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

• for all

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

• Adjoint representation
• Efim Zelmanov
• Engel's theorem

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.