Realization of Lie algebras and superalgebras in terms of creation and annihilation operators: I

For every finite-dimensional nilpotent complex Lie algebra or superalgebra n, we offer three algorithm for realizing it in terms of creation and annih ilation operators. We use these algorithms to realize tie algebras with a m aximal subalgebra of finite codimension. For a simple finite-dimensional g whose maximal nilpotent subalgebra is n, this gives its realization in term s or first-order differential operators on the bi,rr open cell of the flag manifold corresponding to the negative roots of g. For several examples, we executed the algorithms rising the MATHEMATICAL-based package SUPERLie. Th ese realizations form a preparatory step in an explicit construction and de scription of an interesting new class of simple Lie (super)algebras of poly nomial growth, generalizations of the Lie algebra of matrices of complex si ze.