Smith algebras: a tour through symmetry, geometry and representation theory

After some motivation and discussing how they appear naturally through the lens of (noncommutative) geometry, we will discuss Smith algebras and their representations. In particular, following the seminal works of Nilsson, we will discuss representations which are free of finite rank over a subalgebra which plays a role analogous to that of a Cartan subalgebra. 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, China) and E. Mendonça (Lyon, France & USP, Brasil).