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

As 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 $U_q[sl(n+1|m)]$.Comment: 27 pages, LaTeX; to be published in J. Math. Phy

Invitation to the Psalms: A Reader’s Guide for Discovery and Engagement

Invitation to the psalms: a reader\u27s guide for discovery and engagement. Author: Karl Jacobson; Rolf A Jacobson. Publisher: Grand Rapids, Mich. : Baker Academic, 2013. ISBN: 978144125788

Semicrossed products of the disk algebra and the Jacobson radical

We consider semicrossed products of the disk algebra with respect to endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical of these operator algebras. Furthermore, in the case the finite Blaschke product is elliptic, we show that the semicrossed product contains no nonzero quasinilpotent elements. However, if the finite Blaschke product is hyperbolic or parabolic with zero hyperbolic step, the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.Comment: 12 page

Too Good to Miss

Reviewed Book: Jacobson, J. Robert. Too Good to Miss. Calgary: Foothills Lutheran Press, 1977