Finite -algebras and intermediate statistics

Abstract

New realizations of finite W-algebras are constructed by relaxing the usual constraint conditions. Then finite W-algebras are recognized in the Heisenberg quantization recently proposed by Leinaas and Myrheim, for a system of two identical particles in d dimensions. As the anyonic parameter is directly associated to the W-algebra involved in the d = 1 case, it is natural to consider that the W-algebra framework is well adapted for a possible generalization of the anyon statistics.in memory of our friend and colleague Tanguy Altherr

On nonlinear angular momentum theories, their representations and associated Hopf structures

“…In terms of the ladder generators J ± and the diagonal one J 3 , the very well-known linear sl(2) algebra is characterized by the commutation relations [11] [ 2) and by the Casimir operator…”

“…(2.1) and (2.2)), i.e. 8) where the hat indices help us to distinguish these modified structures with respect to the algebra sl (2). In fact, let us define a new basis of the algebra subtended by J ± and J 3 as follows:Ĵ…”

“…In section 3, we give a generalization of nonlinear algebras when the starting point was U q (sl (2)). In section 4, we show that the linear sl(2)-case as well as its q-deformation [1] are particular examples of our developments.…”

$\cw$ -Realization of Lie Algebras: Application to so (4,2) and $\pcr$ Algebras

“…As shown in [1,2], the construction of finite W-algebras achieved in the framework of Hamiltonian reduction [3] also leads to the determination of the commutants, in the enveloping algebra 1 U (G), of particular subalgebras of a simple Lie algebra G.…”

“…Such an approach has already been considered with some success [1] for the study of the Heisenberg quantization for a system of two particles in 1 and 2 dimensions [5]. The corresponding algebras are respectively sp(2) and sp (4), and in each case, it has been possible to relate the anyonic parameter to the eigenvalues of a W-generator.…”