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 ![]() |
---|---|
![]() |
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 ![]() 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.