Partially integrable highest weight modules (Q1268123)

The authors introduce a rather general definition of a Borel subsuperalgebra of an arbitrary complex Lie superalgebra with generalized root decomposition which allows to consider highest weight modules in natural generality and to define the class of weak Kac-Moody algebras which contains the usual Kac-Moody algebras, affine superalgebras and direct limit Lie algebras and superalgebras. The main result of the paper is the complete characterization of integrable directions in certain highest weight modules of weak Kac-Moody superalgebras. Roughly stated, the result gives that the integrable directions form a parabolic subalgebra. Finally, the authors discuss in detail several types of weak Kac-Moody superalgebras and their Borel subsuperalgebras admitting a weak basis: the finite dimensional superalgebras, the Kac-Moody algebras, the affine superalgebras, the Lie algebras \(A(\infty),B(\infty),C(\infty),D(\infty)\) obtained as direct limits of the corresponding series of simple Lie algebras and direct limits of simple Lie superalgebras. In particular, as a consequence of their main theorem, the authors give new proofs of some known results and show that their definition of a Borel subsuperalgebra agrees with other definitions of Borel algebras given before.