Cartan subalgebra

Lie groups and Lie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem
Lie groups in physics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
v t e

In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra ${\displaystyle {\mathfrak {h}}}$ of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ that is self-normalising (if ${\displaystyle [X,Y]\in {\mathfrak {h}}}$ for all ${\displaystyle X\in {\mathfrak {h}}}$, then ${\displaystyle Y\in {\mathfrak {h}}}$). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra ${\displaystyle {\mathfrak {g}}}$ over a field of characteristic ${\displaystyle 0}$.

In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g., ${\displaystyle \mathbb {C} }$), a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements x such that the adjoint endomorphism ${\displaystyle \operatorname {ad} (x):{\mathfrak {g}}\to {\mathfrak {g}}}$ is semisimple (i.e., diagonalizable). Sometimes this characterization is simply taken as the definition of a Cartan subalgebra.^{[1]pg 231}

In general, a subalgebra is called toral if it consists of semisimple elements. Over an algebraically closed field, a toral subalgebra is automatically abelian. Thus, over an algebraically closed field of characteristic zero, a Cartan subalgebra can also be defined as a maximal toral subalgebra.

Kac–Moody algebras and generalized Kac–Moody algebras also have subalgebras that play the same role as the Cartan subalgebras of semisimple Lie algebras (over a field of characteristic zero).