Definition:Commutator

Definition

The commutator of an algebraic structure can be considered a measure of how commutative the structure is.

Groups

Let $\left({G, \circ}\right)$ be a group.

Let $g, h \in G$.

The commutator of $g$ and $h$ is the operation:

$\left[{g, h}\right] := g^{-1} \circ h^{-1} \circ g \circ h$

Rings

Let $\left({R, +, \circ}\right)$ be a ring.

Let $a, b \in R$.

The commutator of $a$ and $b$ is the operation:

$\left[{a, b}\right] := a \circ b + \left({- b \circ a}\right)$

or more compactly:

$\left[{a, b}\right] := a \circ b - b \circ a$

Algebras

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring.

Consider the bilinear mapping $\left[{\cdot, \cdot}\right]: A_R^2 \to A_R$ defined as:

$\forall a, b \in A_R: \left[{a, b}\right] := a \oplus b - b \oplus a$

Then $\left[{\cdot, \cdot}\right]$ is known as the commutator of $\left({A_R, \oplus}\right)$.

Note that trivially if $\left({A_R, \oplus}\right)$ is a commutative algebra, then:

$\forall a, b \in A_R: \left[{a, b}\right] = \mathbf 0_R$