Toral subalgebra

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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).[1] Equivalently, a Lie algebra is toral if it contains no nonzero nilpotent elements. Over an algebraically closed field, every toral Lie algebra is abelian;[1][2] thus, its elements are simultaneously diagonalizable.

In semisimple and reductive Lie algebras

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A subalgebra   of a semisimple Lie algebra   is called toral if the adjoint representation of   on  ,   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,[citation needed] over an algebraically closed field of characteristic 0 is a Cartan subalgebra and vice versa.[3] In particular, a maximal toral Lie subalgebra in this setting is self-normalizing, coincides with its centralizer, and the Killing form of   restricted to   is nondegenerate.

For more general Lie algebras, a Cartan subalgebra may differ from a maximal toral subalgebra.

In a finite-dimensional semisimple Lie algebra   over an algebraically closed field of a characteristic zero, a toral subalgebra exists.[1] In fact, if   has only nilpotent elements, then it is nilpotent (Engel's theorem), but then its Killing form is identically zero, contradicting semisimplicity. Hence,   must have a nonzero semisimple element, say x; the linear span of x is then a toral subalgebra.

See also

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References

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  1. ^ a b c Humphreys 1972, Ch. II, § 8.1.
  2. ^ Proof (from Humphreys): Let  . Since   is diagonalizable, it is enough to show the eigenvalues of   are all zero. Let   be an eigenvector of   with eigenvalue  . Then   is a sum of eigenvectors of   and then   is a linear combination of eigenvectors of   with nonzero eigenvalues. But, unless  , we have that   is an eigenvector of   with eigenvalue zero, a contradiction. Thus,  .  
  3. ^ Humphreys 1972, Ch. IV, § 15.3. Corollary
  • 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