In the theory of Lie algebras, an sl<sub>2</sub>-triple is a triple of elements of a Lie algebra that satisfy the commutation relations between the standard generators of the special linear Lie algebra sl<sub>2</sub>. This notion plays an important role in the theory of semisimple Lie algebras, especially in regard to their nilpotent orbits.
Elements {e,h,f} of a Lie algebra g form an sl<sub>2</sub>-triple if
These commutation relations are satisfied by the generators
of the Lie algebra sl<sub>2</sub> of 2 by 2 matrices with zero trace. It follows that sl<sub>2</sub>-triples in g are in a bijective correspondence with the Lie algebra homomorphisms from sl<sub>2</sub> into g.
The alternative notation for the elements of an sl<sub>2</sub>-triple is {H, X, Y}, with H corresponding to h, X corresponding to e, and Y corresponding to f. H is called a neutral, X is called a nilpositive, and Y is called a nilnegative.
Assume that g is a finite dimensional Lie algebra over a field of characteristic zero. From the representation theory of the Lie algebra sl<sub>2</sub>, one concludes that the Lie algebra g decomposes into a direct sum of finite-dimensional subspaces, each of which is isomorphic to V<sub>j</sub>, the (j + 1)-dimensional simple sl<sub>2</sub>-module with highest weight j. The element h of the sl<sub>2</sub>-triple is semisimple, with the simple eigenvalues j, j − 2, ..., −j on a submodule of g isomorphic to V<sub>j</sub> . 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 JacobsonâÂÂMorozov theorem states that any nilpotent element e of a semisimple Lie algebra g can be included into an sl<sub>2</sub>-triple {e,h,f}, and all such triples are conjugate under the action of the group Z<sub>G</sub>(e), the centralizer of e in the adjoint Lie group G corresponding to the Lie algebra g.
The semisimple element h of any sl<sub>2</sub>-triple containing a given nilpotent element e of g is called a characteristic of e.
An sl<sub>2</sub>-triple defines a grading on g according to the eigenvalues of h:
The sl<sub>2</sub>-triple is called even if only even j occur in this decomposition, and odd otherwise.
If g is a semisimple Lie algebra, then g<sub>0</sub> 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 g<sub>0</sub>.
If the elements of an sl<sub>2</sub>-triple are regular, then their span is called a principal subalgebra.