Lie algebra

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Definition

Suppose R is a commutative unital ring, i.e., an associative ring whose multiplication is commutative and has an identity element.

A Lie algebra over R is a Lie ring L whose additive group is equipped with a R-module structure and whose Lie bracket is R-bilinear.

Explicitly, a Lie algebra over R is a R-module L equipped with a map [ \ , \ ]: L \times L \to L satisfying all the following conditions:

Condition name Explicit identities (all variable letters x,y,z are universally quantified over L and variable r is universally quantified over R)
R-bilinear Additive in left coordinate: [x+y,z] = [x,z] + [y,z]
Additive in right coordinate: [x,y+z] = [x,y] + [x,z]
R-scalars can be pulled out of left coordinate: [rx,y] = r[x,y]
R-scalars can be pulled out of right coordinate: [x,ry] = r[x,y]
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.

Particular cases

Facts

Universal enveloping algebra

Further information: Universal enveloping algebra

Every Lie algebra has a universal enveloping algebra. An enveloping algebra for a Lie algebra is an associative algebra over the same base field which contains the elements of the Lie algebra, such that:

  • The addition in the enveloping algebra is the same as that within the Lie algebra
  • For those elements which are in the Lie algebra, the commutator coincides with the Lie bracket

The universal enveloping algebra is an algebra that is universal among all enveloping algebras.