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
(
G
1
,
G
2
)
in quantum many-body physics. They arise in models whose Hamiltonians
H
have invariance groups
G
i
. Then one can take
G
1
=
G
i
whereas another dual partner
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
g
^
=
g
D
appear in models with essentially nonlinear Hamiltonians. Such an approach leads to a geometrization of model kinematics and dynamics.