#
Fian/td/97-18

Rational Calogero models based on rank-2 root systems:
supertraces on the superalgebras of observables

###### Abstract

It is shown that the superalgebra of observables of the rational Calogero model based on the root system possesses supertraces. Model with three-particle interaction based on the root system belongs to this class of models and its superalgebra of observables has 3 independent supertraces.

## 1 Introduction

In this work we continue to compute the number of supertraces on the superalgebras of observables underlying the rational Calogero models [1] based on the root systems [2]. The results for the root systems of , () and types are listed in [3], [4]. Here we consider the root systems of type. As , , and the case under consideration covers all rank-2 systems. It is shown that the number of supertraces on the superalgebra of observables of Calogero model based on this root system is . In particularly it gives that in the case of the root system there are 3 supertraces.

The definition and some properties of the superalgebra of observables are discussed in the next section. In the section 3 the condition sufficient for existence of the supertraces is formulated and some consequences from the existence of the supertraces are discussed. The superalgebra of observables of the rational Calogero model based on the root system is described in the section 4 and the number of supertraces on this superalgebra is computed in the last section.

## 2 The superalgebra of observables

The superalgebra of observables of the rational Calogero model based on the root system is defined in the following way.

Let us define the reflections , , as follows

(1) |

Here the notation used for the inner product in . We will use also the coordinates of vectors, , where vectors constitute the orthonormal basis in : . The reflections (1) satisfy the properties

(2) |

The finite set of vectors is the root system if is -invariant and the group generated by all reflections with (Coxeter group) is finite. The brief description of the classification of the root systems one can find for instance in [2].

The group acts also on some space of functions on . Let us assume for definiteness that is constituted by all infinitely smooth functions of polynomial growth i.e.that such that . Every function can be also considered as element of i.e.as operator on acting by multiplication: . The action of on such functions has the following form

(3) | |||

(4) |

Dunkl differential-difference operators is defined as [5]

(5) |

where coupling constants are such that

(6) |

These operators are well defined on and commute with each other, .

Due to this property the deformed creation and annihilation operators [6, 7]

(7) |

transform under action of reflections as vectors

(8) |

and satisfy the following commutation relations

(9) |

where is the antisymmetric tensor, .

The operators (, ) together with the elements of the group generate the associative algebra with elements polynomial on . We denote this algebra as and call it the algebra of observables of Calogero model based on the root system . Here the notation stands for a complete set of with .

The commutation relations (9) and (8) allows one to define the parity :

(10) |

and consider as a superalgebra.

Obviously the superalgebra containes as a subalgebra the group algebra of Coxeter group .

An important property of superalgebra is that it has algebra of inner differentiatings with the generators

(11) |

which commute with , , and act on as on -vectors:

(12) |

The restriction of operator on the subspace of -invariant functions is the second-order differential operator which is well-known Hamiltonian of the rational Calogero model [1] based on the root system [2]. One of the relations (11) namely allows one to find the wavefunctions of the equation via usual Fock procedure with the vacuum such that =0 [7]. After -symmetrization these wavefunctions become the wavefunctions of Calogero Hamiltonian .

## 3 Supertraces on

Definition. The supertrace on the superalgebra is a linear complex-valued function such that with definite parity and

(13) |

Every supertrace on generates the invariant bilinear form on

(14) |

It is obvious that if such a bilinear form is degenerated then the null-vectors of this form constitute both-sided ideal .

It was shown that the ideals of this sort are present in the superalgebras (corresponding to the two-particle Calogero model) at [8] and in the superalgebras (corresponding to three-particle Calogero model) at and [9] with every integer and that for all the other values of all supertraces on these superalgebras generate the nondegenerated bilinear forms (14).

The spectrum of -particle rational Calogero Hamiltonian (case ) coincides with the spectrum of system of noninteracting oscillators if the latter is shifted on the constant . It allows one to construct the similarity transformation between operators with different [10]. Nevertheless the previous consideration shows that the corresponding algebras can be nonisomorphic at different values of .

It is easy to describe all supertraces on . Every supertrace on is completely defined by its values on and the function is a central function on i.e.the function on the conjugacy classes.

To formulate the theorem establishing the connection between the supertraces on and the supertraces on let us introduce the grading on the vector space of . The grading of elements is defined as follows. Let be the linear space with basis elements . Consider the subspaces as

(15) |

and put

(16) |

To avoid misunderstanding it should be noticed that is not in general a graded algebra.

The following theorem was proved in [3]
^{1}^{1}1this theorem was proved for the case only
but the proof does not depend on the particular properties
of the symmetric group .:

## 4 Superalgebras

It is convenient to use instead of to describe . The root system contains vectors , . The corresponding Coxeter group has elements, reflections acting on as follows

(19) |

and elements of the form . is the unity in . These elements satisfy the following relations

(20) |

Obviously the reflections lie in one conjugacy class and in another if is even. If is odd then all reflections lie in one conjugacy class.

It is convenient to consider the following basis in

(21) | |||

(22) |

Differential-difference operators have the following form

(23) |

This form unifies both cases of even and odd . If is odd then and depends on the only coupling constant .

The basis in the space is

(24) |

Now we can write down the relation between elements , , and generating the associative superalgebra of polynomials of , with coefficients in :

(25) | |||||

(26) | |||||

(27) | |||||

(28) | |||||

(29) |

(30) |

The terms containing in (4) are absent when is odd and halfinteger indices are senseless, so let us assume that = when is odd.

## 5 The number of supertraces on

In this section the following theorem is proved:

Theorem 2. The superalgebra has supertraces.

It is easy to find the grading :

(31) | |||

and check that satisfies the conditions of Theorem 1, i.e.that

(32) |

and that for even

(33) |

To compute the number of supertraces on the superalgebra we have to find the number of the solutions of the equations (18) which have the following form for the algebra under consideration

(34) |

and

(35) |

The equations (34) lead to

(36) | |||||

and when is even the relations (5) take place and lead to

(37) | |||

(38) |

It is easy to see that equations (37) are consequences of (36). The equations (36) express via and (38) expresses via with when is even. Hence due to theorem 1 every supertrace on is determined completely by its values on with . Since and belong to one conjugacy class if and only if the number of independent supertrace is equal to when is odd and to if is even. It finishes the proof of Theorem 2.

## References

- [1] F. Calogero, J. Math. Phys., 10 (1969) 2191, 2197; ibid 12 (1971) 419.
- [2] M. A.Olshanetsky and A. M. Perelomov, Phys. Rep., 94 (1983) 313.
- [3] S.E. Konstein and M.A. Vasiliev, J. Math. Phys. 37 (1996) 2872.
- [4] S.E.Konstein, Teor. Mat. Fiz., 111 (1997) 592.
- [5] C.F.Dunkl, Trans. Am. Math. Soc. 311 (1989) 167.
- [6] A. Polychronakos, Phys. Rev. Lett. 69 (1992) 703.
- [7] L. Brink, H. Hansson and M.A. Vasiliev, Phys. Lett. B286 (1992) 109.
- [8] M.A. Vasiliev, JETP Letters, 50 (1989) 344-347; Int. J. Mod. Phys. A6 (1991) 1115.
- [9] S.E.Konstein, preprint FIAN/TD/97-17.
- [10] N.Gurappa and P.K.Panigrahi, cond-mat/9710035.