Engel Identity

The Engel identity, named after Friedrich Engel, is a mathematical equation that is satisfied by all elements of a Lie ring, in the case of an Engel Lie ring, or by all the elements of a group, in the case of an Engel group. The Engel identity is the defining condition of an Engel group.

Formal definition

A Lie ring $L$ is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the Jacobi identity with respect to the Lie bracket ,y/math>, defined for all elements $x,y$ in the ring $L$. The Lie ring $L$ is defined to be an n-Engel Lie ring if and only if
* for all $x, y$ in $L$, the n-Engel identity
,[x, \ldots, [x,[x,y\ldots">,_\ldots,_[x,[x,y.html" ;"title=",[x, \ldots, [x,[x,y">,[x, \ldots, [x,[x,y\ldots = 0 (n copies of $x$), is satisfied.

In the case of a group $G$, in the preceding definition, use the definition and replace $0$ by $1$, where $1$ is the identity element of the group $G$.

See also

* Adjoint representation
* Efim Zelmanov">Adjoint representation of a Lie group">Adjoint representation
* Efim Zelmanov
* Engel's theorem

References

Group theory
Lie algebras

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