Affine Lie algebras admit non-classical highest-weight theories through alternative partitions of the root system. Although significant inroads have been made, much of the classical machinery is inapplicable in this broader context, and some fundamental questions remain unanswered. In particular, the structure of the reducible objects in non-classical theories has not yet been fully understood....

We prove a more general version of a result announced without proof in DP], claiming roughly that in a partially integrable highest weight module over a Kac-Moody algebra the integrable directions form a parabolic subalgebra.

We analyze the categories of highest weight modules with a semiregular generalized infinitesimal character for the two exceptional Hermitian symmetric cases. These categories are completely described, and, as a consequence, we see that the combinatorial description of the general (regular integral) categories of highest weight modules previously given in the classical cases holds also in the ex...

The present paper was inspired by the work on polarized embeddings by Cardinali et al. (J. Algebr. Comb. 25(1):7–23, 2007) although some of our results in it date back to 1999. They study polarized embeddings of certain dual polar spaces, and identify the minimal polarized embeddings for several such geometries. We extend some of their results to arbitrary shadow spaces of spherical buildings, ...

We saw in lectures 7 and 8 how Lusztig’s nilpotent variety can be used to realize U−(g) and the crystal B(∞). Last week we saw how to use quiver grassmannians to realize the highest weight modules V (λ) as a quotient of U−(g), and the same construction realizes the crystals B(λ). This week we discuss a more standard approach to realizing V (λ) and B(λ), namely we will use Nakajima’s quiver vari...

For each integer n ≥ 2, we demonstrate that a 2n-dimensional generalized MICZ-Kepler problem with magmatic charge μ = 0 or 1/2 has an Spin(2, 2n+ 1) dynamical symmetry which extends the manifest Spin(2n) symmetry. The Hilbert space of bound states is shown to form a unitary highest weight Spin(2, 2n+1)-module which occurs at the first reduction point in the Enright-Howe-Wallach classification d...

We define regular Kac-Moody superalgebras and classify them using integrable modules. We give conditions for irreducible highest weight modules of regular Kac-Moody superalgebras to be integrable. This paper is a major part of the proof for the classification of finite-growth contragredient Lie superalgebras. The results of this paper are a crucial part of the proof for the classification of co...