a semisimple Lie algebra
Among Lie algebras, some are 'rigid' and beautifully classifiable, while others have a soft, solvable part that complicates everything. Semisimple Lie algebras are the rigid ones — the algebras with no solvable mess, built from simple pieces with nothing abelian sticking out. They are the ones with a complete and elegant classification by Dynkin diagrams, and the natural home of representation theory.
There are several equivalent definitions; pick whichever is most useful. (1) g is semisimple if it has no nonzero solvable ideal (where the radical, the largest solvable ideal, is zero). (2) Cartan's criterion: g is semisimple if and only if its Killing form B is nondegenerate. (3) g is semisimple if and only if it is a direct sum of simple Lie algebras (nonabelian algebras with no proper nonzero ideals). A simple example of what is excluded: any algebra with a center, or the algebra of upper-triangular matrices (solvable), is not semisimple; sl(n), so(n) for n >= 3 (except n=4 which splits), su(n), and sp(n) are semisimple. The complex simple Lie algebras are classified into four infinite families A_n, B_n, C_n, D_n (corresponding to sl(n+1), so(2n+1), sp(2n), so(2n)) plus five exceptionals G_2, F_4, E_6, E_7, E_8 — a list closed in the 1890s.
Semisimplicity is the hypothesis under which Lie theory is cleanest: every finite-dimensional representation is completely reducible (Weyl's theorem), the structure is governed by a root system and a Cartan subalgebra, and compact semisimple groups have bi-invariant metrics from the negated Killing form. The honest fine print: 'semisimple' is strictly stronger than 'simple' for groups but the words tangle — gl(n) is reductive but not semisimple (it has the scalar center), while sl(n) is semisimple. And the general structure theorem (Levi decomposition) says ANY finite-dimensional Lie algebra is a semidirect product of its solvable radical and a semisimple subalgebra, so semisimple algebras are the irreducible 'hard core' that the solvable part wraps around.
sl(2, C), the trace-zero 2-by-2 complex matrices, is the smallest simple (hence semisimple) Lie algebra, with basis H = diag(1,-1), E = [0,1;0,0], F = [0,0;1,0] and relations [H,E]=2E, [H,F]=-2F, [E,F]=H. Its Killing form is nondegenerate (Cartan's criterion confirms semisimplicity), and its representation theory — the 'ladder operators' E, F raising and lowering H-eigenvalues — is the template for all of Lie representation theory and for angular momentum in quantum mechanics.
sl(2,C) with its ladder operators E, F, H is the template case of semisimple representation theory.
Semisimple is not the same as simple, and gl(n) (reductive, with a central scalar) is NOT semisimple though sl(n) is. By Levi decomposition every Lie algebra is solvable-radical plus a semisimple part — semisimplicity isolates the rigid core, it does not describe all algebras.