Sl2-triple

In the theory of Lie algebras, an "sl"₂-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"₂. 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 "sl"₂-triple if

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

These commutation relations are satisfied by the generators

:

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

The alternative notation for the elements of an "sl"₂-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 "sl"₂, 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 "sl"₂-module with highest weight "j". The element "h" of the "sl"₂-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 "sl"₂-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 "sl"₂-triple containing a given nilpotent element "e" of "g" is called a characteristic of "e".

An "sl"₂-triple defines a grading on "g" according to the eigenvalues of "h":

:

The "sl"₂-triple is called even if only even "j" occur in this decomposition, and odd otherwise.

If "g" is a semisimple Lie algebra, then "g"₀ 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

* E. B. Vinberg, V. V. Gorbatsevich, A. L. Onishchik, "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

* E. B. Vinberg, V. L. Popov, "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