Related papers: Rational Top and its Classical R-matrix

In [Frieden, arXiv:1706.02844], we constructed a geometric crystal on the variety $\mathbb{X}_{k} := {\rm Gr}(k,n) \times \mathbb{C}^\times$ which tropicalizes to the affine crystal structure on rectangular tableaux with $n-k$ rows. In this…

We construct a generalised notion of Kac-Moody algebras using smooth maps from the non-compact manifolds ${\cal M}=$SL$(2,\mathbb R)$ and ${\cal M}=$ SL$(2,\mathbb R)/U(1)$ to a finite-dimensional simple Lie group $G$. This construction is…

Let $G$ be a simply connected semisimple algebraic group over $\mathbb{C}$ and let $\rho :G\rightarrow GL(V_\lambda)$ be an irreducible representation of highest weight $\lambda$. Suppose that $\rho$ has finite kernel. Springer defined…

This paper is a continuation of our previous paper \cite{LOSZ}. For simple complex Lie groups with non-trivial center, i.e. classical simply-connected groups, $E_6$ and $E_7$ we consider elliptic Modified Calogero-Moser systems…

In this paper we describe in some detail the representation of the topological $CP^1$ model in terms of a matrix integral which we have introduced in a previous article. We first discuss the integrable structure of the $CP^1$ model and show…

The Hamiltonian symmetry reduction of the geodesics system on a symmetric space of negative curvature by the maximal compact subgroup of the isometry group is investigated at an arbitrary value of the momentum map. Restricting to regular…

The goal of this diploma thesis is to give a detailed description of Kirillov's Orbit Method for the case of compact connected Lie groups. The theory of Kirillov aims at finding all irreducible unitary representations of a given Lie group…

The R-matrix formalism for the construction of integrable systems with infinitely many degrees of freedom is reviewed. Its application to Poisson, noncommutative and loop algebras as well as central extension procedure are presented. The…

We propose an algebraic scheme for quantizing the rational Ruijsenaars-Schneider model in the R-matrix formalism. We introduce a special parameterization of the cotangent bundle over GL(N,C). In new variables the standard symplectic…

We prove the equivalence between two traditional approaches to the classical mechanics of a massive spinning particle in special relativity. One is the spherical top model of Hanson and Regge, recast in a Hamiltonian formulation with…

The category of rational G-equivariant cohomology theories for a compact Lie group $G$ is the homotopy category of rational G-spectra and therefore tensor-triangulated. We show that its Balmer spectrum is the set of conjugacy classes of…

We describe a method for solving linear systems over the localization of a commutative ring $R$ at a multiplicatively closed subset $S$ that works under the following hypotheses: the ring $R$ is coherent, i.e., we can compute finite…

A complete description of the non-dynamical r-matrices of the degenerate Calogero-Moser models based on $gl_n$ is presented. First the most general momentum independent r-matrices are given for the standard Lax representation of these…

Let $k$ be a field with characteristic different from $2$. In this paper, we describe the $k$-rational orbit spaces in some irreducible prehomogeneous vector spaces $(G,V)$ over $k$, where $G$ is a connected reductive algebraic group…

The main result asserts: Let $G$ be a reductive, affine algebraic group and let $(\rho ,V)$ be a regular representation of $G$. Let $X$ be an irreducible $\mathbb{C}^{ \times } G$ invariant Zariski closed subset such that $G$ has a closed…

We propose a construction of 1+1 integrable Heisenberg-Landau-Lifshitz type equations in the ${\rm gl}_N$ case. The dynamical variables are matrix elements of $N\times N$ matrix $S$ with the property $S^2={\rm const}\cdot S$. The Lax pair…

We propose the full system of Baxter Q-functions (QQ-system) for the integrable spin chains with the symmetry of the $D_r$ Lie algebra. We use this QQ-system to derive new Weyl-type formulas expressing transfer matrices in all symmetric and…

In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on $R$-matrix description which provides Lax pairs in terms of quantum and classical $R$-matrices. First, we prove…

We review the separation of variables for the Kowalevski top and for its generalization to the algebra o(4). We notice that the corresponding separation equations allow an interpretation of the Kowalevski top as a B2 integrable lattice.…

We study the matrix factorization problem associated with an SO(2) spinning top by using the algebro-geometric approach. We derive the explicit expressions in terms of Riemann theta functions and discus some related problems including a…