3D binary anti-commutative operadic Lax representations for harmonic oscillator

Quantum counterparts of VII$_{a}$, III$_{a=1}$, VI$_{a\neq1}$ over harmonic oscillator

Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of some real three dimensional Lie algebras. The Jacobi operators of these quantum algebras are studied in semiclassical approximation.

非交换离散第一Painlevé层级：Lax对方法

A non-commutative discrete first Painlevé hierarchy: the Lax pair approach

利用与离散第一Painlevé层级相关的等单子问题的非交换类比，构造了该层级的非交换版本d-PI_m^{nc}，并证明了两个层级均可由Svinin多项式S_s^k(n)表示，同时推导了非交换Volterra晶格层级到d-PI_m^{nc}层级的约化，给出了前三个成员的显式连续极限。

操作素算术用于更高色-动能对偶性

Operadic Calculus for Higher Colour-Kinematics Duality

本文提出了一种同调coexact BV代数的概念，用于研究色-动能对偶性及更高阶结构。

VII$^{\hbar}_a$, III$_{a=1}^{\hbar}$, VI$_{a\neq1}^{\hbar}$

Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of some 3d real Lie algebras in Bianchi classification. The Jacobians of these quantum algebras are studied. It is conjectured that the tangent algebras of these quantum algebras are the Heisenberg algebra. From this it follows that the volume element in $\mathbb{R}^{3}$ is quantized by $|(x,y,z)|=4\sqrt{2}(2n+1)$, ($n=0,1,2,\dots$). Thus, the elementary (minimal) length in this model is $l_{min}=2^{5/6}$.

Dynamical deformations of three-dimensional Lie algebras in Bianchi classification over the harmonic oscillator

Operadic Lax representations for the harmonic oscillator are used to construct the dynamical deformations of three-dimensional (3D) real Lie algebras in the Bianchi classification. It is shown that the energy conservation of the harmonic oscillator is related to the Jacobi identities of the dynamically deformed algebras. Based on this observation, it is proved that the dynamical deformations of 3D real Lie algebras in the Bianchi classification over the harmonic oscillator are Lie algebras.

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 alge…

Non-commutative Painleve' equations and Hermite-type matrix orthogonal polynomials

We study double integral representations of Christoffel-Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the Its-Izergin-Korepin-Slavnov (IIKS) theory with a certain Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax pair whose compatibility conditions lead to a non-commutative version of the Painleve' IV differential equation for each family.

格拉斯曼代数上的缠绕Yang-Baxter映射

Entwining Yang-Baxter maps over Grassmann algebras

本文在n阶格拉斯曼代数上构造了集合论缠绕Yang-Baxter方程的双有理映射新解，通过Lax超矩阵重分解和谱曲线模得到不变量生成函数，并研究了固定阶数n所得映射层级的可积性。

The non-commutative Korteweg--de Vries hierarchy and combinatorial Poppe algebra

We give a constructive proof, to all orders, that each member of the non-commutative potential Korteweg-de Vries hierarchy is a Fredholm Grassmannian flow and is therefore linearisable. Indeed we prove this for any linear combination of fields from this hierarchy. That each member of the hierarchy is linearisable, and integrable in this sense, means that the time evolving solution can be generated from the solution to the corresponding linear dispersion equation in the hierarchy, combined with solving an associated linear Fredholm equation representing the Marchenko equation. Further, we show …