Sl2-triple - Properties

Properties

Assume that g is a Lie algebra over a field of characteristic zero. From the representation theory of the Lie algebra sl₂, one concludes that the Lie algebra g decomposes into a direct sum of finite-dimensional subspaces, each of which is isomorphic to V_j, the j + 1-dimensional simple sl₂-module with highest weight j. The element h of the sl₂-triple is semisimple, with the simple eigenvalues j, j − 2, …, −j on a submodule of g isomorphic to V_j . The elements e and f move between different eigenspaces of h, increasing the eigenvalue by 2 in case of e and decreasing it by 2 in case of f. In particular, e and f are nilpotent elements of the Lie algebra g. Conversely, the Jacobson–Morozov theorem states that any nilpotent element e of a semisimple Lie algebra g can be included into an sl₂-triple {e,h,f}, and all such triples are conjugate under the action of the group Z_G(e), the centralizer of e in the adjoint Lie group G corresponding to the Lie algebra g. The semisimple element h of any sl₂-triple containing a given nilpotent element e of g is called a characteristic of e.

An sl₂-triple defines a grading on g according to the eigenvalues of h:

The sl₂-triple is called even if only even j occur in this decomposition, and odd otherwise.

If g is a semisimple Lie algebra, then g₀ is a reductive Lie subalgebra of g (it is not semisimple in general). Moreover, the direct sum of the eigenspaces of h with non-negative eigenvalues is a parabolic subalgebra of g with the Levi component g₀.