In mathematics, a Cartan subalgebra is a certain kind of subalgebra of a Lie algebra. The subalgebra of a Lie algebra is a Cartan subalgebra if is an abelian subalgebra, with the following property of its adjoint representation: the weighteigenspaces of restricted to diagonalize the representation, and the eigenspace of the zero weight vector is itself. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to A. Since the axioms of algebraic structures in universal algebra are described by equational laws, the... In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... The adjoint representation of a Lie group G is the linearized version of the action of G on itself by conjugation. ... Given a set S of complex matrices, each of which is diagonalizable and any two of which commute under multiplication, it is always possible to diagonalize all the elements of S simultaneously. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, if G is a group and H a subgroup, then for any linear representation ρ of G, we can define the restricted representation ρ|H by simply setting ρ|H(h) = ρ(h). ...
The name is for Élie Cartan. Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ...
For example, for the algebras of n by n matrices of trace zero, the bilinear form is (a, b) = Trace(ab), the Cartan involution is given by minus the transpose, and the grading can be given by "distance from the diagonal" so that the Cartansubalgebra is the diagonal elements.
A generalized Kac-Moody algebra is obtained from a universal one by changing the Cartan matrix, by the operations of killing something in the center, or taking a central extension, or adding outer derivations.
Algebras with Lorentzian Cartansubalgebra whose denominator function is an automorphic form of singular weight.
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