Toral subalgebra

{{Short description|Lie algebra all of which elements are semisimple}}

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).{{harvnb|Humphreys|1972|loc=Ch. II, § 8.1.}} Equivalently, a Lie algebra is toral if it contains no nonzero nilpotent elements. Over an algebraically closed field, every toral Lie algebra is abelian;Proof (from Humphreys): Let $x \in \mathfrak{h}$ . Since $\operatorname{ad}(x)$ is diagonalizable, it is enough to show the eigenvalues of $\operatorname{ad}_{\mathfrak{h}}(x)$ are all zero. Let $y \in \mathfrak{h}$ be an eigenvector of $\operatorname{ad}_{\mathfrak{h}}(x)$ with eigenvalue $\lambda$ . Then $x$ is a sum of eigenvectors of $\operatorname{ad}_{\mathfrak{h}}(y)$ and then $-\lambda y = \operatorname{ad}_{\mathfrak{h}}(y)x$ is a linear combination of eigenvectors of $\operatorname{ad}_{\mathfrak{h}}(y)$ with nonzero eigenvalues. But, unless $\lambda = 0$ , we have that $-\lambda y$ is an eigenvector of $\operatorname{ad}_{\mathfrak{h}}(y)$ with eigenvalue zero, a contradiction. Thus, $\lambda = 0$ . $\square$ thus, its elements are simultaneously diagonalizable.

In semisimple and reductive Lie algebras

A subalgebra $\mathfrak h$ of a semisimple Lie algebra $\mathfrak g$ is called toral if the adjoint representation of $\mathfrak h$ on $\mathfrak g$ , $\operatorname{ad}(\mathfrak h) \subset \mathfrak{gl}(\mathfrak g)$ 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,{{fact|date=January 2020}} over an algebraically closed field of characteristic 0 is a Cartan subalgebra and vice versa.{{harvnb|Humphreys|1972|loc=Ch. IV, § 15.3. Corollary}} In particular, a maximal toral Lie subalgebra in this setting is self-normalizing, coincides with its centralizer, and the Killing form of $\mathfrak g$ restricted to $\mathfrak h$ is nondegenerate.

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

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

References

{{Citation | last1=Borel | first1=Armand | author1-link=Armand Borel | title=Linear algebraic groups | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-97370-8 |mr=1102012 | year=1991 | volume=126}}
{{Citation | last1=Humphreys | first1=James E. | title=Introduction to Lie Algebras and Representation Theory | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90053-7 | year=1972 | url-access=registration | url=https://archive.org/details/introductiontoli00jame }}

Category:Properties of Lie algebras

Toral subalgebra

In semisimple and reductive Lie algebras

See also

References