sl2-triple

In the theory of Lie algebras, an sl₂-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 sl₂. This notion plays an important role in the theory of semisimple Lie algebras, especially in regards to their nilpotent orbits.

Definition

Elements {e,h,f} of a Lie algebra g form an sl₂-triple if

$[h,e]=2e,\quad [h,f]=-2f,\quad [e,f]=h.$

These commutation relations are satisfied by the generators

$h={\begin{bmatrix}1&0\\0&-1\end{bmatrix}},\quad e={\begin{bmatrix}0&1\\0&0\end{bmatrix}},\quad f={\begin{bmatrix}0&0\\1&0\end{bmatrix}}$

of the Lie algebra sl₂ of 2 by 2 matrices with zero trace. It follows that sl₂-triples in g are in a bijective correspondence with the Lie algebra homomorphisms from sl₂ into g.

The alternative notation for the elements of an sl₂-triple is {H, X, Y}, with H corresponding to h, X corresponding to e, and Y corresponding to f.

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:

$g=\bigoplus _{j\in \mathbb {Z} }g_{j},\quad [h,a]=ja{\ \ }{\textrm {for}}{\ \ }a\in g_{j}.$

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 g_0.

References

• A. L. Onishchik, E. B. Vinberg, V. V. Gorbatsevich, Structure of Lie groups and Lie algebras. Lie groups and Lie algebras, III. Encyclopaedia of Mathematical Sciences, 41. Springer-Verlag, Berlin, 1994. iv+248 pp. (A translation of Current problems in mathematics. Fundamental directions. Vol. 41, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990. Translation by V. Minachin. Translation edited by A. L. Onishchik and E. B. Vinberg) ISBN 3-540-54683-9

• V. L. Popov, E. B. Vinberg, Invariant theory. Algebraic geometry. IV. Linear algebraic groups. Encyclopaedia of Mathematical Sciences, 55. Springer-Verlag, Berlin, 1994. vi+284 pp. (A translation of Algebraic geometry. 4, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989. Translation edited by A. N. Parshin and I. R. Shafarevich) ISBN 3-540-54682-0