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| {{Unreferenced|date=November 2009}}
| | If you happen to look at the listing of the Forbes forty richest individuals in Singapore, you will see many who made their fortunes creating and investing in actual property. Or simply go searching you – the average Singaporean's wealth in all probability comes extra from the appreciation of his HDB flat or non-public property than from another asset.<br><br>Some property agents attempt to win your online business by promising a high valuation sale, or a low fee fee. Too excessive a valuation is unrealistic, and might make you lose the chance to promote at a good price while you hold out. Property brokers who quote low commissions often additionally checklist your property at a low worth in an effort to make a fast deal. Good property agents should be sincere about valuations without trying to tug a fast one over the owner. 5. A Good Marketing Technique<br><br>This in-depth coaching permits our associates to attain the best doable level of competence and this is complemented by inside collaboration to ensure that the appropriate individuals are matched to the fitting properties. the person's employer doesn't keep on a business, practise or act as a patent agent; and e. has accomplished internship in patent agency work under the supervision of a registered patent agent, or an individual registered as a patent agent or its equivalent in a rustic or territory, or by a patent workplace, specified in the Fourth. Notwithstanding Rule 6, a person is entitled to be registered as a patent agent if– JLL 2nd Quarter 2014 Real Property Statistics JLL ranked high actual property funding advisor in Asia Pacific Certified Property Agent<br><br>CPD is launched to improve the skilled standard of Real Property Trade in Singapore. CPD hours (points) is required for present salesperson for the renewal of CEA registration. Minimal of 6 hours is required, of at which minimum three hours must be from core topics. Historic Black and White Bungalows vary in size from 15,000 sq ft-270,000 sq ft. Minimum keep at this residence is 2 years. 10 minutes from Orchard Highway and 20 minutes from the Central Enterprise District. Mount Pleasant Road, Singapore. It's FREE to you, if you're a buyer of personal property. HDB resale flat consumers typically pay 1% fee. Should sellers sell their property now or maintain holding on? Potential pitfalls of getting into right into a property sharing settlement Industrial Office Buildings<br><br>Exclusive contracts last three months and the fee for promoting a HDB or non-public property is normally one to 2 per cent of the promoting price. Marcia says that sellers normally fare higher once they work with a trusted unique agent. As he will handle all enquiries, he's in a better place to safe the very best doable value. "When too many agents are advertising the same unit, some might attempt to close the deal quickly by persuading you to seize the primary supply instead of ready for a better worth," she says.<br><br>Service Select Company and brokerage Hospitality and leisure Industrial and logistics agency Funding company Land agency Lease advisory and dispute resolution Office company Residential company Retail agency Consulting Corporate actual estate consulting Growth consulting Hospitality and leisure Planning Retail development consulting Sustainability Investment and asset administration Asset administration Company finance Funding agency Funding management Occupier providers Undertaking management and building consultancy Building consultancy Mission management Property and services administration Corporate actual estate administration Amenities management Property administration Retail administration Analysis Valuations and advisory services Company recovery and restructuring Lease advisory and dispute resolution Rating and statutory valuations Valuations Search Now.<br><br>She is the star in the highly aggressive property agency enterprise." GLOBAL [http://dota.thecollegegamer.com/activity/p/120544/ property listing singapore] STRATEGIC ALLIANCE PTE LTD is accredited by the Council of Estate Brokers underneath the Ministry of Nationwide Improvement. The Graduate Certificates in Intellectual Property Regulation programme (GCIP) is a complete foundation IP regulation course which supplies the fundamental IP knowledge that is wanted by aspiring IP professionals and practitioners for a career within the area of Mental Property. GPS was founded by Dennis Yong and Jeffrey Hong, each of whom are icons of the present day real estate industry. Giants in their very own rights, their perception and views on the native property scene are still extremely valued and wanted. Choose and register for the course on a modular basis |
| 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.
| |
| | |
| ==Notation and conventions==
| |
| 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.
| |
| This article uses the (+) [[sign convention]] for Clifford multiplication so that
| |
| :<math>v^2 = Q(v)\,</math>
| |
| 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'')}}.
| |
| | |
| ==Bott periodicity==
| |
| 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.
| |
| | |
| ==Complex case==
| |
| The complex case is particularly simple: every nondegenerate quadratic form on a complex vector space is equivalent to the standard diagonal form
| |
| :<math>Q(u) = u_1^2 + u_2^2 + \cdots + u_n^2</math>
| |
| 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''').
| |
| | |
| 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
| |
| :<math>P_{\pm} = \frac{1}{2}(1\pm\omega).</math>
| |
| 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''}}):
| |
| :<math>\alpha(C\!\ell_n^\pm(\mathbf{C})) = C\!\ell_n^\mp(\mathbf{C})</math>.
| |
| 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>
| |
| {| border=1 cellpadding=2 style="text-align: center;"
| |
| |-
| |
| | ''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''')
| |
| |}
| |
| </center>
| |
| | |
| 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''')}}.
| |
| | |
| ==Real case==
| |
| 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.
| |
| | |
| ===Classification of quadratic form===
| |
| Firstly, there are non-isomorphic quadratic forms of a given degree, classified by signature.
| |
| | |
| Every nondegenerate quadratic form on a real vector space is equivalent to the standard diagonal form:
| |
| :<math>Q(u) = u_1^2 + \cdots + u_p^2 - u_{p+1}^2 - \cdots - u_{p+q}^2</math>
| |
| 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''').
| |
| | |
| 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]]).
| |
| | |
| To compute the square <math>\omega^2=(e_1e_2\cdots e_n)(e_1e_2\cdots e_n)</math>,
| |
| 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>,
| |
| 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===
| |
| 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>.
| |
| | |
| <center>
| |
| {| border=1 cellpadding=2 style="text-align: center;"
| |
| |-
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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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| |-
| |
| | 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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| |-
| |
| | 6 || − || M(2<sup>(''n''−2)/2</sup>,'''H''')
| |
| | 7 || − || M(2<sup>(''n''−1)/2</sup>,'''C''')
| |
| |}
| |
| </center>
| |
| | |
| 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).
| |
| <center>
| |
| {| border=0 cellpadding=2 style="text-align: center;"
| |
| |-
| |
| | || 8 || 7 || 6 || 5 || 4 || 3 || 2 || 1 || 0 || −1 || −2 || −3 || −4 || −5 || −6 || −7 || −8
| |
| |-
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| | 0 || || || || || || || || || '''R''' || || || || || || || ||
| |
| |-
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| | 1 || || || || || || || || '''R'''<sup>2</sup> || || '''C''' || || || || || || ||
| |
| |-
| |
| | 2 || || || || || || || M<sub>2</sub>('''R''') || || M<sub>2</sub>('''R''') || || '''H''' || || || || || ||
| |
| |-
| |
| | 3 || || || || || || M<sub>2</sub>('''C''') || || M<sub>2</sub><sup>2</sup>('''R''') || || M<sub>2</sub>('''C''') || || '''H'''<sup>2</sup> || || || || ||
| |
| |-
| |
| | 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''') || || || ||
| |
| |-
| |
| | 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''') || || ||
| |
| |-
| |
| | 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''') || ||
| |
| |-
| |
| | 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''') ||
| |
| |-
| |
| | 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''')
| |
| |-
| |
| | || || || || || || || || || || || || || || || || ||
| |
| |-
| |
| | ω<sup>2</sup> || + || − || − || + || + || − || − || + || + || − || − || + || + || − || − || + || +
| |
| |}
| |
| </center>
| |
| | |
| ===Symmetries===
| |
| There is a tangled web of symmetries and relationships in the above table.
| |
| | |
| :<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>
| |
| Going over 4 spots in any row yields an identical algebra.
| |
| | |
| From these Bott periodicity follows:
| |
| :<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>
| |
| If the signature satisfies {{nowrap|1=''p'' − ''q'' ≡ 1 (mod 4)}} then
| |
| :<math>C\ell_{p+k,q}(\mathbf{R}) = C\ell_{p,q+k}(\mathbf{R}) .</math>
| |
| (The table is symmetric about columns with signature 1, 5, −3, −7, and so forth.)
| |
| Thus if the signature satisfies {{nowrap|1=''p'' − ''q'' ≡ 1 (mod 4)}},
| |
| :<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>
| |
| | |
| ==See also==
| |
| *[[Dirac algebra]] Cℓ<sub>1,3</sub>('''C''')
| |
| *[[Pauli algebra]] Cℓ<sub>3,0</sub>('''C''')
| |
| *[[Spacetime algebra]] Cℓ<sub>1,3</sub>('''R''')
| |
| *[[Clifford module]]
| |
| *[[Spin representation]]
| |
| | |
| {{DEFAULTSORT:Classification Of Clifford Algebras}}
| |
| [[Category:Ring theory]]
| |
| [[Category:Clifford algebras]]
| |
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