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arXiv · hep-th/9310125

Hidden algebra of the $N$-body Calogero problem

Alexander Turbiner
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Abstract

A certain generalization of the algebra $gl(N,{\bf R})$ of first-order differential operators acting on a space of inhomogeneous polynomials in ${\bf R}^{N-1}$ is constructed. The generators of this (non)Lie algebra depend on permutation operators. It is shown that the Hamiltonian of the $N$-body Calogero model can be represented as a second-order polynomial in the generators of this algebra. Given representation implies that the Calogero Hamiltonian possesses infinitely-many, finite-dimensional invariant subspaces with explicit bases, which are closely related to the finite-dimensional representations of above algebra. This representation is an alternative to the standard representation of the Bargmann-Fock type in terms of creation and annihilation operators.

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Alexander Turbiner. 1993-10-24. Hidden algebra of the $N$-body Calogero problem. https://doi.org/10.1016/0370-2693(94)90657-2

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