Idempotents of Clifford Algebras

Abstract. It is known that Clifford (geometric) algebra offers a geometric interpretation for square roots of −1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Research has been done [1] on the biquaternion roots of −1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cℓ3 of R 3 . All these roots of −1 find immediate applications in the construction of new types of geometric Clifford Fourier transformations.We now extend this research to general algebras Cℓp,q. We fully derive the geometric roots of −1 for the Clifford (geometric) algebras with p + q ≤ 4. Mathematics Subject Classification (2000). Primary 15A66; Secondary 11E88, 42A38, 30G35.

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“…• The interesting relationship with families of idempotents of Clifford geometric algebras [21]. • What is the relationship with combinatorics?…”

Section: Discussionmentioning

confidence: 99%

Geometric Roots of –1 in Clifford Algebras Cℓ p,q with p + q ≤ 4

Hitzer

Abłamowicz

2010

Adv. Appl. Clifford Algebras

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“…Matrices sigma [1], sigma [2] and sigma [3] are the well-known Pauli matrices with entries in the field K: Let's find matrices representing the two basis elements in the spinor ideal S = C 3,0 f. As expected, these matrices over K have the following form: Thus, a spinor s is a complex vector written in terms of the basis {f 1 , f 2 } and its one-column complex matrix with entries in K = {1, e 2 ∧ e 3 } is: [2]:=a*Id+b*e23,c*Id+d*e23; ψ1, ψ2 := a Id + b e23 , c Id + d e23 > s:=f1 &c psi [1] + f2 &c psi [2];#remember that S is a right K-vector space Since CLIFFORD can handle computations with matrices in any Clifford algebra 11 , it can also handle spinor representations in quaternionic spinor spaces and in spinor spaces over dual numbers in the case of semisimple Clifford algebras. [3] …”

Section: Spinor Representation Of C (Q) In Minimal Left Idealsmentioning

confidence: 99%

“…For a complete treatment of this topic we refer to [11]. It is well known [30] that any primitive idempotent f in C p,q is expressible as a product…”

Section: Continuous Families Of Idempotents: Low Dimensional Examplesmentioning

confidence: 99%

“…has led to unexpected results and new insights. [4,6,7,11] Checking the consistency of the software by testing theorems and known results has led them in the foot steps of Pertti Lounesto to counter examples that have made a rethinking and a more careful restatement of those theorems necessary. However, the most striking ability of CLIFFORD is that it is unique in being able to handle Clifford algebras of an arbitrary bilinear form not restricted by symmetry and not directly related to any quadratic form.…”

Section: Introductionmentioning

confidence: 99%

“…As much as Buchberger's celebrated algorithm is indispensable for computing Gröbner bases, this new algorithm described in [9,10] is indispensable for the Clifford product. Having developed this faster algorithm one was able to find the q-Young Clifford idempotents [6] possessing a desired symmetry; explicitly describe the structure of the Spin(3) group; discover continuous families of idempotents in C (Q) [11]; or gain a better understanding of the properties of the dotted wedge product. [9] The present paper brings to the reader a few examples of computations and results derived with CLIFFORD.…”

Section: Introductionmentioning

confidence: 99%

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Computations with Clifford and Grassmann Algebras

Abłamowicz

2009

AACA

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Abstract. Various computations in Grassmann and Clifford algebras can be performed with a Maple package CLIFFORD. It can solve algebraic equations when searching for general elements satisfying certain conditions, solve an eigenvalue problem for a Clifford number, and find its minimal polynomial. It can compute with quaternions, octonions, and matrices with entries in C (B) -the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. It uses standard (undotted) Grassmann basis in C (Q) but when the antisymmetric part of B is non zero, it can also compute in a dotted Grassmann basis. Some examples of computations are discussed. Mathematics Subject Classification (2000). 15A66, 68W30.

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Square Roots of –1 in Real Clifford Algebras

Hitzer

Helmstetter

Abłamowicz

2013

Quaternion and Clifford Fourier Transforms and Wavelets

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It is well known that Clifford (geometric) algebra offers a geometric interpretation for square roots of −1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Systematic research has been done [32] on the biquaternion roots of −1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cℓ(3, 0) of R 3 . Further research on general algebras Cℓ(p, q) has explicitly derived the geometric roots of −1 for p + q ≤ 4 [17]. The current research abandons this dimension limit and uses the Clifford algebra to matrix algebra isomorphisms in order to algebraically characterize the continuous manifolds of square roots of −1 found in the different types of Clifford algebras, depending on the type of associated ring (R, H, R 2 , H 2 , or C). At the end of the paper explicit computer generated tables of representative square roots of −1 are given for all Clifford algebras with n = 5, 7, and s = 3 (mod 4) with the associated ring C. This includes, e.g., Cℓ(0, 5) important in Clifford analysis, and Cℓ(4, 1) which in applications is at the foundation of conformal geometric algebra. All these roots of −1 are immediately useful in the construction of new types of geometric Clifford Fourier transformations.Mathematics Subject Classification (2010). Primary 15A66; Secondary 11E88, 42A38, 30G35.

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Idempotents of Clifford Algebras

Cited by 11 publications

References 11 publications

Geometric Roots of –1 in Clifford Algebras Cℓ p,q with p + q ≤ 4

Geometric Roots of –1 in Clifford Algebras Cℓ p,q with p + q ≤ 4

Computations with Clifford and Grassmann Algebras

Square Roots of –1 in Real Clifford Algebras

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