JOVANA
Library Glossary Getting Started Three Levels Fields How it works Mission
Join the mission
All guides

Semisimple Algebras and the Killing Form

Quotient out the largest solvable ideal — the radical — and you reach the semisimple world, where everything is rigid and computable. Cartan's criterion detects semisimplicity through the Killing form's nondegeneracy; complete reducibility (Weyl) and the decomposition into simple ideals follow. We compute the Killing form of sl(2) explicitly.

The radical and the definition of semisimple

Every finite-dimensional Lie algebra L has a unique maximal solvable ideal, the radical rad(L) (the sum of two solvable ideals is solvable, so a largest one exists). Define L to be semisimple if rad(L)=0 — no nonzero solvable ideal at all. The quotient L/rad(L) is always semisimple, so up to the solvable 'noise' every Lie algebra reduces to a semisimple one. A simple Lie algebra is a nonabelian one with no ideals but 0 and itself; semisimple turns out to mean exactly a direct sum of simple ideals.

The Killing form and Cartan's criterion

The Killing form is the symmetric bilinear form κ(x,y)=tr(ad(x)∘ad(y)). It is built only from the bracket, and it is associative (invariant): κ([x,y],z)=κ(x,[y,z]). This invariance forces its radical {x:κ(x,·)=0} to be an ideal. Cartan's criterion: L is semisimple if and only if κ is nondegenerate. (A companion criterion: L is solvable iff κ(L,[L,L])=0.) So a single bilinear form decides semisimplicity — this is the computational heart of the whole theory.

Killing form of sl(2,C).  Basis (h,e,f), [h,e]=2e, [h,f]=-2f, [e,f]=h.
Write ad(x) as a 3x3 matrix in the ordered basis (h,e,f).

ad(h): h->0, e->[h,e]=2e, f->[h,f]=-2f
   ad(h) = [0, 0, 0; 0, 2, 0; 0, 0, -2]
ad(e): h->[e,h]=-2e, e->[e,e]=0, f->[e,f]=h
   ad(e) = [0, 0, 1; -2, 0, 0; 0, 0, 0]
ad(f): h->[f,h]=2f, e->[f,e]=-h, f->0
   ad(f) = [0, -1, 0; 0, 0, 0; 2, 0, 0]

kappa(x,y) = tr( ad(x) ad(y) ):
  kappa(h,h) = tr(ad(h)^2) = 0^2 + 2^2 + (-2)^2 = 8
  kappa(e,f) = tr(ad(e) ad(f)):
     ad(e)ad(f) = [0,0,1;-2,0,0;0,0,0][0,-1,0;0,0,0;2,0,0]
                = [2, 0, 0; 0, 2, 0; 0, 0, 0],  trace = 4
  kappa(h,e)=kappa(h,f)=kappa(e,e)=kappa(f,f)=0.

Gram matrix in (h,e,f):  [8, 0, 0; 0, 0, 4; 0, 4, 0]
  det = 8 * ( 0*0 - 4*4 ) = 8 * (-16) = -128  != 0.
Nondegenerate  ==>  sl(2,C) is semisimple (indeed simple).
Killing form of sl(2,ℂ): Gram matrix [8,0,0;0,0,4;0,4,0], determinant −128 ≠ 0, so κ is nondegenerate and sl(2) is semisimple.

What nondegeneracy buys you

  1. Decomposition: a semisimple L is a direct sum of simple ideals L = L_1 ⊕ … ⊕ L_r, orthogonal under κ, and the decomposition is unique.
  2. L = [L,L]: a semisimple algebra equals its own derived algebra, so it has no nonzero abelian quotient.
  3. Weyl's theorem: every finite-dimensional representation of a semisimple L is completely reducible — a direct sum of irreducibles.
  4. ad is injective: Z(L)=0, so L embeds in gl(L); every derivation is inner. Semisimple algebras are rigid.

The complete-reducibility statement — complete reducibility of representations — is the analogue of Maschke's theorem for finite groups, and it is what lets us classify *representations* of semisimple algebras by their highest weights. But before representations, we need a coordinate system inside L itself. That coordinate system is the Cartan subalgebra and the root system of the final guide; the nondegenerate Killing form is precisely the inner product that makes roots into honest geometric vectors.