Representation theory of finite-dimensional graded algebras with triangular decomposition

Many interesting algebras appearing in the context of algebraic Lie theory admit a “triangular” decomposition, a classical example being the universal enveloping algebra of a semisimple complex Lie algebra. The decomposition controls the representation theory to some extend and yields interesting combinatorics. In several examples one can obtain a finite-dimensional “restricted” version of the algebra of interest, still having a triangular decomposition and containing useful information. In my talk I will focus on such algebras in a general setting and discuss some representation-theoretic properties. The talk is addressed to a more general audience. This is based on joint work with Gwyn Bellamy (Glasgow).