Lie algebra: Difference between revisions

Latest revision as of 17:08, 15 August 2013

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: $[r x, y] = r [x, y]$ $R$ -scalars can be pulled out of right coordinate: $[x, r y] = r [x, y]$
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

In the case that $R = Z$ , the notion of $R$ -Lie algebra coincides with the usual notion of Lie ring.

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.

@@ Line 1: / Line 1: @@
 ==Definition==
-A '''Lie algebra''' over a [[field]] <math>k</math> is defined as a set <math>V</math> equipped with the following structures:
+Suppose <math>R</math> is a [[commutative unital ring]], i.e., an [[associative ring]] whose multiplication is commutative and has an identity element.
-* A vector space structure over <math>k</math>
+A '''Lie algebra''' over <math>R</math> is a [[defining ingredient::Lie ring]] <math>L</math> whose additive group is equipped with a <math>R</math>-module structure and whose Lie bracket is <math>R</math>-bilinear.
-* A <math>k</math>-bilinear map <math>[ , ]: V \times V \to V</math> called the '''Lie bracket'''
-satisfying the following compatibility conditions:
+Explicitly, a '''Lie algebra''' over <math>R</math> is a <math>R</math>-module <math>L</math> equipped with a map <math>[ \ , \ ]: L \times L \to L</math> satisfying '''all''' the following conditions:
-* <math>[x,x] = 0 \forall x \in V</math>
+{| class="sortable" border="1"
-* <math>[x,y] + [y,x] = 0 \forall x,y \in V</math>
+! Condition name !! Explicit identities (all variable letters <math>x,y,z</math> are universally quantified over <math>L</math> and variable <math>r</math> is universally quantified over <math>R</math>)
-* <math>[[x,y],z] + [[y,z],x] + [[z,x],y] = 0 \forall x,y,z \in V</math> (this is called the Jacobi identity)
+|-
+| <math>R</math>-bilinear || Additive in left coordinate: <math>[x+y,z] = [x,z] + [y,z]</math><br>Additive in right coordinate: <math>[x,y+z] = [x,y] + [x,z]</math><br><math>R</math>-scalars can be pulled out of left coordinate: <math>[rx,y] = r[x,y]</math><br><math>R</math>-scalars can be pulled out of right coordinate: <math>[x,ry] = r[x,y]</math>
-We note that the first condition will imply the second ( take <math>x+y</math> for <math>x</math>) but the second will imply the first only when the field is of characteristic 2.
+|-
+| alternating (hence skew-symmetric) || Alternation: <math>[x,x] = 0</math><br>Skew symmetry: <math>[x,y] + [y,x] = 0</math><br>The second condition (skew symmetry) follows from the first (alternation); the reverse implication holds only if <math>L</math> is 2-torsion-free.<br>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: <math>[[x,y],z] + [[y,z],x] + [[z,x],y] = 0</math><br>Right-normed version: <math>[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0</math><br>The two versions are equivalent by skew symmetry.
+|}
+==Particular cases==
+* In the case that <math>R = \mathbb{Z}</math>, the notion of <math>R</math>-[[Lie algebra]] coincides with the usual notion of [[Lie ring]].
 ==Facts==
-===Lie algebra of a Lie group===
-{{fillin}}
 ===Universal enveloping algebra===