TY - JOUR

T1 - A Casimir operator for a Calogero W algebra

AU - Correa, Francisco

AU - Leal, Gonzalo

AU - Lechtenfeld, Olaf

AU - Marquette, Ian

N1 - F C was supported by Fondecyt Grants 1171475 and 1211356. He thanks the Departamento de Física Teórica, Atómica y Óptica at Universidad de Valladolid and Universidad Austral de Chile, where this project was initiated, for its kind hospitality.

PY - 2024/2/12

Y1 - 2024/2/12

N2 - We investigate the nonlinear algebra W 3 generated by the 9 functionally independent permutation-symmetric operators in the three-particle rational quantum Calogero model. Decoupling the center of mass, we pass to a smaller algebra W 3 ′ generated by 7 operators, which fall into a spin-1 and a spin- 3 2 representation of the conformal sl(2) subalgebra. The commutators of the spin- 3 2 generators with each other are quadratic in the spin-1 generators, with a central term depending on the Calogero coupling. One expects this algebra to feature three Casimir operators, and we construct the lowest one explicitly in terms of Weyl-ordered products of the 7 generators. It is a polynomial of degree 6 in these generators, with coefficients being up to quartic in ℏ and quadratic polynomials in the Calogero coupling ℏ 2 g ( g − 1 ) . Putting back the center of mass, our Casimir operator for W 3 is a degree-9 polynomial in the 9 generators. The computations require the evaluation of nested Weyl orderings. The classical and free-particle limits are also given. Our scheme can be extended to any finite number N of Calogero particles and the corresponding nonlinear algebras W N and W N ′ .

AB - We investigate the nonlinear algebra W 3 generated by the 9 functionally independent permutation-symmetric operators in the three-particle rational quantum Calogero model. Decoupling the center of mass, we pass to a smaller algebra W 3 ′ generated by 7 operators, which fall into a spin-1 and a spin- 3 2 representation of the conformal sl(2) subalgebra. The commutators of the spin- 3 2 generators with each other are quadratic in the spin-1 generators, with a central term depending on the Calogero coupling. One expects this algebra to feature three Casimir operators, and we construct the lowest one explicitly in terms of Weyl-ordered products of the 7 generators. It is a polynomial of degree 6 in these generators, with coefficients being up to quartic in ℏ and quadratic polynomials in the Calogero coupling ℏ 2 g ( g − 1 ) . Putting back the center of mass, our Casimir operator for W 3 is a degree-9 polynomial in the 9 generators. The computations require the evaluation of nested Weyl orderings. The classical and free-particle limits are also given. Our scheme can be extended to any finite number N of Calogero particles and the corresponding nonlinear algebras W N and W N ′ .

KW - Calogero model

KW - Casimir operator

KW - W algebra

UR - http://www.scopus.com/inward/record.url?scp=85187256333&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/ad24ca

DO - 10.1088/1751-8121/ad24ca

M3 - Article

VL - 57

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 0022-3689

IS - 8

M1 - 085203

ER -