Introduction to family algebras

Classical and quantum family algebras, previously introduced by the author and playing an important role in the theory of semi-simple Lie algebras and their representations are studied. Basic properties, structure theorems and explicit fomulas are obtained for both types of family algebras in many significant cases. Exact formulas (based on experimental calculations) for quantum eigenvalues, their multiplicities, and the trace of the so-called matrix of special elements are conjectured.