Atlas of Fundamental Concepts

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Spin group

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Summary

The group constructed in a Clifford algebra that double-covers SO(n)SO(n) in Euclidean signature; in indefinite signature it covers the appropriate identity component according to convention.

Record metadata

Canonical name
Spin group
Concept ID
spin_group
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.
Concept type
Lie group

Carrier(s)

Data

Axioms / constraints

Canonically induces

Notes

The precise target and notation depend on signature and convention; in Euclidean signature Spin(n)\mathrm{Spin}(n) double-covers SO(n)SO(n).

Concept sources

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Clifford algebraSpin group

Permalink to relation

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

Relation ID
math_clifford_algebra_to_spin_group
Relation type
Canonical construction canonical-construction
Direction
source β†’ target
Endpoint roles
source: Canonically constructs; target: Canonically constructed from
Authored annotation
spin double cover

Authored explanation

The spin group is constructed from invertible elements of a Clifford algebra; in Euclidean signature Spin(n)β†’SO(n)\mathrm{Spin}(n)\to SO(n) is a double cover, with signature-dependent variants in the indefinite case.

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

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Spin groupSpinor representation

Permalink to relation

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

Relation ID
math_spin_group_to_spinor_representation
Relation type
Model / approximation method model-realization
Direction
source β†’ target
Endpoint roles
source: Has model / approximation; target: Model / approximation of
Authored annotation
choose a spin representation

Authored explanation

A spinor representation realizes the spin group linearly, usually through a module over the associated Clifford algebra; a particular representation is additional chosen data.

How to interpret this relation type

The target is a model, idealization, restricted ansatz, approximation scheme, phenomenological fit, or historical representation used for the source system or theory. It need not have a universal small error parameter and is not thereby a controlled limit or effective theory. The edge label and detail must identify the precise status.

Relation sources

  • Wikipedia β€” Spinor β€” Spinor Β· encyclopedic reference Β· source ID wp-spinor_representation
  • nLab β€” spinor β€” spinor Β· mathematical reference Β· source ID nlab-spinor_representation