• Source: Toral subalgebra

In mathematics, a toral subalgebra is a Lie subalgebra of a general linear Lie algebra all of whose elements are semisimple (or diagonalizable over an algebraically closed field). Equivalently, a Lie algebra is toral if it contains no nonzero nilpotent elements. Over an algebraically closed field, every toral Lie algebra is abelian; thus, its elements are simultaneously diagonalizable.




In semisimple and reductive Lie algebras


A subalgebra





h




{\displaystyle {\mathfrak {h}}}

of a semisimple Lie algebra





g




{\displaystyle {\mathfrak {g}}}

is called toral if the adjoint representation of





h




{\displaystyle {\mathfrak {h}}}

on





g




{\displaystyle {\mathfrak {g}}}

,



ad
⁡
(


h


)
⊂


g
l


(


g


)


{\displaystyle \operatorname {ad} ({\mathfrak {h}})\subset {\mathfrak {gl}}({\mathfrak {g}})}

is a toral subalgebra. A maximal toral Lie subalgebra of a finite-dimensional semisimple Lie algebra, or more generally of a finite-dimensional reductive Lie algebra, over an algebraically closed field of characteristic 0 is a Cartan subalgebra and vice versa. In particular, a maximal toral Lie subalgebra in this setting is self-normalizing, coincides with its centralizer, and the Killing form of





g




{\displaystyle {\mathfrak {g}}}

restricted to





h




{\displaystyle {\mathfrak {h}}}

is nondegenerate.
For more general Lie algebras, a Cartan subalgebra may differ from a maximal toral subalgebra.
In a finite-dimensional semisimple Lie algebra





g




{\displaystyle {\mathfrak {g}}}

over an algebraically closed field of a characteristic zero, a toral subalgebra exists. In fact, if





g




{\displaystyle {\mathfrak {g}}}

has only nilpotent elements, then it is nilpotent (Engel's theorem), but then its Killing form is identically zero, contradicting semisimplicity. Hence,





g




{\displaystyle {\mathfrak {g}}}

must have a nonzero semisimple element, say x; the linear span of x is then a toral subalgebra.




See also


Maximal torus, in the theory of Lie groups




References



Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012
Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90053-7

Kata Kunci Pencarian:

• Toral subalgebra
• Cartan subalgebra
• Semisimple Lie algebra
• Glossary of Lie groups and Lie algebras
• Maximal torus
• Hermitian symmetric space
• Complexification (Lie group)
• Glossary of commutative algebra