Semisimple

Semisimple is a term used in several areas of algebra to describe objects that decompose into simple pieces.

In module theory, a left R-module M is semisimple (completely reducible) if it is a direct sum of simple submodules. Equivalently, every submodule of M has a complementary submodule, or every short exact sequence of R-modules splits. If M has finite length, these conditions are equivalent.

In ring theory, a ring R is semisimple if it is semisimple as a left R-module. Equivalently, its Jacobson radical J(R) is zero and R is Artinian. The Wedderburn-Artin theorem characterizes semisimple rings as finite products of matrix algebras over division rings: R ≅ ∏ M_{n_i}(D_i).

In Lie theory, a finite-dimensional Lie algebra g over a field of characteristic zero is semisimple if its radical is zero; equivalently, g is a direct sum of simple Lie algebras, and the Killing form is nondegenerate. Semisimple Lie algebras are classified by root systems, with standard examples including sl_n, so_n, and sp_{2n}.

In representation theory and group theory, Maschke's theorem shows that over a field whose characteristic does not divide the order of a finite group G, the group algebra F[G] is semisimple, so every F[G]-module is completely reducible.

Across these contexts, semisimplicity signals the ability to break structures into simple, well-understood components, enabling a detailed and tractable theory of their representations and modules.