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

7 Pages·0.12 MB·English

by  S. E. Konstein

FIAN/TD/97-18
Rational Calogero models based on rank-2 root
systems: supertraces on the superalgebras of
8
9 observables
9
1

n
a S.E.Konstein ∗†
J

1
I.E.Tamm Department of Theoretical Physics,

]
A
P. N. Lebedev Physical Institute,
Q
117924, Leninsky Prospect 53, Moscow, Russia.
.
h
t
a
m
[
Abstract

1
v It is shown that the superalgebra H of observables of the rational Calogero
1 W(I2(n))
model based on the root system I (n) possesses [(n + 1)/2] supertraces. Model
0 2
0 with three-particle interaction based on the root system G2 belongs to this class of
1 models and its superalgebra of observables has 3 independent supertraces.
0
8
9
/
h
t
a
m
:
v
i
X
r
a
∗E-mail: [email protected]
†ThisworkissupportedbytheRussianFundforBasicResearch,Grants96-02-17314and96-15-96463.
1
1 Introduction
In this work we continue to compute the number of supertraces on the superalgebras of
observablesunderlying therationalCalogeromodels[1]basedontherootsystems [2]. The
results for the root systems of A , B (C ) and D types are listed in [3], [4]. Here
N−1 N N N
we consider the root systems of I (n) type. As W(A ) = W(I (3)), W(B ) = W(I (4)),
2 2 2 2 2
W(D ) = W(I (2)) and W(G ) = W(I (6)) the case under consideration covers all rank-2
2 2 2 2
systems. It is shown that the number of supertraces on the superalgebra of observables
of Calogero model based on this root system is [(n+1)/2]. In particularly it gives that
in the case of the root system G there are 3 supertraces.
2
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 H of observables of the rational Calogero model based on the
W(I2(n))
root system I (n) is described in the section 4 and the number of supertraces on this
2
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.
R
Let us define the reflections R ~v , ~v = 0, as follows
~v N
∀ ∈ 6
(~x, ~v) R
R (~x) = ~x 2 ~v ~x . (1)
~v N
− (~v, ~v) ∀ ∈
R
Here the notation ( , ) used for the inner product in . We will use also the coordi-
N
· ·
def R
nates of vectors, v = (~v, ~e ), where vectors ~e constitute the orthonormal basis in :
i i i N
(~e , ~e ) = δ . The reflections (1) satisfy the properties
i j ij
R
R (~v) = ~v, R2 = 1, (R (~x), ~u) = (~x, R (~u)), ~v, ~x, ~u . (2)
~v − ~v ~v ~v ∀ ∈ N
R
The finite set of vectors R is the root system if R is R -invariant ~v R and
N ~v
⊂ ∀ ∈
the group W(R) generated by all reflections R with~v R (Coxeter group) is finite. The
~v
∈
brief description of the classification of the root systems one can find for instance in [2].
R
The group W(R) acts also on some space C of functions on . Let us assume for
N
definiteness that C is constituted by all infinitely smooth functions of polynomial growth
Z
i.e.that f C n(f) such that lim f(~x)/ ~x n(f) = 0. Every function f C
+ ~x→∞
∀ ∈ ∃ ∈ | | ∈
can be also considered as element of EndC i.e.as operator on C acting by multiplication:
g fg g C. The action of R W(R) on such functions has the following form
~v
→ ∀ ∈ ∈
R (f(~x)) = f(R (~x)) when f(~x) C, (3)
~v ~v
∈
R f(~x) = f(R (~x))R when f(~x) EndC. (4)
~v ~v ~v
∈
Dunkl differential-difference operators is defined as [5]
∂ v
i
D = + ν (1 R ), (5)
i ~v ~v
∂x (~x, ~v) −
i ~v∈R
X
2
where coupling constants ν are such that
~v
ν = ν ~u, ~v R. (6)
~v R~u(~v) ∀ ∈
These operators are well defined on C and commute with each other, [D , D ] = 0.
i j
Due to this property the deformed creation and annihilation operators [6, 7]
1
aα = (x +( 1)αD ), α = 0,1, (7)
i √2 i − i
R
transform under action of reflections R ~v R as vectors
~v N
∀ ∈
N v v
R aα = δ 2 i j aαR (8)
~v i ij − (~v, ~v)! j ~v
j=1
X
and satisfy the following commutation relations
v v
[aα,aβ] = εαβ δ +2 ν i j R (9)
i j  ij ~v(~v, ~v) ~v
~v∈R
X
 
where εαβ is the antisymmetric tensor, ε01 = 1.
The operators aα (α = 0,1, i = 1,...,N) together with the elements of the group
i
W(R) generate the associative algebra with elements polynomial on aα. We denote this
i
algebra as HW(R)(ν) and call it the algebra of observables of Calogero model based on the
root system R. Here the notation ν stands for a complete set of ν with ~v R.
~v
∈
The commutation relations (9) and (8) allows one to define the parity π:
π(aα) = 1 α, i, π(g) = 0 g W(R) (10)
i ∀ ∀ ∈
and consider H (ν) as a superalgebra.
W(R)
Obviously the superalgebra H (ν) containes as a subalgebra the group algebra
W(R)
C
[W(R)] of Coxeter group W(R).
An important property of superalgebra H (ν) is that it has sl algebra of inner
W(R) 2
differentiatings with the generators
1 N
Tαβ = aα, aβ (11)
2 i i
Xi=1n o
which commute with C[W(R)], [Tαβ, R ] = 0, and act on aα as on sl -vectors:
~v i 2
Tαβ, aγ = εαγaβ +εβγaα. (12)
i i i
h i
The restriction of operator HˆR (ν) d=ef T01 on the subspace C C of W(R)-
Cal W(R) ⊂
invariant functions is the second-order differential operator which is well-known Hamil-
tonian of the rational Calogero model [1] based on the root system R [2]. One of the
relations (11) namely [HˆR (ν), aα] = ( 1)αaα allows one to find the wavefunctions of
the equation HˆR (ν)ψ =Caǫlψ viaiusua−l F−ock pirocedure with the vacuum 0 such that
Cal | i
a0 0 =0 i [7]. After W(R)-symmetrization these wavefunctions become the wavefunc-
i| i ∀ R
tions of Calogero Hamiltonian HCal(ν)|CW(R).
3
3 Supertraces on H (ν)
W(R)
Definition. The supertrace on the superalgebra is a linear complex-valued function
A
str( ) such that f,g with definite parity π(f) and π(g)
· ∀ ∈ A
str(fg) = ( 1)π(f)π(g)str(gf). (13)
−
Every supertrace str( ) on generates the invariant bilinear form on
· A A
B (f,g) = str(f g). (14)
str
·
It is obvious that if such a bilinear form is degenerated then the null-vectors of this
form constitute both-sided ideal .
I ⊂ A
It was shown that the ideals of this sort are present in the superalgebras H (ν)
W(A1)
(corresponding to the two-particle Calogero model) at ν = k + 1 [8] and in the super-
2
algebras H (ν) (corresponding to three-particle Calogero model) at ν = k + 1 and
W(A2) 2
ν = k 1 [9] with every integer k and that for all the other values of ν all supertraces on
± 3
these superalgebras generate the nondegenerated bilinear forms (14).
The spectrum of N-particle rational Calogero Hamiltonian (case R = A ) coincides
N−1
with the spectrum of system of N noninteracting oscillators if the latter is shifted on the
constant c(ν) = 1N(N 1)ν. It allows one to construct the similarity transformation
2 −
between operators HˆAN−1(ν) C c(ν) with different ν [10]. Nevertheless the previous
Cal | W(R) −
consideration shows that the corresponding algebras H (ν) can be nonisomorphic
W(AN−1)
at different values of ν.
C C
It is easy to describe all supertraces on [W(R)]. Every supertrace on [W(R)] is
C
completely defined by its values on W(R) [W(R)] and the function str is a central
⊂
function on W(R) i.e.the function on the conjugacy classes.
To formulate the theorem establishing the connection between the supertraces on
C
H (ν) and the supertraces on [W(R)] let us introduce the grading on the vector
W(AN)
C C
space of [W(R)]. The grading E of elements g W(R) [W(R)] is defined as
∈ ⊂
follows. Let α be the linear space with basis elements aα, aα, ..., aα. Consider the
HN 1 2 N
subspaces α(g) α as
E ⊂ HN
α(g) = h α : gh = hg , (15)
E { ∈ HN − }
and put
E(g) = dim α(g). (16)
E
C
To avoid misunderstanding it should be noticed that [W(R)] is not in general a graded
algebra.
The following theorem was proved in [3] 1:
1this theorem was proved for the case R=AN only but the proof does not depend on the particular
properties of the symmetric group SN =W(AN−1).
4
C C
Theorem 1. Let (g) be the projector [W(R)] [W(R)] defined as ( α g ) =
P C → P i i i
α g (g W(R), α ). Let the grading E defined in (16) and the subspaces
i:gi6=1 i i i ∈ i ∈ P
α(g) defined in (15) satisfy the equations
PE
C
E( ([h , h ])g) = E(g) 1 g [W(R)], h α(g). (17)
0 1 α
P − ∀ ∈ ∀ ∈ E
C
Then every supertrace on the algebra [W(R)] satisfying the equations
str([h , h ]g) = 0 g W(R) with E(g) = 0 and h α(g), (18)
0 1 α
∀ ∈ 6 ∀ ∈ E
can be extended to the supertrace on H (ν) in a unique way.
W(R)
4 Superalgebras H
W(I (n))
2
C R
It is convenient to use instead of to describe H . The root system I (n)
2 W(I2(n)) 2
contains 2n vectors v = exp(πik/n), k = 0,1,...,2n 1. The corresponding Coxeter
k
− C
group W(I (n)) has 2n elements, n reflections R acting on z, z∗ as follows
2 k
∈
R z = z∗v2R ,
k − k k
R z∗ = zv∗2R , k Z (19)
k − k k ∈ n
and n elements of the form S = R R . S is the unity in W(I (n)). These elements
k k 0 0 2
satisfy the following relations
R R = S , S S = S , R S = R , S R = R . (20)
k l k−l k l k+l k l k−l k l k+l
Obviously the reflections R lie in one conjugacy class and R in another if n is
2k 2k+1
even. If n is odd then all reflections R lie in one conjugacy class.
k
C
It is convenient to consider the following basis in [W(I (n))]
2
n−1 n−1
L = λkpR , Q = λ−kpS , (21)
p k p k
k=0 k=0
X X
Z
where λ = exp(2πi/n), p . (22)
n
∈
Differential-difference operators have the following form
∂ n−1 v n−1 v
2k 2k+1
D = 2 +ν (1 R )+ν (1 R ),
z ∂z∗ 0 zv∗ +z∗v − 2k 1 zv∗ +z∗v − 2k+1
k=0 2k 2k k=0 2k+1 2k+1
X X
∂ n−1 v∗ n−1 v∗
Dz∗ = 2∂z +ν0 zv∗ +2kz∗v (1−R2k)+ν1 zv∗ 2+k+z1∗v (1−R2k+1). (23)
k=0 2k 2k k=0 2k+1 2k+1
X X
This form unifies both cases of even and odd n. If n is odd then Dz and Dz∗ depends
on the only coupling constant ν +ν .
0 1
The basis in the space α is
H2
1
aα d=ef (z +( 1)αD )
z
2 −
1
bα d=ef (z∗ +( 1)αDz∗). (24)
2 −
5
Now we can write down the relation between elements aα, bα, R and S generat-
k k
ing the associative superalgebra H of polynomials of aα, bα with coefficients in
C W(I2(n)) i i
[W(I (n))]:
2
R aα = v 2bαR , R bα = v∗2aαR , (25)
k − k k k − k k
S aα = v∗2aαS , S bα = v2bαS , (26)
k k k k k k
L aα = bαL , L bα = aαL , (27)
p p+1 p p−1
− −
Q aα = aαQ , Q bα = bαQ , (28)
p p+1 p p−1
L L = nδ Q , L Q = nδ L ,
k l k+l l k l k−l l
def
Q L = nδ L , Q Q = nδ Q , where δ = δ , (29)
k l k+l l k l k−l l k k0
ν +ν ν ν
aα, bβ = εαβ 1+ 0 1L 0 − 1L ,
0 n/2
2 − 2
h i (cid:18) (cid:19)
ν +ν ν ν
aα, aβ = εαβ 0 1L 0 − 1L ,
1 n/2+1
2 − 2
h i (cid:18)ν +ν ν ν (cid:19)
bα, bβ = εαβ 0 1L 0 − 1L . (30)
−1 n/2−1
2 − 2
(cid:18) (cid:19)
h i
The terms containing ν ν in (30) are absent when n is odd and halfinteger indices are
0 1
−
senseless, so let us assume that ν =ν when n is odd.
0 1
5 The number of supertraces on H
W(I (n))
2
In this section the following theorem is proved:
Theorem 2. The superalgebra H has n+1 supertraces.
W(I2(n)) 2
It is easy to find the grading E: h i
E(R ) = 1, R (v∗aα +v bα) = (v∗aα +v bα)R ,
k k k k − k k k
n
E(S ) = 0, k = , (31)
k
∀ 6 2
E(S ) = 2, S aα = aαS , S bα = bαS , if n is even and S exists,
n/2 n/2 n/2 n/2 n/2 n/2
− −
and check that H satisfies the conditions of Theorem 1, i.e.that
W(I2(n))
E (v∗a0 +v b0), (v∗a1 +v b1) R = 0 (32)
P k k k k k
(cid:16) (cid:16)h i(cid:17) (cid:17)
and that for even n
E a0, a1 S = E a0, b1 S =
n/2 n/2
P P
(cid:16) (cid:16)h i(cid:17) (cid:17) (cid:16) (cid:16)h i(cid:17) (cid:17)
E b0, a1 S = E b0, b1 S = 1. (33)
n/2 n/2
P P
(cid:16) (cid:16)h i(cid:17) (cid:17) (cid:16) (cid:16)h i(cid:17) (cid:17)
To compute the number of supertraces onthe superalgebra we have to findthe number
of the solutions of the equations (18) which have the following form for the algebra under
consideration
str (v∗a0 +v b0), (v∗a1 +v b1) R = 0 (34)
k k k k k
(cid:16)h i (cid:17)
6
and
str a0, a1 S = str a0, b1 S =
n/2 n/2
(cid:16)h i (cid:17) (cid:16)h i (cid:17)
str b0, a1 S = str b0, b1 S = 0 for even n. (35)
n/2 n/2
(cid:16)h i (cid:17) (cid:16)h i (cid:17)
The equations (34) lead to
ν +ν 1
0 1
str(R ) = str Q + (Q +Q )
k 0 1 −1
− 2 2
(cid:18) (cid:19)
ν ν 1
+ ( 1)k 0 − 1str Q + (Q +Q ) , (36)
n/2 n/2+1 n/2−1
− 2 2
(cid:18) (cid:19)
and when n is even the relations (35) take place and lead to
str (ν +ν )L ( 1)n/2(ν ν )L = 0 (37)
0 1 ±1 0 1 n/2±1
− − −
(cid:16) 1 (cid:17)
str(S ) = str (ν +ν )L ( 1)n/2(ν ν )L . (38)
n/2 0 1 0 0 1 n/2
−2 − − −
(cid:16) (cid:17)
It is easy to see that equations (37) are consequences of (36). The equations (36) express
str(R ) k via str(S ) and (38) expresses str(S ) via str(S ) with j = n/2 when n is
k j n/2 j
∀ 6
even. Hence due to theorem 1 every supertrace on H (ν) is determined completely
W(I1(n))
by its values on S with j = n/2. Since S and S belong to one conjugacy class if and
j j k
6
only if j = k the number of independent supertrace is equal to (n+1)/2 when n is odd
−
and to n/2 if n is even. It finishes the proof of Theorem 2.
For n = 2,3,4 the obtained result are in agreement with the results obtained in [3] and
[4]. The case n = 6 gives that the superalgebra of observables of the rational Calogero
model based on the root system of G type has 3 independent supertraces.
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.
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