Category:Definitions/Commutators

This category contains definitions related to Commutators.
Related results can be found in Category:Commutators.

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:

$\sqbrk {a, b} := a \circ b - b \circ a$

This page has been identified as a candidate for refactoring of medium complexity. In particular: Technically speaking, the following definition is different from that for the group and ring elements. Does it make sense to distinguish between a commutator between elements, and distinguish it from a commutator defined for the entire space? Seems to me that defining the commutator as it is in groups and rings above seems to fiddly to me. It's just an operator, it should apply to all pairs of elements in the square of the object. Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

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 $\struct {A_R, \oplus}$.