Composition series of a module ($0=M_0\subset M_1\subset \cdots \subset M_r=M$)

A composition series of a module is a finite chain of submodules whose nonzero successive quotient modules are simple modules. Every finite-dimensional Lie algebra representation has such a chain, viewed as a module over the universal enveloping algebra.

Every two finite composition series of a module have the same multiset of simple composition factors, up to isomorphism and order. One proves this by induction on the length: if the first simple submodules differ, their intersection is zero, and the quotient by their sum identifies the remaining common comparison problem; if they coincide, pass directly to the quotient. The theorem makes the multiplicity of a simple module independent of a chosen composition series. It does not assert that the module is a direct sum of its composition factors.

A composition factor is an irreducible successive quotient in a composition series of a module. For a finite-dimensional complex Lie algebra representation and a central Casimir operator, its scalar on a composition factor is an eigenvalue of the operator on the original Lie algebra representation.