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 regard to their nilpotent orbits.

Definition

Elements of a Lie algebra ''g'' form an ''sl''₂-triple if

: ,e= 2e, \quad ,f= -2f, \quad ,f= h.

These commutation relations are satisfied by the generators

: $h = \begin 1 & 0\\ 0 & -1 \end, \quad e = \begin 0 & 1\\ 0 & 0 \end, \quad f = \begin 0 & 0\\ 1 & 0 \end$

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 , 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 ''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 , 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'':

: $g = \bigoplus_ g_j,\quad$, a ja \textrm a\in g_j.

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.

If the elements of an ''sl''₂-triple are regular, then their span is called a principal subalgebra.

See also

*Affine Weyl group
*Finite Coxeter group
*Hasse diagram
*Linear algebraic group
*Nilpotent orbit
*Root system
*Special linear Lie algebra
*Weyl group

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) {{isbn, 3-540-54682-0

Lie algebras