Wurzelraum

Wurzelraum (root space) is a term in Lie theory used to denote a specific eigenspace of the adjoint action of a Cartan subalgebra. In German mathematical literature, Wurzelraum is the standard term for a root space.

Let g be a finite-dimensional Lie algebra over a field of characteristic zero, and let h be a Cartan subalgebra of g. For each linear functional α on h, the root space is defined by

g_α = { x in g | [h, x] = α(h) x for all h in h }.

The set Δ = { α ≠ 0 | g_α ≠ 0 } is called the root system. The Lie algebra admits a decomposition g = ⊕_{α ∈ Δ} g_α ⊕ g_0, where g_0 = { x in g | [h, x] = 0 for all h in h }. In semisimple Lie algebras, g_0 equals h and each nonzero root space g_α is one-dimensional, so dim g_α = 1.

Example: sl_3(C) has h as the diagonal traceless matrices. The roots are α_i − α_j (i ≠ j), and the root space g_{α_i−α_j} is spanned by the matrix unit E_{ij}.

Significance: root spaces organize the structure of g, enabling the root system and its symmetries (Weyl group) to describe the algebra's geometry and representation theory. They underpin essential constructions such as the Cartan decomposition, Dynkin diagrams, and the classification of semisimple Lie algebras. Generalizations extend to infinite-dimensional algebras (e.g., affine Kac–Moody algebras), where multiplicities and imaginary roots arise.