In mathematics, an element ''x'' of a

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

or 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 iden ...

is called an ''n''-Engel element, named after Friedrich Engel, if it satisfies the ''n''-Engel condition that the repeated commutator .. ''x'',''y''">..''x'',''y''''y'' ..., ''y''">''x'',''y''''y''">''x'',''y''">..''x'',''y''''y'' ..., ''y''ref>In other words, ''n'' "["s and n copies of y, for example, [x,y],y],y], x,y],y],y],y]. [x,y],y],y],y],y], and so on. with ''n'' copies of ''y'' is trivial (where [''x'', ''y''] means ''x''⁻¹''y''⁻¹''xy'' or the

Lie bracket 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 identi ...

). It is called an Engel element if it satisfies the Engel condition that it is ''n''-Engel for some ''n''. A Lie group or Lie algebra is said to satisfy the Engel or ''n''-Engel conditions if every element does. Such groups or algebras are called Engel groups, ''n''-Engel groups, Engel algebras, and ''n''-Engel algebras. Every

nilpotent group In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . In ...

or Lie algebra is Engel.

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 representation of a Lie algebra">adjoint m ...

states that every finite-dimensional Engel algebra is nilpotent. gave examples of non-nilpotent Engel groups and algebras.

Notes

*{{Citation , last1=Cohn , first1=P. M. , authorlink = Paul Cohn, title=A non-nilpotent Lie ring satisfying the Engel condition and a non-nilpotent Engel group , mr=0071720 , year=1955 , journal=Proc. Cambridge Philos. Soc. , volume=51 , pages=401–405 , doi=10.1017/S0305004100030395 , issue=3, bibcode=1955PCPS...51..401C Group theory Lie algebras