Cartans

Cartans is a surname of French origin and may refer to people bearing the Cartan name, or to mathematical constructs named after Élie Cartan, a French mathematician who helped establish modern Lie theory. The term is most commonly encountered in the context of Lie algebras and Lie groups, where several foundational concepts carry the Cartan name.

Among the best known Cartan-based concepts are Cartan matrices, Cartan subalgebras, and Cartan subgroups. A Cartan matrix is an integer matrix associated with a root system; its entries encode inner products between simple roots and determine the Dynkin diagram that classifies finite-dimensional semisimple Lie algebras. A Cartan subalgebra of a Lie algebra is a maximal subalgebra consisting of elements that act diagonally in a suitable representation, providing a root space decomposition of the algebra. A Cartan subgroup is a maximal torus in a Lie group; in complex semisimple groups these subgroups are connected and abelian and play a central role in the structure and representation theory of the group.

Cartan decompositions and related tools, named after Élie Cartan, are used to analyze the geometry and representation theory of Lie groups and algebras. The surname is associated with several generations of French mathematicians, including Élie Cartan and, in later generations, Henri Cartan, who contributed to algebraic topology and complex analysis.

In summary, Cartans commonly refers to mathematical constructs tied to Élie Cartan’s work in Lie theory, as well as to individuals who bear the Cartan surname.