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

TL;DR

This paper explores dual algebraic pairs and polynomial Lie algebras in quantum physics, highlighting their role in understanding symmetries, invariants, and the geometric structure of models with nonlinear Hamiltonians.

Contribution

It introduces a framework connecting dual algebraic pairs with polynomial Lie algebras, providing a geometric perspective on quantum many-body models with nonlinear Hamiltonians.

Findings

Dual algebraic pairs relate invariance groups and dynamic symmetries.

Polynomial Lie algebras emerge in models with nonlinear Hamiltonians.

The approach offers a geometric understanding of model kinematics and dynamics.

Abstract

We discuss some aspects and examples of applications of dual algebraic pairs $({\cal G}_1,{\cal G}_2)$ in quantum many-body physics. They arise in models whose Hamiltonians $H$ have invariance groups $G_i$. Then one can take ${\cal G}_1 = G_i$ whereas another dual partner ${\cal G}_2= g^D$ 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.