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| {{Unreferenced|date=November 2009}}
| | Authors Bud Grippo from Newcastle, has pastimes which include telescopes, property developers in singapore and crochet. In the recent couple of months has gone to places like San Marino Historic Centre and Mount Titano.<br><br>Feel free to surf to my website [http://www.everfoundmusic.com/forum/kingsfordhillview-peak-launch-now-424746 Http://Www.Everfoundmusic.Com/Forum/Kingsfordhillview-Peak-Launch-Now-424746] |
| In [[abstract algebra]], in particular in the theory of [[Degeneracy (mathematics)|nondegenerate]] [[quadratic form]]s on [[vector space]]s, the structures of [[finite-dimensional]] [[real number|real]] and [[complex number|complex]] [[Clifford algebra]]s have been completely classified. In each case, the Clifford algebra is [[Algebra homomorphism|algebra isomorphic]] to a full [[matrix ring]] over '''R''', '''C''', or '''H''' (the [[quaternion]]s), or to a [[direct sum of algebras|direct sum]] of two such algebras, though not in a [[canonical form|canonical]] way. Below it is shown that distinct Clifford algebras may be algebra isomorphic, as is the case of Cℓ<sub>2,0</sub>('''R''') and Cℓ<sub>1,1</sub>('''R''') which are both isomorphic to the ring of two-by-two matrices over the real numbers.
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| ==Notation and conventions==
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| The [[Clifford product]] is the manifest ring product for the Clifford algebra, and all algebra [[homomorphism]]s in this article are with respect to this ring product. Other products defined within Clifford algebras, such as the [[exterior product]], are not used here.
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| This article uses the (+) [[sign convention]] for Clifford multiplication so that
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| :<math>v^2 = Q(v)\,</math>
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| for all vectors {{nowrap|1=''v'' ∈ ''V''}}, where ''Q'' is the quadratic form on the vector space ''V''. We will denote the algebra of {{nowrap|1=''n''×''n''}} [[matrix (mathematics)|matrices]] with entries in the [[division algebra]] ''K'' by M<sub>''n''</sub>(''K'') or M(''n'',''K''). The [[direct sum of rings|direct sum]] of two such identical algebras will be denoted by {{nowrap|1=M<sub>''n''</sub><sup>2</sup>(''K'') = M<sub>''n''</sub>(''K'') ⊕ M<sub>''n''</sub>(''K'')}}.
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| ==Bott periodicity==
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| Clifford algebras exhibit a 2-fold periodicity over the complex numbers and an 8-fold periodicity over the real numbers, which is related to the same periodicities for homotopy groups of the stable [[unitary group]] and stable [[orthogonal group]], and is called [[Bott periodicity]]. The connection is explained by the [[Bott periodicity theorem#Geometric model of loop spaces|geometric model of loop spaces]] approach to Bott periodicity: there 2-fold/8-fold periodic embeddings of the [[classical group]]s in each other (corresponding to isomorphism groups of Clifford algebras), and their successive quotients are [[symmetric space]]s which are [[homotopy equivalent]] to the [[loop space]]s of the unitary/orthogonal group.
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| ==Complex case==
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| The complex case is particularly simple: every nondegenerate quadratic form on a complex vector space is equivalent to the standard diagonal form
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| :<math>Q(u) = u_1^2 + u_2^2 + \cdots + u_n^2</math>
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| where ''n'' = dim ''V'', so there is essentially only one Clifford algebra in each dimension. We will denote the Clifford algebra on '''C'''<sup>''n''</sup> with the standard quadratic form by Cℓ<sub>''n''</sub>('''C''').
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| There are two separate cases to consider, according to whether ''n'' is even or odd. When ''n'' is even the algebra Cℓ<sub>''n''</sub>('''C''') is [[central simple algebra|central simple]] and so by the [[Artin-Wedderburn theorem]] is isomorphic to a matrix algebra over '''C'''. When ''n'' is odd, the center includes not only the scalars but the [[pseudoscalar (mathematics)|pseudoscalar]]s (degree ''n'' elements) as well. We can always find a normalized pseudoscalar ω such that ω<sup>2</sup> = 1. Define the operators
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| :<math>P_{\pm} = \frac{1}{2}(1\pm\omega).</math>
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| These two operators form a complete set of [[Idempotent_element#Types_of_ring_idempotents|orthogonal idempotent]]s, and since they are central they give a decomposition of Cℓ<sub>''n''</sub>('''C''') into a direct sum of two algebras
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| :<math>C\!\ell_n(\mathbf{C}) = C\!\ell_n^{+}(\mathbf{C}) \oplus C\!\ell_n^{-}(\mathbf{C})</math> where <math>C\!\ell_n^\pm(\mathbf{C}) = P_\pm C\!\ell_n(\mathbf{C})</math>.
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| The algebras Cℓ<sub>''n''</sub><sup>±</sup>('''C''') are just the positive and negative eigenspaces of ω and the ''P''<sub>±</sub> are just the projection operators. Since ω is odd these algebras are mixed by α (the linear map on ''V'' defined by {{nowrap|1=''v'' ↦ −''v''}}):
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| :<math>\alpha(C\!\ell_n^\pm(\mathbf{C})) = C\!\ell_n^\mp(\mathbf{C})</math>.
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| and therefore isomorphic (since α is an [[automorphism]]). These two isomorphic algebras are each central simple and so, again, isomorphic to a matrix algebra over '''C'''. The sizes of the matrices can be determined from the fact that the dimension of Cℓ<sub>''n''</sub>('''C''') is 2<sup>''n''</sup>. What we have then is the following table:
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| <center>
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| {| border=1 cellpadding=2 style="text-align: center;"
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| |-
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| | ''n'' || Cℓ<sub>''n''</sub>('''C''')
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| |-
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| | 2''m'' || M(2<sup>''m''</sup>,'''C''')
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| |-
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| | 2''m''+1 || M(2<sup>''m''</sup>,'''C''') ⊕ M(2<sup>''m''</sup>,'''C''')
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| |}
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| </center>
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| The even subalgebra of Cℓ<sub>''n''</sub>('''C''') is (non-canonically) isomorphic to Cℓ<sub>''n''−1</sub>('''C'''). When ''n'' is even, the even subalgebra can be identified with the block diagonal matrices (when partitioned into 2×2 [[block matrix]]). When ''n'' is odd, the even subalgebra are those elements of {{nowrap|1=M(2<sup>''m''</sup>,'''C''') ⊕ M(2<sup>''m''</sup>,'''C''')}} for which the two factors are identical. Picking either piece then gives an isomorphism with {{nowrap|1=Cℓ<sub>''n''−1</sub>('''C''') ≅ M(2<sup>''m''</sup>,'''C''')}}.
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| ==Real case==
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| The real case is significantly more complicated, exhibiting a periodicity of 8 rather than 2, and there is a 2-parameter family of Clifford algebras.
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| ===Classification of quadratic form===
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| Firstly, there are non-isomorphic quadratic forms of a given degree, classified by signature.
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| Every nondegenerate quadratic form on a real vector space is equivalent to the standard diagonal form:
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| :<math>Q(u) = u_1^2 + \cdots + u_p^2 - u_{p+1}^2 - \cdots - u_{p+q}^2</math>
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| where {{nowrap|1=''n'' = ''p'' + ''q''}} is the dimension of the vector space. The pair of integers (''p'', ''q'') is called the [[metric signature|signature]] of the quadratic form. The real vector space with this quadratic form is often denoted '''R'''<sup>''p'',''q''</sup>. The Clifford algebra on '''R'''<sup>''p'',''q''</sup> is denoted Cℓ<sub>''p'',''q''</sub>('''R''').
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| A standard [[orthonormal basis]] {''e''<sub>''i''</sub>} for '''R'''<sup>''p'',''q''</sup> consists of {{nowrap|1=''n'' = ''p'' + ''q''}} mutually orthogonal vectors, ''p'' of which have norm +1 and ''q'' of which have norm −1.
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| ===Unit pseudoscalar===
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| {{See also|Pseudoscalar}}
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| The unit pseudoscalar in Cℓ<sub>''p'',''q''</sub>('''R''') is defined as
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| :<math>\omega = e_1e_2\cdots e_n.</math>
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| This is both a [[Coxeter element]] of sorts (product of reflections) and a [[longest element of a Coxeter group]] in the [[Bruhat order]]; this is an analogy. It corresponds to and generalizes a [[volume form]] (in the [[exterior algebra]]; for the trivial quadratic form, the unit pseudoscalar is a volume form), and lifts [[reflection through the origin]] (meaning that the image of the unit pseudoscalar is reflection through the origin, in the [[orthogonal group]]).
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| To compute the square <math>\omega^2=(e_1e_2\cdots e_n)(e_1e_2\cdots e_n)</math>,
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| one can either reverse the order of the second group,
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| yielding <math>\mbox{sgn}(\sigma)e_1e_2\cdots e_n e_n\cdots e_2 e_1</math>,
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| or apply a [[perfect shuffle]],
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| yielding <math>\mbox{sgn}(\sigma)(e_1e_1e_2e_2\cdots e_ne_n)</math>.
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| These both have sign <math>(-1)^{\lfloor n/2 \rfloor}=(-1)^{n(n-1)/2}</math>, which is 4-periodic ([[Symmetric group#Elements|proof]]), and combined with <math>e_i e_i = \pm 1</math>, this shows that the square of ω is given by
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| :<math>\omega^2 = (-1)^{n(n-1)/2}(-1)^q = (-1)^{(p-q)(p-q-1)/2} =
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| \begin{cases}+1 & p-q \equiv 0,1 \mod{4}\\ -1 & p-q \equiv 2,3 \mod{4}.\end{cases}</math>
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| Note that, unlike the complex case, it is not always possible to find a pseudoscalar which squares to +1.
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| ===Center===
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| If ''n'' (equivalently, {{nowrap|1=''p'' − ''q''}}) is even the algebra Cℓ<sub>''p'',''q''</sub>('''R''') is [[central simple algebra|central simple]] and so isomorphic to a matrix algebra over '''R''' or '''H''' by the [[Artin–Wedderburn theorem]].
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| If ''n'' (equivalently, {{nowrap|1=''p'' − ''q''}}) is odd then the algebra is no longer central simple but rather has a center which includes the pseudoscalars as well as the scalars. If ''n'' is odd and {{nowrap|1=ω<sup>2</sup> = +1}} (equivalently, if {{nowrap|1=''p'' − ''q'' ≡ 1 (mod 4)}})
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| then, just as in the complex case, the algebra Cℓ<sub>''p'',''q''</sub>('''R''') decomposes into a direct sum of isomorphic algebras
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| :<math>C\ell_{p,q}(\mathbf{R}) = C\ell_{p,q}^{+}(\mathbf{R})\oplus C\ell_{p,q}^{-}(\mathbf{R})</math>
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| each of which is central simple and so isomorphic to matrix algebra over '''R''' or '''H'''.
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| If ''n'' is odd and ω<sup>2</sup> = −1 (equivalently, if {{nowrap|1=''p'' − ''q'' ≡ −1 (mod 4)}}) then the center of Cℓ<sub>''p'',''q''</sub>('''R''') is isomorphic to '''C''' and can be considered as a ''complex'' algebra. As a complex algebra, it is central simple and so isomorphic to a matrix algebra over '''C'''.
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| ===Classification===
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| All told there are three properties which determine the class of the algebra Cℓ<sub>''p'',''q''</sub>('''R'''):
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| * signature mod 2: ''n'' is even/odd: central simple or not
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| * signature mod 4: ω<sup>2</sup> = ±1: if not central simple, center is {{nowrap|1='''R'''⊕'''R'''}} or '''C'''
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| * signature mod 8: the [[Brauer group|Brauer class]] of the algebra (''n'' even) or even subalgebra (''n'' odd) is '''R''' or '''H'''
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| Each of these properties depends only on the signature {{nowrap|1=''p'' − ''q''}} [[Modular arithmetic|modulo]] 8. The complete classification table is given below. The size of the matrices is determined by the requirement that Cℓ<sub>''p'',''q''</sub>('''R''') have dimension 2<sup>''p''+''q''</sup>.
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| <center>
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| {| border=1 cellpadding=2 style="text-align: center;"
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| |-
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| | ''p''−''q'' mod 8 || ω<sup>2</sup>
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| | Cℓ<sub>''p'',''q''</sub>('''R''')<br><small>(''n'' = ''p''+''q'')</small>
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| | ''p''−''q'' mod 8 || ω<sup>2</sup>
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| | Cℓ<sub>''p'',''q''</sub>('''R''')<br><small>(''n'' = ''p''+''q'')</small>
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| |-
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| | 0 || + || M(2<sup>''n''/2</sup>,'''R''')
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| | 1 || + || M(2<sup>(''n''−1)/2</sup>,'''R''')⊕M(2<sup>(''n''−1)/2</sup>,'''R''')
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| |-
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| | 2 || − || M(2<sup>''n/2''</sup>,'''R''')
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| | 3 || − || M(2<sup>(''n''−1)/2</sup>,'''C''')
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| |-
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| | 4 || + || M(2<sup>(''n''−2)/2</sup>,'''H''')
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| | 5 || + || M(2<sup>(''n''−3)/2</sup>,'''H''')⊕M(2<sup>(''n''−3)/2</sup>,'''H''')
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| |-
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| | 6 || − || M(2<sup>(''n''−2)/2</sup>,'''H''')
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| | 7 || − || M(2<sup>(''n''−1)/2</sup>,'''C''')
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| |}
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| </center>
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| It may be seen that of all matrix ring types mentioned, there is only one type shared between both complex and real algebras: the type '''C'''(2<sup>''m''</sup>). For example, Cℓ<sub>2</sub>('''C''') and Cℓ<sub>3,0</sub>('''R''') are both determined to be '''C'''(2). It is important to note that there is a difference in the classifying isomorphisms used. Since the Cℓ<sub>2</sub>('''C''') is algebra isomorphic via a '''C'''-linear map (which is necessarily '''R'''-linear), and Cℓ<sub>3,0</sub>('''R''') is algebra isomorphic via an '''R'''-linear map, Cℓ<sub>2</sub>('''C''') and Cℓ<sub>3,0</sub>('''R''') are '''R'''-algebra isomorphic.
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| A table of this classification for {{nowrap|1=''p'' + ''q'' ≤ 8}} follows. Here {{nowrap|1=''p'' + ''q''}} runs vertically and {{nowrap|1=''p'' − ''q''}} runs horizontally (e.g. the algebra {{nowrap|1=Cℓ<sub>1,3</sub>('''R''') ≅ M<sub>2</sub>('''H''')}} is found in row 4, column −2).
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| <center>
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| {| border=0 cellpadding=2 style="text-align: center;"
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| |-
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| | || 8 || 7 || 6 || 5 || 4 || 3 || 2 || 1 || 0 || −1 || −2 || −3 || −4 || −5 || −6 || −7 || −8
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| |-
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| | 0 || || || || || || || || || '''R''' || || || || || || || ||
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| |-
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| | 1 || || || || || || || || '''R'''<sup>2</sup> || || '''C''' || || || || || || ||
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| |-
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| | 2 || || || || || || || M<sub>2</sub>('''R''') || || M<sub>2</sub>('''R''') || || '''H''' || || || || || ||
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| |-
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| | 3 || || || || || || M<sub>2</sub>('''C''') || || M<sub>2</sub><sup>2</sup>('''R''') || || M<sub>2</sub>('''C''') || || '''H'''<sup>2</sup> || || || || ||
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| |-
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| | 4 || || || || || M<sub>2</sub>('''H''') || || M<sub>4</sub>('''R''') || || M<sub>4</sub>('''R''') || || M<sub>2</sub>('''H''') || || M<sub>2</sub>('''H''') || || || ||
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| |-
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| | 5 || || || || M<sub>2</sub><sup>2</sup>('''H''') || || M<sub>4</sub>('''C''') || || M<sub>4</sub><sup>2</sup>('''R''') || || M<sub>4</sub>('''C''') || || M<sub>2</sub><sup>2</sup>('''H''') || || M<sub>4</sub>('''C''') || || ||
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| |-
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| | 6 || || || M<sub>4</sub>('''H''') || || M<sub>4</sub>('''H''') || || M<sub>8</sub>('''R''') || || M<sub>8</sub>('''R''') || || M<sub>4</sub>('''H''') || || M<sub>4</sub>('''H''') || || M<sub>8</sub>('''R''') || ||
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| |-
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| | 7 || || M<sub>8</sub>('''C''') || || M<sub>4</sub><sup>2</sup>('''H''') || || M<sub>8</sub>('''C''') || || M<sub>8</sub><sup>2</sup>('''R''') || || M<sub>8</sub>('''C''') || || M<sub>4</sub><sup>2</sup>('''H''') || || M<sub>8</sub>('''C''') || || M<sub>8</sub><sup>2</sup>('''R''') ||
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| |-
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| | 8 || M<sub>16</sub>('''R''') || || M<sub>8</sub>('''H''') || || M<sub>8</sub>('''H''') || || M<sub>16</sub>('''R''') || || M<sub>16</sub>('''R''') || || M<sub>8</sub>('''H''') || || M<sub>8</sub>('''H''') || || M<sub>16</sub>('''R''') || || M<sub>16</sub>('''R''')
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| |-
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| | || || || || || || || || || || || || || || || || ||
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| |-
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| | ω<sup>2</sup> || + || − || − || + || + || − || − || + || + || − || − || + || + || − || − || + || +
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| |}
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| </center>
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| ===Symmetries===
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| There is a tangled web of symmetries and relationships in the above table.
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| :<math>C\ell_{p+1,q+1}(\mathbf{R}) = \mathrm{M}_2(C\ell_{p,q}(\mathbf{R}))</math> | |
| :<math>C\ell_{p+4,q}(\mathbf{R}) = C\ell_{p,q+4}(\mathbf{R})</math>
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| Going over 4 spots in any row yields an identical algebra.
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| From these Bott periodicity follows:
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| :<math>C\ell_{p+8,q}(\mathbf{R}) = C\ell_{p+4,q+4}(\mathbf{R}) = M_{2^4}(C\ell_{p,q}(\mathbf{R})) .</math>
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| If the signature satisfies {{nowrap|1=''p'' − ''q'' ≡ 1 (mod 4)}} then
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| :<math>C\ell_{p+k,q}(\mathbf{R}) = C\ell_{p,q+k}(\mathbf{R}) .</math>
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| (The table is symmetric about columns with signature 1, 5, −3, −7, and so forth.)
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| Thus if the signature satisfies {{nowrap|1=''p'' − ''q'' ≡ 1 (mod 4)}},
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| :<math>C\ell_{p+k,q}(\mathbf{R}) = C\ell_{p,q+k}(\mathbf{R}) = C\ell_{p-k+k,q+k}(\mathbf{R}) = \mathrm{M}_{2^k}(C\ell_{p-k,q}(\mathbf{R})) = \mathrm{M}_{2^k}(C\ell_{p,q-k}(\mathbf{R})) .</math>
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| ==See also==
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| *[[Dirac algebra]] Cℓ<sub>1,3</sub>('''C''')
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| *[[Pauli algebra]] Cℓ<sub>3,0</sub>('''C''')
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| *[[Spacetime algebra]] Cℓ<sub>1,3</sub>('''R''')
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| *[[Clifford module]]
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| *[[Spin representation]]
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| {{DEFAULTSORT:Classification Of Clifford Algebras}}
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| [[Category:Ring theory]]
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| [[Category:Clifford algebras]]
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