We will discuss representations of the Smith algebra which are free of finite rank over a subalgebra which plays a role analogous to that of the (enveloping algebra of the) Cartan subalgebra of the simple Lie algebra \( \mathfrak{sl}_2 \). In the case of rank 1, we obtain a full description of the isomorphism classes, a simplicity criterion, and a combinatorial algorithm to produce all composition series and the multiplicities of the simple factors.

This is joint work with V. Futorny (SUSTech & USP) and E. Mendonça (Lyon & USP).