We will introduce an associative algebra, a quotient of the universal enveloping algebra of a graded, possibly infinite-dimensional Lie algebra. This algebra first appeared in the work of Seligman and later in a different context in the work of Chari-Pressley. It controls certain interesting highest weight representations of the Lie algebra in question. I will present some facts and open problems about this algebra.

The talk is directed to researchers in associative algebras, and will only assume basic Lie theory. It is based on my joint paper with Nathan Manning and Hadi Salmasian.