This paper is dedicated to the memory of Donald Higman. Don was especially attracted to methods with some kind of fundamental simplicity but with sufficient substance to be useful and illuminating. This paper is written in the spirit of this lofty aspiration and also somewhat in the direction of Don’s early interests in homological algebra. The paper begins in Section 1 with a foundational disc...

We prove that for simple complex finite dimensional Lie algebras, affine Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro and for Heisenberg algebras.

As shown in the paper of Stembridge [Ste03], crystal graphs can be characterized by their local behavior. In this paper, we observe that a certain local property on crystals forces a more global property. In type A, this statement says that if a node has a single parent and single grandparent, then there is a unique walk from the highest weight node to it. In other classical types, there is a s...

We explain the construction of minimal tilting complexes for objects highest weight categories and we study in detail standard simple objects. For certain representations complex Lie algebras, affine Kac-Moody algebras quantum groups at roots unity, relate multiplicities indecomposable appearing terms these to coefficients Kazhdan-Lusztig polynomials.