Explicit Realization of Induced and Coinduced modules over Lie Superalgebras by Differential Operators

This text is an extended version of a part of my 1980 Trieste preprint [16]. In this preprint I gave, in particular, an expression for the action of a Lie superalgebra g on the g-module coinduced from an h-module V for the case where both g and V are finite-dimensional and g decomposes (as a linear space, not Lie superalgebra) into a direct sum g− ⊕ h with g− a Lie superalgebra. The importance of coinduced and induced g-modules is a consequence of the fact that the vast majority of representations of Lie superalgebras occurring both in mathematical and physical practice, belongs to one of these two classes (e.g., the “universal” g-module U(g) and its dual “couniversal” module U(g), Verma and Harish-Chandra modules, etc.). Nevertheless, even some mathematicians working with these representations do not realize that they are “speaking prose”, to say nothing of physicists who study “representations by creation and annihilation operators” while speaking about Verma modules or their duals. The standard method used by both mathematicians and physicists to calculate the action for basis elements of g− in coinduced modules (where they act as differential operators) in case where g is a finite-dimensional Lie algebra, was to calculate the differential of the induced representation of the corresponding Lie group. This method is inapplicable both if g is infinite-dimensional and if g is a superalgebra, even a finite-dimensional one. In the infinite-dimensional case the reason is that there is no correspondence between abstract infinite-dimensional Lie algebras and infinite-dimensional Lie groups, and no correspon-