  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.
