Abstract:
For every finite-dimensional nilpotent complex Lie algebra or superalgebra $\mathfrak n$, we offer three algorithms for realizing it in terms of creation and annihilation operators. We use these algorithms to realize Lie algebras with a maximal subalgebra of finite codimension. For a simple finite-dimensional $\mathfrak g$ whose maximal nilpotent subalgebra is $\mathfrak n$, this gives its realization in terms of first-order differential operators on the big open cell of the flag manifold corresponding to the negative roots of $\mathfrak g$. For several examples, we executed the algorithms using the MATHEMATICA-based package SUPERLie. These realizations form a preparatory step in an explicit construction and description of an interesting new class of simple Lie (super)algebras of polynomial growth, generalizations of the Lie algebra of matrices of complex size.