# Quasi-invariants and quantum integrals of the deformed Calogero-Moser systems

@article{Feigin2003QuasiinvariantsAQ, title={Quasi-invariants and quantum integrals of the deformed Calogero-Moser systems}, author={Misha Feigin and A P Veselov}, journal={International Mathematics Research Notices}, year={2003}, volume={2003}, pages={2487-2511} }

The rings of quantum integrals of the generalized Calogero-Moser systems related to the deformed root systems An(m) and Cn(m, l) with integer multiplicities and corresponding algebras of quasi-invariants are investigated. In particular, it is shown that these algebras are finitely generated and free as the modules over certain polynomial subalgebras (Cohen-Macaulay property). The proof follows the scheme proposed by Etingof and Ginzburg in the Coxeter case. For two-dimensional systems the…

## 20 Citations

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## References

SHOWING 1-10 OF 20 REFERENCES

### Deformed Quantum Calogero-Moser Problems and Lie Superalgebras

- Mathematics
- 2004

The deformed quantum Calogero-Moser-Sutherland problems related to the root systems of the contragredient Lie superalgebras are introduced. The construction is based on the notion of the generalized…

### Quantum completely integrable systems connected with semi-simple Lie algebras

- Mathematics
- 1977

The quantum version of the dynamical systems whose integrability is related to the root systems of semi-simple Lie algebras are considered. It is proved that the operators Ĵk introduced by Calogero…

### SUPERANALOGS OF THE CALOGERO OPERATORS AND JACK POLYNOMIALS

- Mathematics
- 2000

Abstract. A depending on a complex parameter k superanalog SL of Calogero operator is constructed; it is related with the root system of the Lie superalgebra gl(n|m). For m = 0 we obtain the usual…

### Multidimensional Baker–Akhiezer Functions and Huygens' Principle

- Mathematics
- 1999

Abstract:A notion of the rational Baker–Akhiezer (BA) function related to a configuration of hyperplanes in Cn is introduced. It is proved that the BA function exists only for very special…

### Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group

- Mathematics, Medicine
- 1993

Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale, toute copie ou impressions de ce fichier doit contenir la présente mention de copyright.

### Commutative rings of partial differential operators and Lie algebras

- Mathematics
- 1990

We give examples of finite gap Schrödinger operators in the two-dimensional case.

### Introduction to commutative algebra

- Mathematics
- 1969

* Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings *…

### Solution of the One‐Dimensional N‐Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials

- Mathematics
- 1971

The quantum‐mechanical problems of N 1‐dimensional equal particles of mass m interacting pairwise via quadratic (``harmonical'') and/or inversely quadratic (``centrifugal'') potentials is solved. In…