Sl2-triple

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. H is called a neutral, X is called a nilpositive, and Y is called a nilnegative.

Properties

Assume that g is a finite dimensional 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 Vj, 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 Vj . 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 g0.

If the elements of an sl2-triple are regular, then their span is called a principal subalgebra.

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)
• 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)