In mathematics, an Engel subalgebra of a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
with respect to some element ''x'' is the subalgebra of elements annihilated by some power of ad ''x''. Engel subalgebras are named after
Friedrich Engel. For finite-dimensional Lie algebras over infinite fields the minimal Engel subalgebras are the
Cartan subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
s.
See also
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Engel's theorem
In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra \mathfrak g is a nilpotent Lie algebra if and only if for each X \in \mathfrak g, the adjoint map
:\operatorname(X)\colon \mathfrak \ ...
References
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Lie algebras
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