In this article, we introduce monomial irreducible representations of the special linear Lie algebra $sln$. We will show that this kind of representations have bases for which the action of the Chevalley generators of the Lie algebra on the basis elements can be given by a simple formula.

In this paper, we study the Cartan invariants for rank-one Cartan type Lie algebra W (1, n). This is done by reducing representations of generalized restricted Cartan type Lie algebra W (1, n) to representations of restricted Lie algebras W (1, 1) and of sl(2), and then extending Feldvoss-Nakano’s argument on W (1, 1) to the case W (1, n). Our discussion also shows some surprising difference be...