problem. In the case of curves with a holomorphic involution, it has been presented quite fully in wellknown Fay's Lectures on theta functions. The case of real algebraic curves with involution is presented with less completeness in the literature. We provide a detailed presentation of that case, including the case of real curves of the non-separating type with a holomorphic involution, not considered before with this relation. In particular, we formulate the symmetry of the Prym theta function in this case.

math.AG↗
arXiv2024

Separation of variables for Hitchin systems with the structure group $SO(4)$, on genus two curves

Sets of points giving spectral curves can be regarded as phase coordinates of Hitchin systems. We address the problem of finding out trajectories of Hitchin systems in those coordinates. The problem is being solved for the systems with structure groups $SO(4)$ and $SL(2)$ on genus 2 curves. Our method is a transfer of straight line windings of fibers of the Hitchin map, which are given by Prymians of the spectral curves for the systems with simple classical structure groups. The transfer is carried out by means of an analog of the Jacobi inversion map, which does not exist for Prymians in general but can be defined in the two cases in question.

math-ph↗
arXiv2020

Quantization of Lax integrable systems and Conformal Field Theory

We present the correspondence between Lax integrable systems with spectral parameter on a Riemann surface, and Conformal Field Theories, in quite general set-up suggested earlier by the author. This correspondence turns out to give a prequantization of the integrable systems in question.

math-ph↗
arXiv2017

Some reductions of rank 2 and genera 2 and 3 Hitchin systems

Certain reductions of the rank 2, genera 2 and 3 Hitchin systems are considered, which are shown to give an integrable system of 2, resp. 3, interacting points on the line. It is shown that the reduced systems are particular cases of a certain universal integrable system related to the Lagrange interpolation polynomial. Admissibility of the reduction is proved using computer technique. The corresponding codes are given in the text.

math-ph↗
arXiv2017

Matrix divisors on Riemann surfaces and Lax operator algebras

Matrix divisors are introduced in the work by A.Weil (1938) which is considered as a starting point of the theory of holomorphic vector bundles on Riemann surfaces. In this theory matrix divisors play the role similar to the role of usual divisors in the theory of line bundles. Moreover, they provide explicit coordinates (Tyurin parameters) in an open subset of the moduli space of stable vector bundles. These coordinates turned out to be helpful in integration of soliton equations. We would like to gain attention to one more relationship between matrix divisors of vector G-bundles (where G is a complex semi-simple Lie group) and the theory of integrable systems, namely to the relationship with Lax operator algebras. The result we obtain can be briefly formulated as follows: the moduli space of matrix divisors with certain discrete invariants and fixed support is a homogeneous space. Its tangent space at the unit is naturally isomorphic to the quotient space of M-operators by L-operators, both spaces essentially defined by the same invariants (the result goes back to Krichever, 2001). We give one more description of the same space in terms of root systems.

math.AG↗
arXiv2016

Lax operator algebras and integrable systems

A new class of infinite-dimensional Lie algebras given a name of Lax operator algebras, and the related unifying approach to finite-dimensional integrable systems with spectral parameter on a Riemann surface, such as Calogero--Moser and Hitchin systems, are presented. In particular, our approach includes (the non-twisted) Kac--Moody algebras and integrable systems with rational spectral parameter. The presentation is based on the quite simple ideas related to gradings of semisimple Lie algebras, and their interaction with the Riemann--Roch theorem. The basic properties of the Lax operator algebras, as well as the basic facts of the theory of integrable systems in question, are treated (and proven) from this general point of view, in particular existence of commutative hierarchies and their Hamiltonian properties. We conclude with an application of the Lax operator algebras to prequantization of finite-dimensional integrable systems.

math-ph↗
arXiv2014

Lax operator algebras and gradings on semi-simple Lie algebras

A Lax operator algebra is constructed for an arbitrary semi-simple Lie algebra over $\mathbb C$ equipped with a $\mathbb Z$-grading, and arbitrary compact Riemann surface with marked points. In this set-up, a treatment of almost graded structures, and classification of the central extensions of Lax operator algebras are given. A relation to the earlier approach based on the Tyurin parameters is established.

math.RA↗
arXiv2011

Lax equations and Knizhnik-Zamolodchikov connection

Given a Lax system of equations with the spectral parameter on a Riemann surface we construct a projective unitary representation of the Lie algebra of Hamiltonian vector fields by Knizhnik-Zamolodchikov operators. This provides a prequantization of the Lax system. The representation operators of Poisson commuting Hamiltonians of the Lax system projectively commute. If Hamiltonians depend only on action variables then the corresponding operators commute.

math.RT↗
arXiv2010

Lax operator algebras

In this paper we develop a general concept of Lax operators on algebraic curves introduced in [1]. We observe that the space of Lax operators is closed with respect to their usual multiplication as matrix-valued functions. We construct the orthogonal and symplectic analogs of Lax operators, prove that they constitute almost graded Lie algebras and construct local central extensions of those Lie algebras.

math.RT↗
arXiv2009

Lax operator algebras and Hamiltonian integrable hierarchies

We consider the theory of Lax equations in complex simple and reductive classical Lie algebras with the spectral parameter on a Riemann surface of finite genus. Our approach is based on the new objects -- the Lax operator algebras, and develops the approach of I.Krichever treating the $\gl(n)$ case. For every Lax operator considered as the mapping sending a point of the cotangent bundle on the space of extended Tyrin data to an element of the corresponding Lax operator algebra we construct the hierarchy of mutually commuting flows given by Lax equations and prove that those are Hamiltonian with respect to the Krichever-Phong symplectic structure. The corresponding Hamiltonians give integrable finite-dimensional Hitchin-type systems. For example we derive elliptic $A_n$, $C_n$, $D_n$ Calogero-Moser systems in frame of our approach.

math.AG↗
arXiv2007

Central extensions of Lax operator algebras

Lax operator algebras were introduced by Krichever and Sheinman as a further development of I.Krichever's theory of Lax operators on algebraic curves. These are almost-graded Lie algebras of current type. In this article local cocycles and associated almost-graded central extensions are classified. It is shown that in the case that the corresponding finite-dimensional Lie algebra is simple the two-cohomology space is one-dimensional. An important role is played by the action of the Lie algebra of meromorphic vector fields on the Lax operator algebra via suitable covariant derivatives.

math.QA↗
arXiv2003

The Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras, II

This paper continues the same-named article, Part I (math.QA/9812083). We give a global operator approach to the WZWN theory for compact Riemann surfaces of an arbitrary genus g with marked points. Globality means here that we use Krichever-Novikov algebras of gauge and conformal symmetries (i.e. algebras of global symmetries) instead of loop and Virasoro algebras (which are local in this context). The elements of this global approach are described in Part I. In the present paper we give the construction of conformal blocks and the projective flat connection on the bundle constituted by them.

math.AG↗
arXiv1998

The Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras, I

Elements of a global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus $g$ are given. Sheaves of representations of affine Krichever-Novikov algebras over a dense open subset of the moduli space of Riemann surfaces (respectively of smooth, projective complex curves) with $N$ marked points are introduced. It is shown that the tangent space of the moduli space at an arbitrary moduli point is isomorphic to a certain subspace of the Krichever-Novikov vector field algebra given by the data of the moduli point. This subspace is complementary to the direct sum of the two subspaces containing the vector fields which vanish at the marked points, respectively which are regular at a fixed reference point. For each representation of the affine algebra $3g-3+N$ equations $(\partial_k+T[e_k])Φ=0$ are given, where the elements $\{e_k\}$ are a basis of the subspace, and $T$ is the affine Sugawara representation of the centrally extended vector field algebra. For genus zero one obtains the Knizhnik-Zamolodchikov equations in this way. The coefficients of the equations for genus one are found in terms of Weierstraß-$σ$ function.

math.QA↗
arXiv1995

Sugawara Construction and Casimir Operators for Krichever-Novikov Algebras

We show how to obtain from highest weight representations of Krichever-Novikov algebras of affine type (also called higher genus affine Kac-Moody algebras) representations of centrally extended Krichever-Novikov vector field algebras via the Sugawara construction. This generalizes classical results where one obtains representations of the Virasoro algebra. Relations between the weights of the corresponding representations are given and Casimir operators are constructed. In an appendix the Sugawara construction for the multi-point situation is done.

q-alg↗