Conformal transformations and Clifford algebras

Lounesto, Pertti; Latvamaa, Esko

doi:10.1090/s0002-9939-1980-0572296-4

Cited by 27 publications

(4 citation statements)

References 5 publications

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“…Remark 2.4. The geometric significance of the conjugation Φ u (x) = uxu −1 is well explained in the paper by Lounesto and Latvamaa [18]. For example, when Q is the quadratic form of signature (p, q) corresponding to the Clifford algebra C p,q and uũ = 1, the following exact sequences exist:…”

Section: Clifford Operator Calculusmentioning

confidence: 92%

Kravchuk Polynomials and Induced/Reduced Operators on Clifford Algebras

Staples

2014

Complex Anal. Oper. Theory

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Kravchuk polynomials arise as orthogonal polynomials with respect to the binomial distribution and have numerous applications in harmonic analysis, statistics, coding theory, and quantum probability. The relationship between Kravchuk polynomials and Clifford algebras is multifaceted. In this paper, Kravchuk polynomials are discovered as traces of conjugation operators in Clifford algebras, and appear in Clifford Berezin integrals of Clifford polynomials. Regarding Kravchuk matrices as linear operators on a vector space V , the action induced on the Clifford algebra over V is equivalent to blade conjugation, i.e., reflections across sets of orthogonal hyperplanes. Such operators also have a natural interpretation in terms of raising and lowering operators on the algebra. On the other hand, beginning with particular linear operators on the Clifford algebra C Q(V), one obtains Kravchuk matrices as operators on the paravector space V * through a process of operator grade-reduction. Symmetric Kravchuk matrices are recovered as representations of grade-reductions of maps induced by negative-definite quadratic forms on V .

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Section: Clifford Operator Calculusmentioning

confidence: 92%

Kravchuk Polynomials and Induced/Reduced Operators on Clifford Algebras

Staples

2014

Complex Anal. Oper. Theory

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“…The smooth composition with translations is best done in the conformal model [1,8,15,17,18,27] of Euclidean space (in the GA of R 4,1 ), which adds two null-vector dimensions for the origin e 0 and infinity e ∞ X = x + 1 2 x 2 e ∞ + e 0 , e 2 0 = e 2 ∞ = X 2 = 0, X · e ∞ = −1.…”

Section: Space Groupsmentioning

confidence: 99%

“…Intuitively all symmetry properties of crystals depend on these vectors. Indeed, the geometric product of vectors [7] combined with the conformal model of three-dimensional (3D) Euclidean space [1,8,15,17,18,27] yield an algebra fully expressing crystal point and space groups [10,13,[19][20][21]. Two successive reflections at (non-) parallel planes express (rotations) translations, etc.…”

Section: Introductionmentioning

confidence: 99%

Interactive 3D Space Group Visualization with CLUCalc and the Clifford Geometric Algebra Description of Space Groups

Hitzer

Perwass

2010

Adv. Appl. Clifford Algebras

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Abstract.A new interactive software tool is described, that visualizes 3D space group symmetries. The software computes with Clifford (geometric) algebra. The space group visualizer (SGV) originated as a script for the open source visual CLUCalc, which fully supports geometric algebra computation.Selected generators (Hestenes and Holt, JMP, 2007) form a multivector generator basis of each space group. The approach corresponds to an algebraic implementation of groups generated by reflections (Coxeter and Moser, 4th ed., 1980). The basic operation is the reflection. Two reflections at non-parallel planes yield a rotation, two reflections at parallel planes a translation, etc. Combination of reflections corresponds to the geometric product of vectors describing the individual reflection planes.We first give some insights into the Clifford geometric algebra description of space groups. We relate the choice of symmetry vectors and the origin of cells in the geometric algebra description and its implementation in the SGV to the conventional crystal cell choices in the International Tables of Crystallography (T . Hahn, Springer, 2005). Finally we briefly explain how to use the SGV beginning with space group selection. The interactive computer graphics can be used to fully understand how reflections combine to generate all 230 three-dimensional space groups. Mathematics Subject Classification (2000). Primary 20H15; Secondary 15A66, 74N05, 76M27, 20F55 .

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“…The main difficulty is related to the fact that the point-wise stabilizer of a complete geodesic in Isom + (H 4 ) is isomorphic to SO(3), which is non-commutative, as opposed to the case of H 3 , where the point-wise stabilizer of a complete geodesic in Isom + (H 3 ) is isomorphic to SO(2) and is commutative. To a certain extent, this non-commutativity is reflected by the non-commutativity of A 2 , so that the representation of elements of Isom + (H 4 ) by Vahlen matrices (2 × 2 matrices with entries in A 2 satisfying certain conditions, following Vahlen [27] and more recently Ahlfors and his collaborators and several others ([1]- [6], [21], [28], [29], [30])), is particularly useful and appropriate. This is the approach adopted here.…”

Section: Introductionmentioning

confidence: 99%

Delambre-Gauss Formulas for Augmented, Right-Angled Hexagons in Hyperbolic 4-Space

Tan,

Wong,

Zhang

2011

Preprint

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We study the geometry of oriented right-angled hexagons in H 4 , the hyperbolic 4-space, via Clifford numbers or quaternions. We show how to augment alternate sides of such a hexagon so that for the non-augmented sides, we can define quaternion half side-lengths whose angular parts are obtained from half the Euler angles associated to a certain orientation-preserving isometry of the Euclidean 3-space. This generalizes the complex half side-lengths of oriented right-angled hexagons in H 3 . We also define appropriate complex half side-lengths for the augmented sides of the hexagon. We further explain how to geometrically read off the quaternion half side-lengths for a given oriented, augmented, right-angled hexagon in H 4 . Our main result is a set of generalized Delambre-Gauss formulas for oriented, augmented, right-angled hexagons in H 4 , involving the quaternion half side-lengths and the complex half side-lengths. We also show in the appendix how the same method gives Delambre-Gauss formulas for oriented right-angled hexagons in H 3 , from which the well-known sine and cosine laws can be deduced. These formulas generalize the classical Delambre-Gauss formulas for spherical/hyperbolic triangles.

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Conformal transformations and Clifford algebras

Cited by 27 publications

References 5 publications

Kravchuk Polynomials and Induced/Reduced Operators on Clifford Algebras

Kravchuk Polynomials and Induced/Reduced Operators on Clifford Algebras

Interactive 3D Space Group Visualization with CLUCalc and the Clifford Geometric Algebra Description of Space Groups

Delambre-Gauss Formulas for Augmented, Right-Angled Hexagons in Hyperbolic 4-Space

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