Weight modules of infinite-dimensional Lie algebras and Lie superalgebras

We study various weight representation of infinite-dimensional Lie algebras and Lie superalgebras. For the simple direct limit Lie algebras {dollar}A(infty), B(infty), C(infty){dollar} and {dollar}D(infty),{dollar} we prove that the shadow of any irreducible weight module is well-defined. This means that for a given root {dollar}alpha{dollar} the intersection of the ray {dollar}lambda+IRsb+alpha{dollar} with the support of M, suppM, is either finite for all {dollar}lambdain{lcub}rm supp{rcub}M{dollar} or infinite for all {dollar}lambdain{lcub}rm supp{rcub}M.{dollar} Using this remarkable property of the support, we assign to M a canonical decomposition of g into four subalgebras and a parabolic subalgebra {dollar}{lcub}bf p{rcub}sb{lcub}M{rcub}{dollar} of g. We then consider in more detail the integrable weight representations of {dollar}A(infty), B(infty), C(infty){dollar} and {dollar}D(infty){dollar} and single out the finite integrable modules, which are analogous to finite-dimensional modules. We are able to present a rather explicit description of all irreducible finite integrable modules.; The second part of the dissertation is devoted to a study of the integrable directions of an arbitrary highest weight module V over certain class of infinite-dimensional Lie superalgebras. This study is motivated by the fact, established by V. Kac and M. Wakimoto, that certain most natural affine Lie superalgebras do not admit integrable highest weight modules other than the trivial one. Ivan Penkov and I have initiated a systematic study of partially integrable highest weight modules over various infinite-dimensional Lie algebras and Lie superalgebras. In this dissertation theorems describing the structure of the integrable sl(2)-directions of highest weight modules are obtained for Kac-Moody algebras, affine Lie superalgebras and the algebras {dollar}A(infty), B(infty), C(infty){dollar} and {dollar}D(infty).{dollar}; It is also worth mentioning that a general interrelationship between Borel sub-superalgebras of an arbitrary Lie superalgebra, the {dollar}IR{dollar}-linear orders on {dollar}langleDeltaranglesbIR{dollar} and the orriented complete generalized flags in {dollar}langleDeltaranglesbIR{dollar} is established.