Toral subalgebra
Template:Short description 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
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,Template:Fact 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
- Maximal torus, in the theory of Lie groups
References
- β 1.0 1.1 1.2 Template:Harvnb
- β 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, .
- β Template:Harvnb