Jacobson generators of the quantum superalgebra Uq[sl(n+1|m)] and Fock representations

Abstract

Jacobson generators of the quantum superalgebra AbstractAs an alternative to Chevalley generators, we introduce Jacobson generators for the quantum superalgebra U q [sl(n + 1|m)]. The expressions of all Cartan-Weyl elements of U q [sl(n + 1|m)] in terms of these Jacobson generators become very simple. We determine and prove certain triple relations between the Jacobson generators, necessary for a complete set of supercommutation relations between the Cartan-Weyl elements. Fock representations are defined, and a substantial part of this paper is devoted to the computation of the action of Jacobson generators on basis vectors of these Fock spaces. It is also determined when these Fock representations are unitary. Finally, Dyson and Holstein-Primakoff realizations are given, not only for the Jacobson generators, but for all Cartan-Weyl elements of U

Deformed CliffordClq(n m) and orthosymplecticUq[osp(2n 1 2m)] superalgebras and their root of unity representations

“…A set of Cartan-Weyl elements of U q [sl(m|n)] has been considered in [17], and consists of m + n − 1 "Cartan" elements Hi = e 11 − (−1) θi+1 e i+1,i+1 = ĥ1 + (−1)…”

“…Setting (see (3)- (5)) A set of Cartan-Weyl elements of U q [sl(m|n)] has been considered in [17], and consists of m + n − 1…”

Fock representations of the superalgebrasl(n+1|m), its quantum analogueUq[sl(n+1|m)] and related quantum statistics

“…This assumption holds for the classes A , B , C , D of simple Lie algebras [13]. It was studied in detail for the Lie algebras sl(n + 1) (A−statistics) [14], for the LSs sl(1|m) (A−superstatistics) [8,15] and recently for the classical Lie superalgebra q(n + 1) [16].…”

“…We write down directly the analogue of the relations (16). To this end introduce first the following Cartan generators:…”