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

Canonical static concept record

Representation of a Lie algebra

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Summary

A Lie algebra homomorphism into the endomorphism Lie algebra of a vector space.

Record metadata

Canonical name: Representation of a Lie algebra
Concept ID: lie_algebra_representation
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Representation theory
Domains: Representation theory, Algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

Lie algebra $\mathfrak{g}$ over a field $k$
$k$ -vector space $V$

Data

$k$ -linear Lie homomorphism $\mathfrak{g}\to\mathfrak{gl}_k(V)$

Concept sources

Wikipedia — Lie algebra representation — Lie algebra representation · encyclopedic reference · source ID wp-j-lie-rep
Mathlib — Lie algebras and Lie modules — Lie algebras and Lie modules · formal reference · source ID mathlib-lie-module

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Each relation below ends at this concept.

Lie algebra + vector space action(construction junction)Representation of a Lie algebra

Permalink to relation

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

Relation ID: j_lie_rep__out
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: impose compatibility

Authored explanation

Choose a $k$ -linear action $\mathfrak g\to\operatorname{End}_k(V)$ and require $[x,y]$ to act as the commutator of the endomorphisms assigned to $x$ and $y$ .

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Mathlib — Lie algebras and Lie modules — Lie algebras and Lie modules · formal reference · source ID mathlib-lie-module

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.