# Lie ring

From Groupprops

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 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 ) |
---|---|

-bilinear, also known as biadditive, also known as left and right distributive | Additive in left coordinate: Additive in right coordinate: |

alternating (hence skew-symmetric) | Alternation: Skew symmetry: The second condition (skew symmetry) follows from the first (alternation); the reverse implication holds only if 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: Right-normed version: 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 is a commutative unital ring and is additionally equipped with the structure of a -module, and the Lie bracket of is -bilinear, then this makes a -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.