Lie ring

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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

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.