Definition:Commutator
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Definition
The commutator of an algebraic structure can be considered a measure of how commutative the structure is.
Groups
The commutator of $g$ and $h$ is the element of $G$ defined and denoted:
- $\sqbrk {g, h} := g^{-1} \circ h^{-1} \circ g \circ h$
Rings
Let $\struct {R, +, \circ}$ be a ring.
Let $a, b \in R$.
The commutator of $a$ and $b$ is the operation:
- $\sqbrk {a, b} := a \circ b + \paren {-b \circ a}$
or more compactly:
- $\sqbrk {a, b} := a \circ b - b \circ a$
Algebras
Let $\struct {A_R, \oplus}$ be an algebra over a ring.
Consider the bilinear mapping $\sqbrk {\, \cdot, \cdot \,}: A_R^2 \to A_R$ defined as:
- $\forall a, b \in A_R: \sqbrk {a, b} := a \oplus b - b \oplus a$
Then $\sqbrk {\, \cdot, \cdot \,}$ is known as the commutator of $\struct {A_R, \oplus}$.
Note that trivially if $\struct {A_R, \oplus}$ is a commutative algebra, then:
- $\forall a, b \in A_R: \sqbrk {a, b} = \mathbf 0_R$
Also see
- Results about commutators can be found here.