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

Exterior algebra

Open this concept in the interactive graph Read the Markdown equivalent

Summary

The graded algebra $\Lambda V$ generated by a vector space $V$ subject to the alternating relations $v\wedge v=0$ .

Record metadata

Canonical name: Exterior algebra
Concept ID: exterior_algebra
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebra
Domains: Algebra, Differential geometry
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Graded algebra

Carrier(s)

vector space $V$

Data

graded algebra $\Lambda V$
canonical map $V\to\Lambda^1V$

Axioms / constraints

$v\wedge v=0$ for every $v$
universal property for alternating algebra maps

Canonically induces

exterior powers $\Lambda^kV$
wedge product

Notes

In characteristic not equal to 2, the alternating relation implies graded anticommutativity for degree-one generators; the alternating formulation is characteristic-independent.

Concept sources

Wikipedia — Exterior algebra — Exterior algebra · encyclopedic reference · source ID wp-exterior_algebra
nLab — exterior algebra — exterior algebra · mathematical reference · source ID nlab-exterior_algebra

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Vector space Exterior algebra

Permalink to relation

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

Relation ID: math_vector_space_to_exterior_algebra
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: exterior algebra

Authored explanation

The exterior algebra is the quotient of the tensor algebra by the two-sided ideal generated by all $v\otimes v$ .

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 — Exterior algebra — Exterior algebra · encyclopedic reference · source ID wp-exterior_algebra
nLab — exterior algebra — exterior algebra · mathematical reference · source ID nlab-exterior_algebra

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.