a Cartan subalgebra and roots
/ kar-TAHN (Élie Cartan) /
To understand a complicated symmetry algebra, first find the largest collection of symmetries that all commute with each other — these can be diagonalized simultaneously, like choosing axes. Then watch how everything else fails to commute with them: the failure patterns are discrete 'directions' called roots, and they encode the entire algebra in a finite picture of vectors and angles. This decomposition turns the soft problem of a Lie algebra into the hard combinatorics of a root system.
For a complex semisimple Lie algebra g, a Cartan subalgebra h is a maximal abelian subalgebra whose elements act diagonalizably (on g via ad); its dimension is the rank of g. Because the operators ad(H) for H in h commute and are diagonalizable, they share eigenspaces, giving the root-space decomposition g = h (+) (sum over roots alpha of g_alpha), where for a nonzero linear functional alpha in h^* the root space is g_alpha = {X in g : [H, X] = alpha(H) X for all H in h}, a (typically 1-dimensional) common eigenspace. The nonzero alpha that actually occur are the roots, a finite set forming a root system in h^*: it is symmetric (alpha a root implies -alpha a root) and closed under the reflections generated by the roots (the Weyl group). For sl(n+1) the roots are e_i - e_j (the A_n system); the angles and lengths among the roots, recorded in a Dynkin diagram, determine g up to isomorphism. The bracket of root vectors obeys [g_alpha, g_beta] subset g_(alpha+beta), so the root system is a complete addition table for the algebra.
Cartan subalgebras and roots are the engine of the classification of semisimple Lie algebras and of their representations (weights are the analogous eigenvalues for any representation, not just the adjoint one). In a compact group, exp of a Cartan subalgebra is a maximal torus, and characters of representations restrict to it. Two honesty notes. First, this clean theory is for SEMISIMPLE (or reductive) algebras over an algebraically closed field of characteristic zero; over R the picture splits into real forms and is subtler. Second, all Cartan subalgebras of a given g are conjugate, so the rank and root system are well defined, but a Cartan subalgebra is not canonical — you choose one, just as you choose coordinate axes.
In sl(2, C), take the Cartan subalgebra h = span{H} with H = diag(1, -1) (rank 1). Then [H, E] = 2E and [H, F] = -2F, so E spans the root space for the root alpha (alpha(H) = 2) and F spans the root space for -alpha. The root system is {+2, -2} on the line h^* — the smallest root system, A_1.
sl(2,C): one Cartan generator H and the two roots +-2 — the rank-1 root system A_1.
The clean root-space theory is for semisimple (reductive) algebras over an algebraically closed field of characteristic 0. Over R one must pass to real forms. All Cartan subalgebras are conjugate (rank is well defined) but no single one is canonical.