Toral Lie algebra

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In mathematics, a toral Lie algebra 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.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

In linear algebra, a square matrix is called diagonalizable or nondefective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix such that is a diagonal matrix. If is a finite-dimensional vector space, then a linear map is called diagonalizable if there exists an ordered basis of with respect to which is represented by a diagonal matrix. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. A square matrix that is not diagonalizable is called defective.

In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0.


Semisimple and reductive Lie algebras

A subalgebra H of a semisimple Lie algebra L is called toral if the adjoint representation of H on L, ad(H)gl(L) is a toral Lie algebra. 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 L restricted to H is nondegenerate.

Semisimple Lie algebra Direct sum of simple Lie algebras

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras whose only ideals are {0} and itself. It is important to emphasize that a one-dimensional Lie algebra is by definition not considered a simple Lie algebra, even though such an algebra certainly has no nontrivial ideals. Thus, one-dimensional algebras are not allowed as summands in a semisimple Lie algebra.

In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, whence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: there are alternative characterizations, given below.

Cartan subalgebra Nilpotent subalgebra of a Lie algebra

In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is self-normalising. They were introduced by Élie Cartan in his doctoral thesis.

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

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Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups.

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The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.