cartan subalgebra

In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra





h




{\displaystyle {\mathfrak {h}}}

of a Lie algebra





g




{\displaystyle {\mathfrak {g}}}

that is self-normalising (if



[
X
,
Y
]
∈


h




{\displaystyle [X,Y]\in {\mathfrak {h}}}

for all



X
∈


h




{\displaystyle X\in {\mathfrak {h}}}

, then



Y
∈


h




{\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





g




{\displaystyle {\mathfrak {g}}}

over a field of characteristic



0


{\displaystyle 0}

.
In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g.,




C



{\displaystyle \mathbb {C} }

), a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements x such that the adjoint endomorphism



ad
⁡
(
x
)
:


g


→


g




{\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.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).

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