Select any concept, construction junction, or annotated relation by pointer, touch, search, or keyboard.

Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Operator commutator

Open this concept in the interactive graph Read the Markdown equivalent

Summary

For operators $A$ and $B$ with a common meaningful domain, the commutator is $[A,B]=AB-BA$ ; domain issues are essential for unbounded operators.

Record metadata

Canonical name: Operator commutator
Concept ID: operator_commutator
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebra
Domains: Algebra, Analysis
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Bilinear operation

Carrier(s)

operators $A,B$ with compatible products

Data

operator $[A,B]=AB-BA$ on a common domain

Axioms / constraints

defined only where both compositions make sense

Canonically induces

Lie-algebra structure in bounded/algebraic settings
uncertainty relations

Notes

For unbounded operators, domain questions are essential.

Concept sources

Wikipedia — Commutator — Commutator · encyclopedic reference · source ID wp-operator-commutator
nLab — commutator — commutator · mathematical reference · source ID nlab-operator-commutator

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Linear operator Operator commutator

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: linearops_to_commutator
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: noncommutative product difference

Authored explanation

Whenever both products are defined on a suitable common domain, operator composition defines the commutator $AB-BA$ .

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Commutator — Commutator · encyclopedic reference · source ID wp-operator-commutator
nLab — commutator — commutator · mathematical reference · source ID nlab-operator-commutator

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.