Editing Clifford algebra (section)

{{Use American English|date=January 2019}}
{{Short description|Algebra based on a vector space with a quadratic form}}
{{About|the orthogonal Clifford algebra|the symplectic Clifford algebra|Weyl algebra}}

{{Ring theory sidebar|expanded=Noncommutative}}

In [[mathematics]], a '''Clifford algebra'''{{efn|Also known as a ''geometric algebra'' (especially over the real numbers)}} is an [[algebra over a field|algebra]] generated by a [[vector space]] with a [[quadratic form]], and is a [[Unital algebra|unital]] [[associative algebra]] with the additional structure of a distinguished subspace. As [[algebra over a field|{{math|''K''}}-algebras]], they generalize the [[real number]]s, [[complex number]]s, [[quaternion]]s and several other [[hypercomplex number]] systems.{{sfn|Clifford|1873|pages=381–395|ps=none}}{{sfn|Clifford|1882|ps=none}} The theory of Clifford algebras is intimately connected with the theory of [[quadratic form]]s and [[orthogonal group|orthogonal transformation]]s. Clifford algebras have important applications in a variety of fields including [[geometry]], [[theoretical physics]] and [[digital image processing]]. They are named after the English mathematician [[William Kingdon Clifford]] (1845–1879). 

The most familiar Clifford algebras, the '''orthogonal Clifford algebras''', are also referred to as (''pseudo-'')''Riemannian Clifford algebras'', as distinct from ''symplectic Clifford algebras''.{{efn|See for ex. {{harvnb|Oziewicz|Sitarczyk|1992}}}}