Engel identity explained

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

L

is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the Jacobi identity with respect to the Lie bracket

[x,y]

, 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

x,y

in

L

, the n-Engel identity

[x,[x,\ldots,[x,[x,y]]\ldots]]=0

(n copies of

x

), is satisfied.[1]

G

, in the preceding definition, use the definition and replace

0

by

1

, where

1

is the identity element of the group

G

.[2]

See also

Notes and References

  1. Engel Lie-Algebras. Traustason. Gunnar. Quart. J. Math. Oxford. 1993. 44. 3. 355–384. 10.1093/qmath/44.3.355.
  2. Web site: Engel groups (a survey). Traustason, Gunnar.