Igor M. Krichever, Oleg K. Sheinman
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.
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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 wit…
Lax operator algebras for the root system $G_2$, and arbitrary finite genus Riemann surfaces and Tyurin data on them are constructed.
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.
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.
$W$-algebras are certain algebraic structures associated to a finite dimensional Lie algebra $\mathfrak g$ and a nilpotent element $f$ via Hamiltonian reduction. In this note we give a review of a recent approach to the study of (classical affine and quantum finite) $W$-algebras based on the notion of Lax type operators. For a finite dimensional representation of $\mathfrak g$ a Lax type operator for $W$-algebras is constructed using the theory of generalized quasideterminants. This operator carries several pieces of information about the structure and properties of the $W$-algebras and shows …
Finite $W$-algebra invariants via Lax type operators
本文利用 Lax 型算子的变体，为 Dynkin 分次为偶数的 B、C、D 型有限 $W$-代数构造了显式生成元。
Recently, Lax operator algebras appeared as a new class of higher genus current type algebras. Based on I.Krichever's theory of Lax operators on algebraic curves they were introduced by I. Krichever and O. Sheinman. These algebras are almost-graded Lie algebras of currents on Riemann surfaces with marked points (in-points, out-points, and Tyurin points). In a previous joint article of the author with Sheinman the local cocycles and associated almost-graded central extensions are classified in the case of one in-point and one out-point. It was shown that the almost-graded extension is essential…
Lax functoriality of Hochschild cochain complex
本文证明dg范畴的Hochschild上链复形具有松弛函子性，推广了代数中心松弛函子性及Keller的受限函子性至全局情形。
Using the Lax operator formalism, we construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables and of two parameters $q,t$ are their eigenfunctions. We express our operators in terms of the Hall-Littlewood symmetric functions of the same variables and of the parameter $t$ corresponding to the partitions with one part only. Our expression is based on the notion of Baker-Akhiezer function.
For a reductive Lie algebra g, its nilpotent element f and its faithful finite dimensional representation, we construct a Lax operator L(z) with coefficients in the quantum finite W-algebra W(g,f). We show that for the classical linear Lie algebras gl_N, sl_N, so_N and sp_N, the operator L(z) satisfies a generalized Yangian identity. The operator L(z) is a quantum finite analogue of the operator of generalized Adler type which we recently introduced in the classical affine setup. As in the latter case, L(z) is obtained as a generalized quasideterminant.
We present a family of novel Lax operators corresponding to representations of the RTT-realisation of the Yangian associated with $D$-type Lie algebras. These Lax operators are of oscillator type, i.e. one space of the operators is infinite-dimensional while the other is in the first fundamental representation of $\mathfrak{so}(2r)$. We use the isomorphism between the first fundamental representation of $D_3$ and the $\mathbf{6}$ of $A_3$, for which the degenerate oscillator type Lax matrices are known, to derive the Lax operators for $r=3$. The results are used to generalise the Lax matrices …
Jack Littlewood-Richardson Coefficients and the Nazarov-Sklyanin Lax Operator
本文研究 Nazarov-Sklyanin 量子 Lax 算子多项式本征函数性质，通过考虑本征函数乘积推广 Kerov 公式，推导出基于分区乘法运算的 Jack Littlewood-Richardson 系数约束系统。
We construct a family of $GL_n$ rational and trigonometric Lax matrices $T_D(z)$ parametrized by $Λ^+$-valued divisors $D$ on $\mathbb{P}^1$. To this end, we study the shifted Drinfeld Yangians $Y_μ(\mathfrak{gl}_n)$ and quantum affine algebras $U_{μ^+,μ^-}(L\mathfrak{gl}_n)$, which slightly generalize their $\mathfrak{sl}_n$-counterparts. Our key observation is that both algebras admit the RTT type realization when $μ$ (respectively, $μ^+$ and $μ^-$) are antidominant coweights. We prove that $T_D(z)$ are polynomial in $z$ (up to a rational factor) and obtain explicit simple formulas for those…
A new Lax pair for the sixth Painlevé equation $P_{VI}$ is constructed in the framework of the loop algebra $\mathfrak{so}(8)[z,z^{-1}]$. The whole affine Weyl group symmetry of $P_{VI}$ is interpreted as gauge transformations of the corresponding linear problem.
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