Engel identity
This article needs additional citations for verification. (June 2012) |
The Engel identity, named after
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
is defined as a anticommutative and satisfies the Jacobi identity with respect to the Lie bracket
, defined for all elements in the ring . The Lie ring is defined to be an n-Engel Lie ring if and only if
- for all in , the n-Engel identity
(n copies of ), is satisfied.[1]
In the case of a group , in the preceding definition, use the definition [x,y] = x−1 • y−1 • x • y and replace by , where is the identity element of the group .[2]
See also
- Adjoint representation
- Efim Zelmanov
- Engel's theorem
References
- .
- ^ Traustason, Gunnar. "Engel groups (a survey)" (PDF).
{{cite journal}}
: Cite journal requires|journal=
(help)