Groupprops, The Group Properties Wiki (pre-alpha)

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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 G equipped with the structure of an Abelian group (operation denoted + ) and a bracket operation (called a Lie bracket) such that:

• Left and right distributivity laws hold, viz.:

[x,y + z] = [x,y] + [x,z]

and:

[x + y,z] = [x,z] + [y,z]

• The Lie bracket is anticommutative, viz., the following two laws hold:

[x,x] = 0

and:

[x,y] + [y,x] = 0

• The Lie bracket satisfies the Jacobi identity:

[[x,y],z] + [[y,z],x] + [[z,x],y] = 0

A Lie ring which is also an algebra over a field is termed a Lie algebra.