Related papers: Flag Integrable Models and Generalized Graded Alge…
Grassmann and flag varieties lead many lives in pure and applied mathematics. Here we focus on the algebraic complexity of solving various problems in linear algebra and statistics as optimization problems over these varieties. The measure…
Aspects of the algebraic structure and representation theory of the quantum affine superalgebras with symmetrizable Cartan matrices are studied. The irreducible integrable highest weight representations are classified, and shown to be…
In this paper we define a new class of the quantum integrable systems associated with the quantization of the cotangent bundle $T^*(GL(N))$ to the Lie algebra $\frak{gl}_N$. The construction is based on the Gelfand-Zetlin maximal commuting…
A $Z_3$-graded Hopf algebra structure of exterior algebra with two generators is introduced. Two covariant differential calculus on the $Z_3$-graded exterior algebra are presented. Using the generators and their partial derivatives a…
The Adler-Kostant-Symes $R$-bracket scheme is applied to the algebra of pseudo-differential operators to relate the three integrable hierarchies: KP and its two modifications, known as nonstandard integrable models. All three hierarchies…
A new (in)finite dimensional algebra which is a fundamental dynamical symmetry of a large class of (continuum or lattice) quantum integrable models is introduced and studied in details. Finite dimensional representations are constructed and…
We compute the automorphism groups of finite and cofinite ind-grassmannians, as well as of the ind-variety of maximal flags indexed by Z_{>0}. We pay special attention to differences with the case of ordinary flag varieties.
We interpret the equivariant cohomology algebra H^*_{GL_n\times\C^*}(T^*F_\lambda;\C) of the cotangent bundle of a partial flag variety F_\lambda parametrizing chains of subspaces 0=F_0\subset F_1\subset\dots\subset F_N =\C^n, \dim…
A special case of the Fateev-Zamolodchikov model is studied resulting in a solution of the Yang-Baxter equation with two spectral parameters. Integrable models from this solution are shown to have the symmetry of the Drinfeld double of a…
We consider some algebraic and geometric aspects of the theory of integrable systems in finite dimensions, associated with the existence of a classical $r$-matrix, first introduced by Sklyanin as the classical analogue of the quantum…
The direct sum of irreducible level one integrable representations of affine Kac-Moody Lie algebra of (affine) type $ADE$ carries a structure of $P/Q$-graded vertex operator algebra. There exists a filtration on this direct sum studied by…
Given a vector bundle $A\to M$ we study the geometry of the graded manifolds $T^*[k]A[1]$, including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical…
We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relation with Gromov--Witten theory and integrable systems.
In this letter, a new generalized matrix spectral problem of Dirac type associated with the super Lie algebra $\mathcal{B}(0,1)$ is proposed and its corresponding super integrable hierarchy is constructed.
I extend the ramified geometric Satake equivalence of Zhu from tamely ramified groups to include the case of general connected reductive groups. As a prerequisite I prove basic results on the geometry of affine flag varieties.
The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Determinantal formulas, relation to conformal field models and the theory of Generalized Kontsevich model are discussed in some…
We propose a gl(r)-covariant parameterisation of Bethe algebra appearing in so(2r) integrable models, demonstrate its geometric origin from a fused flag, and use it to compute the spectrum of periodic rational spin chains, for various…
We explore integrable Landau-Zener-type Hamiltonians through the framework of Lie algebraic structures. By reformulating the classic two-level Landau-Zener model as a Lax equation, we show that higher-spin generalizations lead to exactly…
We introduce a novel machine learning based framework for discovering integrable models. Our approach first employs a synchronized ensemble of neural networks to find high-precision numerical solution to the Yang-Baxter equation within a…
A new class of associative algebras referred to as affine walled Brauer algebras are introduced. These algebras are free with infinite rank over a commutative ring containing 1. Then level two walled Brauer algebras over C are defined,…