In this talk, I will review our recent work on a
Lie-algebraic approach to various quantum-mechanical
problems. The first part will be devoted to non-equilibrium
driven dynamics of closed quantum systems. It will be
emphasized that mathematically  a non-equilibrium
Hamiltonian represents a trajectory in a Lie algebra, while
the evolution operator is a trajectory in a Lie group
generated by the underlying algebra via exponentiation. This
turns out to be a constructive statement that establishes in
particular the fact that classical and   quantum unitary
evolutions are two sides of the same coin determined
uniquely by the same dynamic  generators in the group. An
equation for the generators, dubbed dual Schrodinger-Bloch
equation, will be derived and analyzed. In the second part
of the talk, I will extend the Lie algebraic approach to
many-particle quantum systems in equilibrium, in particular
to Hubbard-like models and quantum spin models. The
canonical problem of determining the partition function of
these models will be put in the context  of classification
of the Lie algebras. Based on these observations, examples
of statistical-transmutation and new spin-to-fermion
transforms will be constructed.