Lie ring

This article defines a non-associative ring property: a property that an be evaluated to true or false for any non-associative ring.
View other non-associative ring properties

Definition

A Lie ring is a set $L$ equipped with the structure of an abelian group (operation denoted $+$) and a bracket operation $[\ ,\ ]$ (called a Lie bracket) satisfying the following additional conditions:

Condition name	Explicit identities (all variable letters are universally quantified over $L$)
$\mathbb{Z}$-bilinear, also known as biadditive, also known as left and right distributive	Additive in left coordinate: $[x+y,z] = [x,z] + [y,z]$ Additive in right coordinate: $[x,y+z] = [x,y] + [x,z]$
alternating (hence skew-symmetric)	Alternation: $[x,x] = 0$ Skew symmetry: $[x,y] + [y,x] = 0$ The second condition (skew symmetry) follows from the first (alternation); the reverse implication holds only if $L$ is 2-torsion-free. Note also that skew symmetry means that we need assume only one of the two additivity identities and it implies the other.
Jacobi identity	Left-normed version: $[[x,y],z] + [[y,z],x] + [[z,x],y] = 0$ Right-normed version: $[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$ The two versions are equivalent by skew symmetry.

A Lie ring which is also an algebra over a field (or a commutative unital ring) is termed a Lie algebra over that field (or commutative unital ring).

If $R$ is a commutative unital ring and $L$ is additionally equipped with the structure of a $R$-module, and the Lie bracket of $L$ is $R$-bilinear, then this makes $L$ a $R$-Lie algebra.

Viewpoints

• The Lie rings form a variety of algebras called the variety of Lie rings.
• The Jacobi identity for Lie rings is a close counterpart to associativity in groups.
• The associated Lie ring for a strongly central series and the associated Lie ring for a group form a convenient way to pass from groups to Lie rings.
• The Malcev correspondence and its generalization, the Lazard correspondence, form convenient ways to move back and forth between groups and Lie rings without loss of information.

Related notions

• Lie algebra is a variation where the underlying group is a module over a specified commutative unital ring and the Lie bracket is a bilinear map.
• Multiplicative Lie ring is a generalization to cases where the underlying group is possibly non-abelian.