In
mathematics, a toral subalgebra is a
Lie subalgebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of a general linear Lie algebra all of whose elements are
semisimple
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
(or
diagonalizable
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
over an algebraically closed field).
Equivalently, a Lie algebra is toral if it contains no nonzero
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
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
of a
semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra is ...
is called toral if the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
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, over an algebraically closed field of characteristic 0 is a
Cartan subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
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
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.
In fact, if
has only nilpotent elements, then it is
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
(
Engel's theorem
In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra \mathfrak g is a nilpotent Lie algebra if and only if for each X \in \mathfrak g, the adjoint map
:\operatorname(X)\colon \mathfrak \ ...
), 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 mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.
A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefor ...
, in the theory of Lie groups
References
*
*{{Citation , last1=Humphreys , first1=James E. , title=Introduction to Lie Algebras and Representation Theory , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, New York , isbn=978-0-387-90053-7 , year=1972 , url-access=registration , url=https://archive.org/details/introductiontoli00jame
Properties of Lie algebras