Clifford algebra

E801053

algebra over a field associative algebra

Clifford algebra is an associative algebraic framework that generalizes complex numbers and quaternions to describe geometric transformations and quadratic forms in various dimensions.

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Observed surface forms (2)

Surface form	Occurrences
"Clifford Algebra to Geometric Calculus"	1
Clifford calculus	1

Statements (47)

Predicate	Object
instanceOf	algebra over a field ⓘ associative algebra ⓘ
basedOn	bilinear form ⓘ quadratic form ⓘ
definedOver	field ⓘ vector space ⓘ
generalizes	complex numbers ⓘ exterior algebra ⓘ geometric algebra NERFINISHED ⓘ quaternions ⓘ
hasComponent	even subalgebra ⓘ odd subspace ⓘ
hasOperation	Clifford conjugation ⓘ Clifford product NERFINISHED ⓘ grade involution ⓘ reversion ⓘ
hasProperty	Z2-graded algebra ⓘ associative multiplication ⓘ finite-dimensional when base space is finite-dimensional ⓘ unital algebra ⓘ
hasSpecialCase	complex numbers as Clifford algebra of a 1-dimensional space with negative quadratic form ⓘ geometric algebra as real Clifford algebra ⓘ quaternions as Clifford algebra Cl(0,2) or Cl(3,0) up to isomorphism ⓘ
introducedIn	19th century ⓘ
namedAfter	William Kingdon Clifford NERFINISHED ⓘ
relatedTo	Clifford module ⓘ Dirac operator NERFINISHED ⓘ Lie groups NERFINISHED ⓘ Pin group NERFINISHED ⓘ orthogonal group NERFINISHED ⓘ spin group ⓘ spinor bundle ⓘ
satisfiesRelation	v·v = Q(v)·1 for all vectors v ⓘ
usedFor	Dirac equation NERFINISHED ⓘ describing geometric transformations ⓘ encoding quadratic forms ⓘ representing reflections ⓘ representing rotations ⓘ spinor representations ⓘ
usedIn	computer graphics ⓘ control theory ⓘ differential geometry ⓘ quantum mechanics ⓘ relativity theory ⓘ robotics ⓘ signal processing ⓘ theoretical physics ⓘ

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

spacetime algebra → basedOn → Clifford algebra ⓘ

geometric calculus → usesConcept → Clifford algebra ⓘ

geometric calculus → formalizedIn → Clifford algebra ⓘ

this entity surface form: "Clifford Algebra to Geometric Calculus"

geometric calculus → relatedTo → Clifford algebra ⓘ

this entity surface form: Clifford calculus

foundations of geometric algebra as a unified language for physics → hasCoreConcept → Clifford algebra ⓘ

subject surface form: Foundations of geometric algebra as a unified language for physics

Dirac operator → relatedTo → Clifford algebra ⓘ