Cartan

Cartan may refer to several topics in mathematics, most notably connected with the French mathematician Élie Cartan (1869–1951). Cartan was a pioneer of modern differential geometry and played a foundational role in the development of Lie groups and Lie algebras. His work established methods and perspectives that influenced many areas of geometry and representation theory, including the moving frame method and Cartan’s method of equivalence for analyzing geometric structures.

In mathematics, several constructs bear the Cartan name. A Cartan subalgebra is a maximal nilpotent self-normalizing subalgebra of a Lie algebra; in semisimple Lie algebras it is a maximal abelian subalgebra consisting of semisimple elements. The Cartan matrix is an integer matrix that encodes the inner products of simple roots in a finite-dimensional semisimple Lie algebra, and it serves as the basis for the Dynkin diagram classification of these algebras. The Cartan decomposition refers to a specific splitting of a Lie algebra, typically g = k ⊕ p under an involution, which underlies the theory of symmetric spaces. Cartan geometry and Cartan connections generalize classical geometries by modeling spaces on homogeneous spaces G/H and describing them with a Cartan connection, a type of geometric gauge that extends notions of curvature and parallel transport. Cartan’s magic formula, L_X = i_X d + d i_X, relates Lie derivatives to exterior differentiation and interior multiplication and is a key identity in differential geometry.

Beyond Élie Cartan’s contributions, the name appears as a surname for various individuals in mathematics and science. The Cartan framework remains influential in geometry, representation theory, and mathematical physics.