Spectral theory within the framework of locally solvable Lie algebras

It is going on lecture notes of my talk in the special session of Seminar Sophus Lie -- ``Workshop on Lie groups and Lie algebras'', September 4--15, 2000, Bedlewo, Poland.

The exposition is an invitation for the reader to explore a field where Lie theory and operator theory are interacting. The discussion is concerned with some of the main achievements related to applications of Lie algebras to the problem of finding satisfactory non-commutative versions of the several variable spectral theory constructed by J.L. Taylor in the seventies. Virtually everybody could understand what is going on, in view of the illustrating examples developed in a self-contained manner.