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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.

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) 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

@@ Line 30: / Line 30: @@
 Assume that ''g'' is a Lie algebra over a [[field (mathematics)|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 [[eigenvalue]]s ''j'', ''j'' &minus; 2, &hellip;, &minus;''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 orbit|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''<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''.
@@ Line 39: / Line 41: @@
 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 [[Semisimple Lie algebra|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</sup>.
+If the elements of an ''sl''<sub>2</sub>-triple are [[regular element of a Lie algebra|regular]], then their span is called a [[principal subalgebra]].
 == References ==