Dynamical symmetries in a spherical geometry. I

The tetrahexahedric angular Calogero model

“…Secondly, this radial scaling suggests a further codimension-one reduction to S n−2 , which defines the 'angular (relative) rational Calogero model'. 2 This system describes a particle moving on S n−2 in a collection of 1 2 n(n−1) Higgs-oscillator potentials [20,21], each one centered at a positive root of A n−1 (and its antipode) and blowing up at the intersection of the reflection hyperplane with the unit sphere. These singular loci tessellate the (n−1)-sphere into n!…”

Finite W-algebras and intermediate statistics

“…As remarked in [3], this algebra has already been considered by P.W. Higgs as an algebra of conserved quantities for a Coulombic central force problem in a space of constant curvature [4].…”

“…As in the d = 1 case, the (x j , p j ) realization as given in (3.13-3.15) is a particular one, associated with special eigenvalues of the sp(4, R) fundamental invariants ∆ (2) and ∆ (4) . But one can proceed as before in order to get a more general representation.…”

“…Secondly one can note that any state in a sp(4) representation is completely specified by six numbers, each eigenvalue of one of six simultaneously commuting operators: usually are considered the two sp(4) fundamental invariants ∆ (2) and ∆ (4) , then one can choose the two Casimir and two Cartan generators of the two sℓ(2) forming the maximal sℓ(2) ⊕ sℓ(2) subalgebra in sp (4). From the algebra (3.28), µ, S and the gℓ(2)-like Casimir operator…”

“…Thus the four operators associated with the finite W-algebra are naturally used to obtain, after modification of the expressions of four sp(4) generators, the general "coordinate" realization of sp (4). As developed in [5], matrix representations of the gℓ(2)-like algebra, in which the wave functions behave as vectors, have to be considered.…”