Dual Algebraic Pairs and Polynomial Lie Algebras in Quantum Physics: Foundations and Geometric Aspects

Abstract

We discuss some aspects and examples of applications of dual algebraic pairs $({\mathcal{G}}_1,{\mathcal{G}}_2)$ in quantum many-body physics. They arise in models whose Hamiltonians $H$ have invariance groups $G_i$. Then one can take ${\mathcal{G}}_1=G_i$ whereas another dual partner ${\mathcal{G}}_2=g^D$ is generated by $G_i$ invariants, possesses a Lie-algebraic structure and describes dynamic symmetry of models; herewith polynomial Lie algebras $\hat{g}=g^D$ appear in models with essentially nonlinear Hamiltonians. Such an approach leads to a geometrization of model kinematics and dynamics.

10 pages, LATEX; submitted to Proceedings of the Workshop "Contemporary Geometry and Related Topics" (Belgrade, May 15-21, 2002)