Abstract
The generators of the algebra <em>gl<sub>n+1</sub></em> in the form of differential operators of the first order acting on <strong>R</strong><sup><em>n</em></sup> with matrix coefficients are explicitly written. The algebraic Hamiltonians for matrix generalization of 3−body Calogero and Sutherland models are presented.
Highlights
This work has a certain history related to Miloslav Havlicek
The algebra gln of differential operators plays the role of a hidden algebra for all An, Bn, Cn, Dn, BCn Calogero-Moser Hamiltonians, both rational and trigonometric, with the Weyl symmetry of classical root spaces
We have described a procedure which, in our opinion, should carry the name of the Havlicek procedure, to construct the algebra gln of the matrix differential operators
Summary
This work has a certain history related to Miloslav Havlicek. On the important occasion of Miloslav’s 75th birthday, we think this story should be revealed. AVT studied them for many years, at first separately and together with the first author (YuFS), who happened to have the same set of preprints The results of these studies are presented below. While carrying out these studies, we always kept in mind that a constructive answer exists and is known to Miloslav. Our main goal is to find a mixed representation of the algebra gln+1 which contains both matrices and differential operators in a non-trivial way. To generalize it to a polynomial algebra which we call g(m) (see below, Section 4). Another goal is to apply the obtained representations for a construction of the algebraic forms of (quasi)-exactly-solvable matrix Hamiltonians
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