$q$-Deformed Classical Lie Algebras and their Anyonic Realization

Abstract

All classical Lie algebras can be realized à la Schwinger in terms of fermionic oscillators. We show that the same can be done for their $q$-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on a two dimensional lattice. The deformation parameter $q$ is a phase related to the anyonic statistical parameter. A crucial rôle in this construction is played by a sort of bosonization formula which gives the generators of the quantum algebras in terms of the underformed ones. The entire procedure works even on one dimensional chains; in such a case $q$ can also be real.

in Plain TeX; one figure and two tables in LaTeX are not included but available under request , 22 pages, DFTT 16/93

References in corpus (3)

• An introduction to non-commutative differential geometry on quantum groups
• Anyons and Quantum Groups
• Anyonic Realization of $S{U}_q(N)$ Quantum Algebra

Cited by in corpus (6)

• On the Fock space for nonrelativistic anyon fields and braided tensor products
• Anyonic Realization of the Quantum Affine Lie Algebra U_q(A_N)
• Anyonic Realization of the Quantum Affine Lie Superalgebra U_q(A(M,N)^{(1)})
• Anyonic Realization of the Quantum Affine Lie Algebras
• Exotic Particles and $w_{\infty }$-Algebras in Two- and High-Dimensional Spaces
• Multiparameter Deformations of the Algebra $g{l}_n$ in Terms of Anyonic Oscillators