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Subalgebras, Ideals, and the Bridge to Lie Groups

The internal anatomy of a Lie algebra — subalgebras, ideals, quotients, homomorphisms — mirrors group theory exactly, with the bracket replacing conjugation. We then make precise the slogan that a Lie algebra is the tangent space of a Lie group at the identity, with the bracket as infinitesimal commutator.

Subalgebras versus ideals

A subspace M⊆L is a subalgebra if [M,M]⊆M, and an ideal if the stronger [L,M]⊆M holds. The distinction is the exact analogue of subgroup versus normal subgroup: an ideal is closed under bracketing with *everything*, just as a normal subgroup is closed under conjugation by everything. And just as with groups, you can form the quotient L/M precisely when M is an ideal, with bracket [x+M,y+M]=[x,y]+M.

Three ideals you always have: the center Z(L)={x:[x,y]=0 for all y} (the kernel of ad); the derived algebra [L,L], spanned by all brackets, the analogue of the commutator subgroup; and L itself together with 0. A homomorphism φ:L→L′ has kernel an ideal and image a subalgebra, and the isomorphism theorems hold verbatim: L/ker φ ≅ im φ, and so on. If you know the group versions, you already know these.

Example: the upper-triangular and strictly-upper-triangular subalgebras of gl(3).

b = { upper triangular } = span of E_11, E_22, E_33, E_12, E_13, E_23
n = { strictly upper triangular } = span of E_12, E_13, E_23

Is n an ideal of b?  Check [b, n] subset of n.
  Recall [E_ij, E_kl] = delta_jk E_il - delta_li E_kj.
  [E_11, E_12] = E_12 - 0 = E_12   (in n)
  [E_22, E_12] = 0 - E_12 = -E_12  (in n)
  [E_12, E_23] = E_13 ,  [E_13, anything in n] = 0 here
Every bracket [diagonal or upper, strictly upper] lands in n. So n is an ideal of b.

Quotient b/n:  brackets of diagonal parts vanish mod n, so
  b/n  is abelian, dim 3 (the diagonal h).
Meanwhile [b,b] = n exactly:  the derived algebra of b is n.
n (strictly upper-triangular) is an ideal of b (upper-triangular); the quotient b/n is the abelian diagonal. This is the smallest nontrivial flag picture.

The bridge to Lie groups

Here is the picture that names the whole subject. A Lie group G is a group that is also a smooth manifold, with smooth multiplication and inversion — think GL(n,ℝ), SO(3), the unitary group U(n). Its Lie algebra is the tangent space at the identity, L=T_eG, equipped with a bracket. The bracket is not arbitrary: it is the *infinitesimal commutator*. If you take two near-identity elements, conjugate, and expand to second order, the leading nonabelian term is exactly [X,Y].

  1. Each X in L gives a one-parameter subgroup t↦exp(tX) in G; the matrix exponential exp(X)=I+X+X²/2+… is the prototype.
  2. Conjugation in G differentiates to the adjoint representation Ad of G on L; differentiating Ad again gives ad, and ad(X)(Y)=[X,Y].
  3. The Baker–Campbell–Hausdorff formula exp(X)exp(Y)=exp(X+Y+½[X,Y]+…) reassembles the group product from bracket data alone.