· algebras, some Ore extensions, Weyl algebras and their quantizations, etc. Examples of GR-algebras, which are either G algebras or are isomorphic to quotient algebras of a G-algebra

International Advisory Committee

R. Ablamowicz A. Jadczyk I. PorteousUSA France UK

P. Anglés B. Jancewicz J. RyanFrance Poland USA

W. Baylis J. Keller I. ShestakovCanada Mexico Brazil

E. Bayro J. Ławrynowicz F. SommenMexico Poland Belgium

L. Dabrowski A. Micali G. SommerItaly France Germany

T. Dray Z. Oziewicz W. SprößigUSA Mexico Germany

B. Fauser J.M. Parra V. SoucekGermany Spain Czech Rep.

J. Helmstetter M. PavšicFrance Slovenia

Local Organizing Committee

J. C. Gutiérrez R. da RochaUSP UFABC

P. Koshlukov W. A. Rodrigues Jr.∗UNICAMP UNICAMP

R. Mosna F. ToppanUNICAMP CBPF

E. C. de Oliveira J. Vaz Jr.∗UNICAMP UNICAMP

∗Chairman

ABSTRACTS

Abłamowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Batard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Baylis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Bayro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Brachey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Brackx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Conradt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Czachor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16De Melo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17De Schepper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Degimerci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Demir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Eriksson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Fioresi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Franssens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Gürlebeck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Helmstetter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Hestenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Hitzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Hitzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Hitzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Hoefel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Jancewicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Jardim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Krump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Kuznetsova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Lasenby . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Lavor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Ławrynowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Leão . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Limoncu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Loya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Lundholm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Macías . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Marmolejo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Micali . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Mosna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Notte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Pavšic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Perotti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Pinotsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Reséndiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Rocha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Rochon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Rodrigues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Santhanam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Schmeikal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Selig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Smid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Snygg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Sobczyk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Souza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Sprößig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Staples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Stolfi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Sweetser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Tolksdorf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Toppan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Tremblay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Trovon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Vergara . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Wills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Computation of Non Commutative Gröbner Bases

in Grassmann and Clifford Algebras

Rafał Abłamowicz

It is well known that tensor algebras, Clifford algebras, and Grassmman and su-per Grassmann algebras belong to a wide class of non-commutative algebras thathave a Poincaré-Birkhoff-Witt (or, PBW for short) “monomial" basis. The necessaryand sufficient condition for an algebra to have such basis have been established byV. Levandovskyy as the so called “nondegeneracy condition". This has led him toa re-discovery of the so called G-algebras (previously introduced by J. Apel) andGR-algebras (Gröbner-ready algebras) and their classification. It was Teo Mora whoalready in the 90’s considered a comprehensive and algorithmic approach to Gröb-ner bases for commutative and non-commutative algebras. It was T. Stokes who 18years ago introduced Gröbner left bases (GLB) and Gröbner left ideal basis, with thelatter solving an ideal membership problem. Thus, a natural question is to first seekGröbner bases with respect to a suitable admissible monomial order for ideals intensor algebras T and then consider quotient algebras T/I. It was shown by Levan-dovskyy that these quotient algebras possess a PBW basis if and only if the ideal Ihas a Gröbner basis. Of course, these quotient algebras are of great interest because,in particular, Grassmann and Clifford algebras of a quadratic form arise that way.Examples of G-algebras include quasi-commutative polynomial rings, such as, forexample, the quantum plane, universal enveloping algebras of finite dimensional Liealgebras, some Ore extensions, Weyl algebras and their quantizations, etc. Examplesof GR-algebras, which are either G algebras or are isomorphic to quotient algebrasof a G-algebra modulo a proper two-sided ideal, include Grassmann and Cliffordalgebras, and other finite dimensional associative algebras. After recalling basic con-cepts behind the theory of commutative Gröbner bases, a review of the Gröbner innon-commutative algebras will be given with a special emphasis on computation ofsuch bases in Grassmann and Clifford algebras.

8

nD Images as Clifford Bundles Sections -

Application to Segmentation

Thomas Batard, Christophe Saint Jean and Michel Berthier

We present a new theoretical framework for nD image processing using Cliffordalgebras. Multidimensionnal images are considered as sections of a trivial Cliffordbundle (CT(D), π, D), endowed with a riemannian fiber metric.

Due to the triviality, any covariant derivative ∇ on this bundle is the sum of theusual derivative with ω, a one-form on D with values in End(CT(D)). We show thatvarying ω and derivating well-chosen sections with respect to ∇ provides all theinformation needed to perform various kind of segmentation. We present severalillustrations of our results, dealing in particular with color (n=3) and color/infrared(n=4) images. As an example, let us mention the problem of detecting homogeneousregions of a given hue with constraints on temperature; the segmentation resultsfrom the computation of ∇(IB), where I is the image section and B is the bivectorsection coding the given hue.

9

Quantum/Classical Interface:

A Classical Geometric Origin of Fermion Spin

W. E. Baylis, R. Cabrera and D. Keselica

Although intrinsic spin is usually viewed as a purely quantum property with noclassical analog, we present evidence here that fermion spin has a classical originrooted in the geometry of three-dimensional physical space. Our approach to thequantum/classical interface is based on a formulation of relativistic classical me-chanics that uses spinors. Spinors and projectors arise naturally in the Clifford’sgeometric algebra of physical space and not only provide powerful tools for solvingproblems in classical electrodynamics, but also reproduce a number of quantum re-sults. In particular, many properites of elementary fermions, as spin-1/2 particles,are obtained classically and relate spin, the associated g-factor, its coupling to anexternal magnetic field, and its magnetic moment to Zitterbewegung and de Brogliewaves. Spinors are also amplitudes that can undergo quantum-like interference. Therelationship of spin and geometry is further strengthened by the fact that physicalspace and its geometric algebra can be derived from fermion annihilation and cre-ation operators. The approach resolves Pauli’s argument against treating time as anoperator by recognizing phase factors as projected rotation operators.

10

Conformal Geometric Algebra for Robot Physics

E. Bayro-Corrochano

In this talk we use as a mathematical framework the conformal geometric alge-bra for applications in computer vision, graphics engineering, learning, control androbotics. We will show that this mathematical system keeps our intuitions and in-sight of the geometry of the problem at hand and it helps us to reduce considerablythe computational burden of the problems. Surprisingly as opposite to the standardprojective geometry, in conformal geometric algebra we can deal simultaneouslywith incidence algebra operations (meet and join) and conformal transformationsrepresented effectively using spinors (quaternions, dual quaternions, etc). In thisregard, surprisingly this framework appears promising for dealing with kinematics,dynamics and projective geometry problems without the need to abandon the math-ematical system (as current approaches). We present some real tasks of perceptionand action treated in a very elegant and efficient way: sensor-body calibration, 3Dreconstruction and robot navigation and visually guided 3D object grasping makinguse of the directed distance , algebra of incidence and conformal transformations.For a real time probabilistic geometric framework in tracking we use the Motor(dual quaternion) extended Kalman filter and for control problems we reformulatethe differential geometry and the Jacobian based control rule for n D. O. F. robotarms using conformal geometric algebra. The authors believe that the framework ofgeometric algebra can be in general of great advantage for applications in image pro-cessing, stereo vision, range data, laser, omnidirectional and odometry based roboticsystems, kinematics and dynamics of robot mechanisms, humanoids and advancednonlinear control techniques.

11

Algorithms for Computation of Gröbner Bases

in Grassmann Algebras

Troy Brachey

Algorithms for computation of Gröbner bases in Grassmann algebras are pre-sented. The author illustrates his own procedures as part of a Maple package forcomputation in Grassmann algebras. Examples of computation will be shown todemonstrate effectiveness of algorithms and procedures. Emphasis will be placedon the ideal membership problem.

Keywords: Gröbner basis, Grassmann algebra, Clifford algebra, ideal membership,Maple, Gröbner left basis, Gröbner left ideal basis

12

Hermitean Clifford Analysis

Fred Brackx

Euclidean Clifford analysis is nowadays a well established branch of classical anal-ysis centred around the notion of monogenic functions, in particular null solutions ofthe rotation invariant Dirac operator. Recently so–called Hermitean Clifford analysisemerged as a refinement of Euclidean Clifford analysis. Hermitean Clifford analysisis based on the introduction of an additional datum, a so–called complex structure,in order to bring the notion of monogenicity closer to complex analysis. A complexstructure J on a euclidean space E should be compatible with the Euclidean structureon E, i.e. J ∈ SO(E), and J2 = −1E. It is seen at once that the dimension of E is thenforced to be even: m = 2n. The subgroup of SO(E) preserving the complex structure,it means commuting with J, turns out to be isomorphic with the unitary group U(n).The complex structure J induces an associated, so–called twisted, Dirac operator ∂J .Hermitean Clifford analysis then focusses on Hermitean monogenic functions, i.e.simultaneous null solutions of both operators ∂ and ∂J , in this way breaking downthe rotational invariance of the Dirac operator, reducing it to U(n)-symmetry for theconsidered system.

This talk is focussed on the justification of the Hermitean Dirac system. Firstit is shown how the Hermitean Dirac operators originate quite naturally as gener-alized gradients in the sense of Stein and Weiss, when projecting the gradient onU(n)-invariant subspaces generated by the projection operators 1

2 (1± i J). Next it isshown how under the action of U(n), the space of spinor valued polynomials on R2n

decomposes into a sum of irreducible subspaces of R2n ∼= Cn, which is however notmultiplicity free. By complementing the U(n)–action by a new, hidden, symmetrycommuting with it, the resulting decomposition becomes multiplicity free; this ideais the well-known Howe Dual Pair from representation theory. It is shown that thedecompositions obtained exactly correspond to the fundamental Fischer decompo-sitions for Hermitean monogenic functions.

13

The conceptual meaning of Hermitean monogenicity is further unravelled by study-ing possible splittings of the Hermitean Dirac first order system into independentparts without changing the properties of the solutions. In this way connections withholomorphic functions of several complex variables are established. These connec-tions also become apparent when studying the Cauchy integral formulae. In factthe Hermitean Clifford analysis function theory is in full development. Its currentstate of affairs is presented: Martinelli-Bochner formula, Hilbert transform, Taylorexpansion, Bergman kernel,etc.

14

From Projective to Geometric Algebra

Oliver Conradt

An (outer) algebra without an one element is developed from projective geometry.The main characteristics of projective geometry such as the incidence relations, theoperations of connecting and intersecting and the complete principle of duality arereflected by this Projective Algebra. Following Arthur Cayley (1821-1895) and theErlanger Programm by Felix Klein (1849-1925) a generic metric is introduced makingProjective to a Geometric Clifford Algebra. Remaks on the history of mathematics,on space and counterspace and its application to geometry and physics complete thetalk.

15

Geometric-Algebra Teleportation

of Geometric Structures

Marek Czachor

“Cartoon-computation” is a formalism based on geometric algebra coding thatallows for a geometric analogue of quantum computation. The geometric productreplaces here the tensor product, and entangled states are replaced by multivectors.One does not need quantum mechanics and yet all the quantum algorithms canbe reformulated in this language. In particular, since teleportation protocols canbe formulated in terms of networks of elementary gates and all the quantum gatesgave geometric-algebra analogues, it follows that teleportation can be formulated inpurely geometric terms. I will show on explicit examples how it works.

[1] D. Aerts, M. Czachor, “Cartoon computation: Quantum-like algorithms without quantum me-chanics”, J. Phys. A: Math. Theor. 40, F259-F266 (2007), Fast Track Communication, quant-ph/0611279.

[2] M. Czachor, “Elementary gates for cartoon computation”, J. Phys. A: Math. Theor. 40, F753-F759(2007), Fast Track Communication, arXiv:0706.0967 [quant-ph].

[3] D. Aerts, M. Czachor, “Tensor-product vs. geometric-product coding”, Phys. Rev. A 77, 012316(2008), arXiv:0709.1268 [quant-ph].

16

Variational Formulation for

Quaternionic Quantum Mechanics

C. A. M. de Melo and B. M. Pimentel

A quaternionic version of Quantum Mechanics is achieved using theSchwinger’s formulation based on measurements and a Variational Principle. Com-mutation relations and evolution equations are provided, and the results are com-pared with other formulations.

17

The Hilbert–Dirac Operator on R2n

in Hermitean Clifford Analysis

Hennie De Schepper

Hermitean Clifford analysis is a recent branch of Clifford analysis, refining thestandard Euclidean case. It focusses on the simultaneous null solutions, called Her-mitean monogenic functions, of two complex Dirac operators which are invariantunder the action of the unitary group. The specificity of the framework, introducedby means of a complex structure creating a Hermitean space, forces the underlyingvector space to be even dimensional.

In engineering sciences, and in particular in signal analysis, the Hilbert transformof a real signal u(t) of a one-dimensional time variable t has become a fundamentaltool. The multidimensional approach to the Hilbert transform usually is a tensorialone, considering the so-called Riesz transforms in each of the variables separately.As opposed to these tensorial approaches, Clifford analysis is particularly suited fora treatment of multidimensional phenomena, encompassing all dimensions at thesame time as an intrinsic feature.

In this contribution, we devote ourselves to the introduction of a Hilbert transformon R2n in the Hermitean setting. Due to the forced even dimension of all vectorspaces involved, any Hilbert convolution kernel in R2n should originate from thenon-tangential boundary limits of a corresponding Cauchy kernel in R2n+2. Weshow that the difficulties posed by this inevitable dimensional jump can be overcomeby following a matrix approach. The resulting matrix Hermitean Hilbert transformalso gives rise, through composition with the matrix Dirac operator, to a HermiteanHilbert-Dirac convolution operator “factorizing” the Laplacian and being closelyrelated to Riesz potentials.

18

Spinors on 2n-dimensional Manifolds

with Structure Group SO(n,C)

N. Degirmenci and S. Karapazar

It is known that spinors are important geometric objects on manifolds as tensors.In this work we construct a new kind of spinors on a 2n-dimensional manifold Mwith structure group SO(n,C). The complex spin group Spin(n,C) is the universalcovering group of the complex orthogonal group SO(n,C). We use the spinor repre-sentation of Spin(n,C) for the construction of the spinors on M. Then we define theircovariant derivative and study some properties of them. Lastly we consider Diracoperator on such spinors.

19

Hyperbolic Quaternion Formulation

of Electromagnetism

Suleyman Demir, Murat Tanıslı and Nuray Candemir

Many papers in literature have been demonstrated that hypercomplex numbersystems with nonreal square root +1 have a wide potential to investigate the physi-cal theories in different areas. Hyperbolic quaternions are one of the non-associativehyperbolic number systems that are very suitable for the investigation of space-timetheories. Unfortunately, this system is 4-dimensional. By using the same idea on theconstruction of the complex quaternions, we combined two hyperbolic quaternionto express up to 8-dimensional physical quantities. Hyperbolic quaternion formu-lation of electromagnetism was absent in literature. Therefore, this work fills a gapand contains useful results. Maxwell’s equations and relevant field equations areinvestigated with hyperbolic quaternions, and these equations have been given incompact, simpler and elegant forms. Derived equations are compared with theirvectorial, complex quaternionic, dual quaternionic and octonionic representations,as well.

20

Hyperbolic Function Theory

Sirkka-Liisa Eriksson

The aim of this talk is to consider the hyperbolic version of the standard Cliffordanalysis. The need for such a modification arises when one wants to make surethat the power function xm is included. H. Leutwiler noticed in 1990 that the powerfunction is the conjugate gradient of a harmonic function, defined with respect tothe hyperbolic metric of the upper half space. The theory was extended to the totalClifford algebra valued functions called hypermonogenic in 2000 by H. Leutwilerand S.-L. Eriksson. The integral formula in the upper half space was proved in2004. We give a new formulation of the integral theorem, where the kernel functionsare hypermonogenic. We consider also the power series presentations and relatedresults.

21

Supersymmetry and Supergeometry

Rita Fioresi

Supersymmetry has been the driving force to develop supergeometry: how canwe express symmetries which go beyond the ordinary groups? In supergeometrythe underlying topological space of a supermanifold or a supervariety are only partof the story, we also need a supersheaf to describe the geometric objects.The functorof points approach to supergeometry brings back the geometric intuition and recov-ers all the ordinary constructions in the much richer setting of supergeometry.Asexamples of this phylosophy we will describe the construction of tangent spaces toa supermanifold and to a Lie supergroup, the quotient of Lie supergroups and theglobal Frobenious theorem.

22

Clifford Analysis Solution of the

Electromagnetic Boundary Value Problem

in a Gravitational Background Vacuum

Ghislain R. Franssens

We formulate and solve the boundary value problem for electromagnetic radia-tion in a vacuum with given arbitrary gravitational background, in terms of CliffordAnalysis based on the algebra Cl1,3 (R) over a pseudo-Riemannian manifold (M, g)with signature (1, 3). It is found that the general solution for the full electromag-netic field can be obtained by analytical means, once a fundamental solution of theLaplace-de Rham scalar wave equation is known. As a by product of our method,we obtain the generalization of the Sommerfeld radiation condition in a gravitationalbackground vacuum.

The considered problem is also instructional for fine tuning Clifford Analysis overmanifolds, so that it becomes better suited to solve a larger set of physics prob-lems. In particular, we show the naturalness of imposing a Clifford structure on thecotangent bundle, as opposed to the tangent bundle.

23

On Some Mapping Properties of

Monogenic Functions

K. Gürlebeck and J. Morais

The aim of this contribution is to study if monogenic functions can be defined bytheir geometric mapping properties. At first monogenic functions are considered asgeneral quasi-conformal mappings. Dilatations and distortions of these mappingsare estimated in terms of the hypercomplex derivative. This includes the descriptionof the interplay between the Jacobian determinant and the hypercomplex deriva-tive of such monogenic functions. It will be shown that both concepts can be usedto characterize quasi-conformal monogenic functions. Pointwise estimates from be-low and from above are given by using a generalized Bohr theorem and a Borel-Caratheodory theorem for monogenic functions. Then it will be studied if a subclassof quasi-conformal mappings exists that can be used for a “geometric” definitionof monogenic functions via their mapping properties. Main goal is to find out ifall functions from this subclass are monogenic and if monogenic functions must bein this subclass of quasi-conformal mappings. Finally it will be checked if thesefunctions belong to some recently studied weighted spaces of monogenic functions.

Keywords: monogenic functions, quasi-conformal mappings, geometric mappingproperties

24

Lipschitz Groups in Meson Algebras

Jacques Helmstetter

Let B(M, f ) be the meson algebra associated with the symmetric bilinear formM×M→ K over a field K of characteristic 6= 2.

Jacobson discovered the algebra morphism D from B(M, f ) into the non-twistedtensor product C`(M, f ) ⊗ C`(M, f ) that maps every a ∈ M to (a ⊗ 1 + 1⊗ a)/2 ;from the knowledge of the graded structure of B(M, f ), a much easier and shorterproof of the injectiveness of D can be derived. Its injectiveness proves that the evensubalgebra B0(M, f ) is provided with a parity subgrading B0,0(M, f )⊕ B0,1(M, f ) .

We still assume f to be nondegenerate. Every reflection r in (M, f ) is determinedby a non-isotropic vector d ∈ M, and for all a ∈ M,

r(a) = a− 2 f (a, d) df (d, d)

= −zaz−1 if z =2d2

f (d, d)− 1 ∈ B0,1(M, f ) .

The multiplicative group G generated by all these factors z is isomorphic to theorthogonal group GO(M, f ) by an isomorphism that maps every x ∈ G to the trans-formation a 7−→ xax−1 if x ∈ B0,0(M, f ), or a 7−→ −xax−1 if x ∈ B0,1(M, f ).

Two applications follow. First we can define mesonic Lipschitz groups and mesonicLipschitz monoids that satisfy the usual properties. Secondly we can prove that Duf-fin’s wave equation for meson particles is invariant by the action of GO(M, f ).

25

The Geometry of the Electron Clock

David Hestenes

In his seminal contribution to quantum mechanics, Louis de Broglie conjecturedthat the electron has an internal clock oscillating with frequency (B = mc2/h).This idea was soon forgotten with the invention of wave mechanics, wherein thede Broglie frequency (B is interpreted exclusively as the frequency of a wave. Re-cent confluence of a new electron theory with experimental evidence suggests thatde Broglie may well have been right in the first place. This lecture explains howthat came about. Geometric Algebra played a crucial role in creating a new theorywherein the electron is the seat of an electric dipole that oscillates with an ultrahighfrequency called zitter that is close to twice the de Broglie frequency. Direct detec-tion of the zitter appears to be possible by channeling electrons along crystal axes.Theory predicts a resonant interaction between zitter and crystal periodicity at about80 MeV/c. An exploratory experiment has reported a positive result. If the existenceof zitter is confirmed, it will have major implications for quantum mechanics. Inparticular, it requires a subtle modification of the Dirac equation with a surprisingconnection to electroweak theory.

26

Geometric Roots of −1

Eckhard Hitzer

It is known that Clifford geometric algebra offers a geometric interpretation forsquare roots of -1 in the form of blades that square to minus 1. This extends to ageometric interpretation of quaternions as the side face bivectors of a unit cube. Re-search has been done on the biquaternion roots of -1 (Sangwine, 2006), abandoningthe restriction to blades. Biquaternions are isomorphic to Cl(3,0). All these roots of-1 find immediate applications in the construction of new types of geometric CliffordFourier transformations.

We now extend this research to general algebras Cl(p,q). We will fully derive thegeometric roots of -1 for the Clifford geometric algebras with p+q<=3, and explainthe resulting solution manifolds.

27

The Interactive 3D Space Group Visualizer

Based on Clifford Geometric Algebra

Description of Space Groups

Eckhard Hitzer and C. Perwass

A new interactive software tool is demonstrated, that visualizes 3D space groupsymmetries. The software computes with Clifford (geometric) algebra. The spacegroup visualizer (SGV) is a script for the open source visual CLUCalc, which fullysupports geometric algebra computation.

Selected generators (Hestenes & Holt, JMP, 2007) form a multivector generatorbasis of each space group. The approach corresponds to an algebraic implemen-tation of groups generated by reflections (Coxeter and Moser, 4th ed., 1980). Thebasic operation is the reflection. Two reflections at non-parallel planes yield a rota-tion, two reflections at parallel planes a translation, etc. Combination of reflectionscorresponds to the geometric product of vectors describing the individual reflectionplanes.

In our presentation we will first give some insights into the Clifford geometricalgebra description of space groups. The symmetry generation data are stored in anXML file, which is read by a special CLUScript in order to generate the visualization.Then we will use the Space Group Visualizer to demonstrate space group selectionand give a short interactive computer graphics presentation on how reflections com-bine to generate all 230 three-dimensional space groups.

28

Geometric Algebra Feature

Extraction and Classification

Minh Tuan Pham, Kanta Tachibana, Eckhard Hitzer,

Sven Buchholz, Tomohiro Yoshikawa and Takeshi Furuhashi

This research proposes to use geometric algebra to systematically extract geomet-ric features from data given in a vector space. We show the results of classification ofhand-written digits, which were classified by feature extraction with the proposedmethod. Given a set of spatial vectors ξ = pl ∈ Rn, l = 1, . . . , m we extract k-vectors of different grades k; which encode the variations of the features.

Figure 0.1: Examples of the handwritten digit ‘1’, shown with straight line segments,rescaled to square, different from real pen curves.

Assuming ξ is a series of n-dimensional vectors, n′ + 1 feature extractions are de-rived where n′ = minn, m. For k = 1, . . . , n′, we write k′ = k− 1,

f0 (ξ) = 〈plpl+1〉, l = 1, . . . , n′ − 1 ∈ Rm−1,

fk (ξ) = 〈pl . . . pl+k′e−1I 〉, I ∈ Ik, l = 1, . . . , n′ − k′ ∈ R(m−k′)|Ik|.

29

Figure 0.2: Correct classification rate with f1 and mixture of experts.

Figure 0.3: Flow of multi-class classification. The top diagram shows the training ofthe GMM for class C ∈ ‘0’, . . . , ‘9’. The D1C denotes a subset of trainingsamples whose label is C. The f : ξ 7→ x shows feature extraction. Eitherof f1, f2, f0 is chosen as f . The bottom diagram shows estimation by thelearned GMMs. The same f chosen for training is used here. The GMMC

outputs p (ξ | C). The final estimation is C∗ = arg maxC p (ξ | C) P (C),where P (C) is the prior distribution. The set D3 consists of independenttest data.

30

Figure 0.4: Mixture of GMMs. Three GMMs via different feature extractions aremixed to yield output p (ξ | C).

|Ik| is the cardinality of the set Ik of all combinations of k elements from n elements.For I = i1 . . . ik, e−1

I = eik . . . ei2ei1 .

We performed the feature extractions fk, k ∈ 0, 1, 2 for hand-written digit data ofthe UCI Repository, see Fig. 0.1. In this Pendigits dataset 7494 samples were writtenby 30 people, divided into learning data D1, and validation data D2. The 3498 re-maining samples were written by 14 other people and are used as test data D3. Theflow of training and estimation of hand-written digit classification, as an exampleof multi-class classification, is shown in Fig. 0.3. Hyperparameters like the mixturenumber and the cutoff coefficient for each Gaussian mixture model (GMM) are de-cided by validation with dataset D2. The classification precision for D3 using onlycoordinate feature extraction f1 decreased remarkably with increasing ε, the randomrotation range parameter, Fig. 0.2. On the other hand, the classification precision us-ing mixture of experts (as in Fig. 0.4) did not decrease that much. The rotations hadno influence in the cases of f0 and f2, which use inner and outer products, respectively.

Our results confirm that the strategy to mix different GA feature extractions issuperior in both classification precision and robustness when compared with purecoordinate value features, which is the most often used conventional method.

Grant-in-Aid for COE program Frontiers of Computational Science (Nagoya University), and for

Young Scientists (B) #19700218.

31

On the Cohomology of g-algebras

Eduardo Hoefel

A g-algebra consists a triple (g, A, ρ) where g is a Lie algebra, A is an associativealgebra and ρ : g⊗ A → A is a Lie algebra action by derivations. The cohomologyof g-algebras has been introduced by Flato, Gerstenhaber and Voronov in 1995. Inthis talk I will show how the cohomology of g-algebras can be defined throughKoszul operads. The operad of g-algebras is related to Kajiura-Stasheff’s Open-Closed Homotopy Algebra (OCHA) and to Voronov’s swiss-cheese operad

32

The Energy-Momentum Tensor

in Premetric Electrodynamics

Bernard Jancewicz

The electromagnetic theory in large part is metric independent. It is called premet-ric electrodynamics. Metric enters the constitutive relation. The energy-momentumis a one-form, the energy-momentum tensor is an energy-momentum three-dimensionaldensity, therefore is has to be a mapping of volume trivectors into one-forms. WhenF and G are well known two-forms describing the electromagnetic field, V is avolume trivector, the energy momentum tensor is the following linear mappingV → T(V) = 1

2 [Gb(VbF)− Fb(VbG)].

33

Twisted Dirac Operators

on Asymptotically Locally Flat

Gravitational Instantons

Marcos Jardim

Asymptotically locally flat (ALF) gravitational instantons are an important classof non-compact manifolds, both from the physical and mathematical point of view.We will discuss instantons over ALF gravitational instantons and the correspondingtwisted Dirac operators. In particular, we give a condition for these twisted Diracoperators to be Fredholm and compute their index.

34

Three Dirac Operators in the Stable Rank

Lukas Krump

For many years, there is a constant interest in understanding a structure of a reso-lution starting with the Dirac operator in several Clifford variables. The case of threevariables in the stable rank was studied by several methods.

Recently, the Penrose transform method was successfully applied in low dimen-sions and similarly for two variables in higher dimensions. We shall show that thePenrose transform methods can be applied also for three variables in higher dimen-sions, giving comparable results and yielding a perspective way to a general case.

35

Clifford Algebras and Non (anti)commutative

Deformations of Supersymmetry

Zhanna Kuznetsova

Supergroups with Grassmann parameters are replaced by odd Clifford parame-ters. The connection with non-anticommutative supersymmetry is discussed. Thefollowing topics will be covered: Berezin-like calculus, non (anti)commutative su-persymmetric quantum mechanics, Drinfeld twist deformations.

36

Applications of Geometric Algebra

in Cosmology and Physics

Anthony Lasenby

Geometric Algebra (GA) is a powerful tool in many branches of physics and engi-neering. Here we will summarise some selected results obtained in cosmology andphysics using a GA approach.

The cosmology topics to be discussed centre on three linked areas: (1) a confor-mal geometric algebra (CGA) approach to Bianchi cosmology, with applications toa novel nonsingular Bianchi IX universe; (2) the use of CGA to give a boundarycondition on the total elapsed conformal time in the Universe. This provides anunexpected linkage between the value of the cosmological constant and the numberof e-folds of inflation [1]; (3) the role of spinning fluids and torsion in cosmology. Inthe former category we look at Weyssenhoff fluids[2, 3], and in the latter we examineElko spinors (as introduced in [4] and discussed further in [5]) from a GA point ofview.

The physics topics link to the above via conformal geometric algebra and a back-ground embedding in a scale-free 5-dimensional spacetime that in 4d constitutes deSitter space. Some aspects of propagators in the curved background space are dis-cussed, and applications made in electromagnetism, quantum mechanics and fluidflow. (A preliminary discussion of applications to electromagnetism was given in[6].) If time permits, some recent applications of CGA in rigid body motion will alsobe discussed.

[1] A. Lasenby and C. Doran. Closed universes, de Sitter space, and inflation. Phys.Rev.D, 71(6):063502,2005. astro-ph/0307311.

[2] J. Weyssenhoff and A. Raabe. Relativistic dynamics of spin-fluids and spin-particles. Acta Phys.Pol., 9:7, 1947.

37

[3] S. D. Brechet, M. P. Hobson, and A. N. Lasenby. Weyssenhoff fluid dynamics in general relativityusing a 1 + 3 covariant approach. Classical and Quantum Gravity, 24:6329-6348, 2007. arXiv:0706.2367.

[4] D. V. Ahluwalia-Khalilova and D. Grumiller. Spin-half fermions with mass dimension one: the-ory, phenomenology, and dark matter. Journal of Cosmology and Astro-Particle Physics, 7:12, 2005.arXiv:hep-th/0412080.

[5] R. da Rocha and W.A. Rodrigues Jr. Where are ELKO spinor fields in Lounesto spinor fieldclassification? Mod. Phys. Lett A, 21:65-74, 2006. arXiv:math-ph/0506075.

[6] A.N. Lasenby. Recent applications of conformal geometric algebra. In H. Li, P.J. Olver, and G.Sommer, editors, Computer Algebra and Geometric Algebra with Applications (Lecture Notes inComputer Science), page 298. Springer, Berlin, 2005.

38

Clifford Algebra Applied to

Grover’s Algorithm

Rafael Alves and Carlile Lavor

Grover’s algorithm is a quantum algorithm for searching in unstructured databases.Due to the properties of quantum mechanics, it provides a quadratic speedup overtheir classical counterparts. Using Clifford algebra, we present a new way to under-stand and simplify the ideas of Grover’s algorithm.

39

An Approach to Two- and Three-Dimensional

Models of Order-Disorder Transition

and Simple Orthorhombic Ising Lattices

Julian Ławrynowicz

The famous works by L. Onsager [8] and Z.-D. Zhang [10] give quaternion basedtwo- and three-dimensional models of order-disorder transition and simple orthorhom-bic Ising [2] lattices. The way of applying the quaternion structure can be mademore elegant and simple by the use of Clifford structures and the related P. Jordanstructures. In particular, four sequences of Jordan algebras can be constructed [3] inrelation to R, C, H and O, and the H-sequence appears to be crucial for approachingin four directions: 1) Onsager-Zhang, 2) fractals determining the binary and ternaryalloy structures [5, 6, 7], and 3) optimizing the quantum mechanics framework [1, 4].Besides, the Onsager-Zhang techniques and Clifford-Jordan structures approach in-volve and stimulate a quick development of the Toeplitz forms theory as it had beensuccessfully initiated by G. Szegö [9].

[1] S. I. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford Univ. Press 1966.

[2] E. Ising, Z. Phys. 31 (1925), 253–162.

[3] P. Jordan, J. von Neumann, E. Wigner, Ann. of Math. 35 (1934), 29–64.

[4] J. Ł., Osamu Suzuki, Internat. J. of Theor. Phys. 40 (2001), 387–397.

[5] J. Ł., O. S., Internat. J. of Pure and Appl. Math. 24, no. 2 (2005), 181–209.

[6] J. Ł., S. Marchiafava, S. Nowak-Kepczyk, Internat. J. of Geom. Meth. in Modern Phys. 3 (2006),1167–1197.

[7] J. Ł., M. N.-K., O. S., Internat. J. of Pure and Appl. Math., to appear.

[8] L. Onsager, Phys. Rev. 65 (1944), 117–149.

[9] G. Szegö, Communications du séminaire mathématique de l’université de Lund, tome supplémentairedédié à Marcel Riesz (1952), 228–238.

[10] Z.-D. Zhang, Philosophical Magazine 87 (2007), 5309–5419.

40

On the Spectrum of the Twisted

Dolbeault Laplacian over Kähler Manifolds

Marcos Jardim and Rafael Leão

We use Dirac operators techniques to improve the estimates for the first eigenval-ues of the Dolbeault Laplacian twisted by a Hermitian-Einstein connection on Kählermanifolds with positive scalar curvature obtained using the Kähler identities for theconnection.

Keywords: Twisted Dolbeault Laplacian; Hermitian-Einstein connections; holomor-phic vector bundles.

41

Degenerate Spin Groups

as Semi-Direct Products

T. Dereli, S. Koçak and M. Limoncu

Let Q be a symmetric bilinear form on Rn=Rp+q+r with corank r, rank p + q andsignature type (p, q), p resp. q denoting positive resp. negative dimensions. Weconsider the degenerate spin group Spin(Q) = Spin(p, q, r) in the sense of Crumey-rolle and prove that this group is isomorphic to the semi-direct product of thenondegenerate and indefinite spin group Spin(p, q) with the additive matrix groupMat

((p + q), r

).

42

Witten’s Holonomy Theorem

on Manifolds with Corners

Paul Loya

In the 1980’s Daniel Quillen introduced determinant line bundles and about thesame time Edward Witten derived a remarkable formula for the holonomy of the de-terminant line bundle of a Dirac operator using something called the “eta invariant”of Atiyah, Patodi, and Singer. In the physics literature, the holonomy of the determi-nant line bundle is called the “global anomaly". Witten’s derivation was later maderigorous by Bismut and Freed and also by Cheeger.

In this talk I will give an introduction to eta invariants and Witten’s holonomytheorem, and then I will discuss recent work concerning generalizations of this the-orem to situations quite different from the original results. This talk will be suitablefor a general audience.

[1] J.-M. Bismut and D.S. Freed, The analysis of elliptic families. Dirac operators, eta invariants, andthe holonomy theorem, Comm. Math. Phys.107 (1986), no.1, 103-163.

[2] J. Cheeger, η-invariants, the adiabatic approximation and conical singularities. I. The adiabaticapproximation, J. Differential Geom. 26 (1987), no. 1, 175–221.

[3] E. Witten, Global gravitational anomalies, Comm. Math. Phys. 100 (1985), no.2, 197-229.

43

On the Geometry of Supersymmetric

Quantum Mechanical Systems

Douglas Lundholm

We consider some simple examples of supersymmetric quantum mechanical sys-tems and explore their possible geometric interpretation with the help of geometricaspects of real Clifford algebras. This leads to natural extensions of the consideredsystems to higher dimensions and more complicated potentials.

44

On the Notions of Hyperderivative

and n-dimensional Directional Derivatives

M. E. Luna-Elizarrarás, M. A. Macías-Cedeño and M. Shapiro

The aim of this talk is to introduce the notion of n-dimensional directional deriva-tive in the context of Clifford analysis and to establish its relations with the hyper-derivative of a hyperholomorphic function introduced by Gürlebeck and Malonekin [1]. These relations are similar to those existing between the derivative of a holo-morphic function and its one-dimensional directional derivatives, in one complexvariable. Thus, there are extended onto the Clifford Analysis case the correspond-ing notions from [2]. All this applies to the problem of the hyperderivability of theClifford-Cauchy-type integral.

[1] H. Malonek, K. Gürlebeck, A Hypercomplex Derivative of Monogenic Functions in Rn+1 and its Appli-cations. Complex Variables, Vol. 39, pp. 199–228, 1999.

[2] I. Mitelman, M. Shapiro, Differentiation of the Martinelli-Bochner Integrals and the Notion of Hyper-derivability. Math. Nachr. 172, pp. 211–238, 1995.

45

Hardy Spaces, Singular Integrals

and the Geometry of Euclidean Domains

Emilio Marmolejo Olea

We study the interplay between the geometry of Hardy spaces and functionalanalytic properties of singular integral operators(SIO’s), such as Riesz transforms aswell as Cauchy-Clifford and harmonic double layer operator, on the one hand and,on the other hand the regularity and geometric properties of domains. Among otherthings, we give several characterizations of Euclidean balls, their complements, andHalf-spaces, in terms of the aforementioned SIO’s.

(This is joint work with Steve Hoffman, Marius Mitrea, Salvador Perez-Esteva andMichael Taylor.)

46

Dirac and Semi-Dirac Pairs

of Differential Operators

Mircea Martin

The standard Dirac operator D = Deuc,n on the Euclidean space Rn, n ≥ 2, isdefined as a first-order homogeneous differential operator on Rn with coefficients inthe real Clifford algebra An(R) associated with Rn by assuming that DD† = D†D =∆euc,n, where ∆euc,n stands for the Laplace operator on Rn, and D† = −D.

As a generalization of this specific class of differential operators we will inves-tigate pairs (D, D†) of first-order homogeneous differential operators on Rn withcoefficients in a real unital Banach algebra A, such that either

DD† = µL∆euc,n, D†D = µR∆euc,n,

orDD† + D†D = µ∆euc,n,

where µL, µR, or µ are some elements of A. Every pair (D, D†) that has the for-mer property is called a Dirac pair of differential operators, and each pair (D, D†)with the latter property is called a semi-Dirac pair. The two typical examples of aDirac, or semi-Dirac pair of differential operators on Rn are given by D = d + d∗

and D† = −(d + d∗), or D = d and D† = −d∗, where d is the operator of exteriordifferentiation acting on smooth differential forms on Rn, and d∗ is its formal adjoint.

Our main goal is to prove that for any Dirac pair, or semi-Dirac pair, (D, D†),we have two Cauchy-Pompeiu type, and two Bochner-Martinelli-Koppelman type inte-gral representation formulas in several real variables, one for D and, as expected,another for D†, respectively. In addition, we are going to show that the existence ofsuch integral representation formulas characterizes the two classes of pairs of differ-ential operators.

47

As a final comment, we should point out that the concept of a Dirac, or semi-Dirac,pair of differential operators has natural extensions in several complex variablesand in the setting of differential operators on a Clifford bundle over an orientedRiemannian manifold.

48

The Graded Structure

of Nondegenerate Meson Algebras

Artibano Micali

In 1938 Duffin proposed a wave equation for meson particles which looked likeDirac’s wave equation for electrons:(

β41c

∂

∂t− β1

∂

∂x− β2

∂

∂y− β3

∂

∂z+

imch

)ψ = 0 ;

yet the four matrices β j satisfied other relations than Dirac’s relations; if we setβ′4 = β4 and β′j = −β j for j = 1, 2, 3, Duffin’s relations are:

β jβ′kβl + βl β

′kβ j = δj,kβl + δl,kβ j .

Whereas the universal algebra associated with Dirac’s relations is a Clifford algebra,the universal algebras associated with Duffin’s relations and with all analogous re-lations are called Duffin-Kemmer algebras, or shortly meson algebras.

Let M be a vector space of finite dimension n over a field K, and f a symmetricbilinear form M × M → K. The meson algebra B(M, f ) is the associative algebragenerated by M (and the unit element 1 ) with the only relations aba = f (a, b) a forall a, b ∈ M. As a consequence of these relations, we also have

abc + cba = f (a, b) c + f (c, b) a .

Like all Clifford algebras, B(M, f ) is provided with a parity grading B(M, f ) =B0(M, f ) ⊕ B1(M, f ) and with a reversion. The dimensions of B0(M, f ), B1(M, f )and B(M, f ) are (2n

n ) , ( 2nn−1) and (2n+1

n ) .

To get a precise description of B(M, f ) when f is nondegenerate, we define agraded representation B(M, f )→ End(E) with

E =∧

(M)⊕∧

(M) , E0 =∧

(M)⊕ 0 and E1 = 0⊕∧

(M) ;

49

with every a ∈ M is associated this odd operator in End(E) :

(u, v) 7−→ ( a ∧ v , a y f u ) for all u, v ∈∧

(M) .

We prove that the image of B(M, f ) in End(E) is the subalgebra of all endomor-phisms of E satisfying these two properties: they leave invariant all

∧k(M)⊕∧k−1(M)(for k = 0, 1, 2, ..., n + 1), and they commute with the natural action of the gradedalgebra

∧0(M)⊕∧n(M) in E.

Some applications to Duffin’s wave equation follow.

50

Latent Symmetries of the Dirac-Kähler Equation

and Electroweak Interactions

Ricardo Mosna and Jayme Vaz Jr.

The solutions of the Dirac-Kähler equation in flat spacetime are known to pos-sess a fourfold degeneracy which can be used to define, in a certain sense, fouruncoupled Dirac equations on minimal left ideals of the Dirac algebra (complexifiedClifford algebra of spacetime). The arbitrariness in choosing this set of ideals givesrise to a global symmetry of the Dirac-Kähler Lagrangian. We gauge this symmetryby considering sets of (minimal left) ideals varying from point to point in space-time. The resulting gauge fields then couple, in an essential way, the different idealsof the algebra. The structure of the interactions imply that the gauge fields onlycouple states with the same handedness, so that gauge interactions between left-handed and right-handed states are naturally suppressed. Moreover, the formalismautomatically gives rise to a term in the Lagrangian corresponding to the associ-ated antiparticles, with the correct handednesses. By restricting the interactions tothose conserving electric charge, the resulting model recovers the left-right symmet-ric model of electroweak interactions, provided that we identify the different idealswith leptons (or quarks) of a given generation of particles. When the symmetry isbroken, so that the ideals corresponding to the right-handed (left-handed) neutrino(antineutrino) remain fixed, the Glashow-Weinberg-Salam model is recovered. Inthis context, the Higgs field can be essentially thought of as defining a parametriza-tion of the set of ideals associated with the different particles. We finally considerpossible applications of this formalism to lattice field theory.

51

The Square of the Dirac and Spin-Dirac Operators

on a Riemann-Cartan Space

Eduardo Notte-Cuello

In this work we introduce the Dirac and spin-Dirac operators associated to a con-nection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac opera-tors associated with a Levi-Civita connection on a Riemannian (Lorentzian) space(time)and calculate the square of these operators, which play an important role in severaltopics of modern Mathematics. We obtain a generalized Lichnerowicz formula, de-compositions of the Dirac and spin-Dirac operators and their squares in terms of thestandard Dirac and spin-Dirac operators and using the fact that spinor fields (sectionsof a spin-Clifford bundle) have representatives in the Clifford bundle we present alsoa noticeable relation involving the spin-Dirac and the Dirac operators.

(This work is in conjuntion with W. A. Rodrigues Jr. and Q. A. G. Souza and waspartially supported by the DIULS of the La Serena University)

52

On the Unification of Interactions

by Clifford Algebra

Matej Pavšic

In current approaches to quantum gravity the starting point is often in assumingthat at short distances there exists an underlying structure, based, e.g., on stringsand branes, or spin networks and spin foams. It is then expected that the smoothspacetime manifold M4 of classical general relativity will emerge as a sufficientlygood approximation at large distances. However, it is feasible to assume that whatwe have, even at large distances, is in fact not spacetime, but a more general space.One possibility is in assuming that the long distance approximation to a more fun-damental structure is the space of extended events, corresponding to points, lines,areas, 3-volumes, and 4-volumes in M4. All those objects can be elegantly repre-sented by Clifford numbers xMγM ≡ xµ1...µR γµ1...µR , R = 0, 1, 2, 3, 4. This leads to theconcept of the so called Clifford space C, a 16-dimensional manifold whose tangentspace is Clifford algebra C`(1, 3). We assume that C has in general non vanishingcurvature. The connection and vielbein of Clifford space are determined by solu-tions to the generalized Einstein equations, and contain not only the 4-dimensionalgravitational field, but also other gauge fields, thus enabling a unification of interac-tions similar to that in Kaluza-Klein theories.

We consider the generalized Dirac equation for the Clifford algebra valued fieldΨ(X) that depends on position in C. At every point X ∈ C the field Ψ can be decom-posed into four independent geometric spinors belonging to the left minimal idealsof C`(1, 3). We explore such a system and argue that it is promising for the unifi-cation of the Standard model particles and gauge fields. Usually it is believed thatC`(1, 3) is not sufficient, therefore higher dimensional Clifford algebras are consid-ered. But in those approaches the fields depend on position in M4, not in C. Havinga 16-dimensional manifold C, we can exploit the possibility that C admits isome-tries. The corresponding conserved charges turn out to have two contributions: onefrom the ‘orbital’ angular momentum in the ‘internal’ part of C, and the other one

53

from the ‘internal spin’. This brings into the game additional quantum numbers thatenlarge the set of basis states for our system.

54

On Directional Quaternionic Hilbert Operators

Alessandro Perotti

The talk discusses harmonic conjugate functions and Hilbert operators in the spaceof Fueter regular functions of one quaternionic variable. We consider left-regularfunctions in the kernel of the Cauchy-Riemann operator

D =∂

∂x0+ i

∂

∂x1+ j

∂

∂x2− k

∂

∂x3= 2

(∂

∂z1+ j

∂

∂z2

).

Let J1, J2 be the complex structures on the cotangent bundle of H ' C2 inducedby left multiplication by i and j, and set J3 = J1 J2. For every complex structureJp = p1 J1 + p2 J2 + p3 J3 (p ∈ S2 a imaginary unit), let ∂p = 1

2

(d + pJp d

)be the

Cauchy-Riemann operator w.r.t. the structure Jp.Let Cp = 〈1, p〉 ' C. If Ω satisfies a geometric condition, for every Cp-valued

function f1 in a Sobolev space on the boundary ∂Ω, we obtain a function Hp( f1) :∂Ω → C⊥p , such that f = f1 + Hp( f1) is the trace of a regular function on Ω. Thefunction Hp( f1) is uniquely characterized by L2(∂Ω)-orthogonality to the space ofCR-functions w.r.t. the structure Jp.

In this way we get, for every direction p ∈ S2, a bounded, linear Hilbert operatorHp, with the property that H2

p = id− Sp, where Sp is the Szegö projection w.r.t. thestructure Jp.

55

Quaternions, Boundary Value Problems

and Evaluation of Integrals

Dimitrios Pinotsis

We present two novel applications of the theory of Quaternions: (a) The solutionof certain boundary value problems for linear elliptic Partial Differential Equations(PDEs) in four dimensions. (b) The explicit computation of certain three dimensionalintegrals without integrating with respect to the real variables. Both applications arebased on an important formalism in complex analysis, the so called Dbar formalism,and its quaternionic generalizations. The relevant results have been published in[1,2].

[1] D.A. Pinotsis, The Dbar Formalism, Quaternions and Applications, PhD Thesis, University ofCambridge (2006)

[2] A.S. Fokas and D.A. Pinotsis, Quaternions, Evaluation of Integrals and Boundary Value Problems,Computational Methods and Function Theory (to appear)

56

Oscillatory Movements and Dual Quaternions

Rafael Reséndiz

This paper shows how dual quaternions can be a way to describe oscillatory move-ments. Dual quaternions are the algebraic counterpart of screws. This fact enables toget an alternative description of an harmonic oscillatory motion. In this way, we canarrive to model more complicated oscillatory movements for example, it is possibleto get the solution of a classic PDE: the Wave Equation. Thus, following this idea wecan describe the kinetic behavior (position, velocity and time) by a dual quaternion.

57

Geometric Aspects of ELKO Spinor Fields:

Pure Spinors, Supergravity and Flagpoles

Roldão da Rocha

Dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields)belong, together with Majorana spinor fields, to a wider class of spinor fields, theso-called flagpole spinor fields, corresponding to the class (5), according to Lounestospinor field classification based on the relations and values taken by their associatedbilinear covariants. There exists only six such disjoint classes: the first three cor-responding to Dirac spinor fields, and the other three respectively correspondingto flagpole, flag-dipole and Weyl spinor fields. We also investigate and provide thenecessary and sufficient conditions to naturally extend the Standard Model to spinorfields possessing mass dimension one. As ELKO is a prime candidate to describedark matter. Also, we show that the Einstein-Hilbert, the Einstein-Palatini, and theHolst actions can be derived from the Quadratic Spinor Lagrangian (QSL), when thethree classes of Dirac spinor fields, under Lounesto spinor field classification, areconsidered. To each one of these classes, there corresponds a unique kind of actionfor a covariant gravity theory. Any other class of spinor field (Weyl, Majorana, flag-pole, or flag-dipole spinor fields) yields a trivial (zero) QSL, up to a boundary term.Finally it is shown how to express ELKO spinor fields uniquely in terms of purespinor fields.

58

On a Generalized Fatou-Julia Theorem

in Multicomplex Spaces

Dominic Rochon

In this talk, we present an overview of the hypercomplex 3D fractals generatedfrom Multicomplex Dynamics. In particular, we give a multicomplex (i.e. bicomplex,tricomplex, etc.) version of the so-called Fatou-Julia theorem. More precisely, wepresent a complete topological characterization in R2n

of the multicomplex filled-Julia set for a quadratic polynomial in multicomplex numbers of the form w2 + c. Wealso present a simple method to explore and infinitely approach these hypercomplex3D fractals.

59

Killing Vector Fields, Maxwell Equations

and Lorentzian Spacetimes

Waldyr A. Rodrigues Jr.

In this talk we first analyze the structure of Maxwell equations in a Lorentzianspace when the potential obeys the Lorenz gauge. We show that imposition of theLorenz gauge can only be done if the spacetime has Killing vector fields, and inthis case the potential must be a (dimensional) constant multiple of a the 1-formfield physically equivalent to a Killing vector field. Moreover we determine the formof the current associated with this potential showing that it is proportional to thepotential, i.e., given by 2AβRβ, where the Rβ are the Ricci 1-form fields. Finally westudy the structure of the spacetime generated by the coupled system consisting ofa electromagnetic field F = dA, (with the electromagnetic potential A satisfying theLorenz gauge) an ideal charged fluid with dynamics described by an action functionS and the gravitational field. We show that Einstein equations is then equivalent toMaxwell equations with a current given by f FAF (the product meaning the Cliffordproduct of the corresponding fields), where f is a scalar function which satisfies awell determined algebraic quadratic equation.

60

Generalized Clifford Algebra,

Unbiased Quantum States, and Quantum Information

T. S. Santhanam

Weyl’s algebra of unitary operators in ray space is a special case of a General-ized Clifford Algebra. Two unitary operators of Generalized Clifford Algebra arecalled “complementary” if their eigenvectors (in n-dimensions) satisfy the relation|(ej, ek)| = 1√

n . independent of j and k. In other words, the connecting matrix has

all elements of modulus 1√n . Such bases are called “mutually unbiased”. These

states play a very important role in quantum information and quantum communi-cation. What this means is that for a given input all outputs are equally probable.We will discuss in this talk a method of constructing these mutually unbiased statesof quantum mechanics using the representations of Generalized Clifford Algebras.

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Pregeometry of Extensions and Eigenfields∗

Bernd Schmeikal

In this lecture it is attempted to derive geometric properties of space from thoseof the fields. This can be done by realizing the meaning of extensions within non-commutative geometry. The mystery lies in a special togetherness of commutativityand anticommutativity, of 1-norm and 2-norm. We are familar with the Z2 -gradingof the Clifford algebra and the double cover of orthogonal groups. With this weassociate the projector equation and decomposition of unity 1 = P1 + P0 accordingto spin decomposition or chirality. A binary decomposition like that is characteristicfor quantumelectrodynamics (qed). It has first been used by John von Neumannand interpreted by von Weizsäcker as logic alternative. Weyl has for some timepondered over the meaning of the Klein-4 group and rays as compared with vec-tors. Then he could not yet realize the importance of a quaternary decomposition ofunity 1 = P0 + P1 + P2 + P3 and the K4-grading. This equation characterizes quan-tumchromodynamics (qcd) in quite general algebras. What a binary decompositionis for qed, the quaternary is for qcd. In this paper the algebraic foundations are givenby what is called here a maximal ternary Cartan decomposition in noncommutative al-gebras. The natural norm of a Cartan extension is not derived from the Minkowskimetric, but from the fact that, within the extension subspaces, the Clifford productbecomes the inner product while the exterior product vanishes.

Keywords: noncommutative geometry, graded algebra, Neumann ring, Weyl field,stochastic graded field, extension, Cartan extension, ternary extension, eigenfield,pure state, standard model, Majorana spinor, Clifford algebra, 1-norm, stratifiedspace, K4-grading, quantumchromodynamics

∗written in memory of Carl Friedrich von Weizsäcker

62

Exponential and Cayley maps

for Dual Quaternions

Jon Selig

Dual quaternions were introduced by Clifford in [1] to transform what he calledrotors. These were vectors bound to points in space, essentially directed lines withan associated magnitude. These rotors were intended to model angular velocitiesand wrenches. The sum of two rotors is in general a motor, what would now becalled a twist. In this work the term dual quaternion will be used rather than Clif-ford’s name ‘biquaternion’ since this seems to refer to several possible cases.

In modern notation dual quaternions can be thought of as elements of the degener-ate Clifford algebra C`(0; 2; 1) or perhaps more conveniently as the even subalgebraof C`(0; 3; 1). The Spin group for this algebra is the double cover of the group ofproper Euclidean motions. The group of proper Euclidean motions itself can berealised as a quadric in the projectivisation of the Clifford algebra, this quadric isusually known as the Study quadric.

The Lie algebra of both groups groups also lies in the Clifford algebra. Lie algebraelements, sometimes called twists, are represented by dual pure quaternions. That isdual quaternions of the form s = q0 + εq1, where ε is the dual unit satisfying ε2 = 0and q0, q1 are quaternions with no real part.

These twist satisfy a degree 4 relation, namely

s4 + 2θ2s2 + θ4 = 0,

where θ2 = q0q−1 . Using this relation it is possible to find a system of idempotentsand nilpotents P+, P−, N+, N− satisfying,

P2+ = P+, P2

− = P−, P+N+ = N+, P−N− = N−, N2+ = N2

− = 0,

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and all other products are zero, [4]. This system can be used to write infinite powerseries in s as cubic polynomials in s.

In particular the exponential map, defined by the familiar McLauren series but fordual pure quaternions, is a map from the lie algebra to the group. Using the abovemethod we obtain the following formula,

es =12(2 cos θ + θ sin θ)− 1

2θ(θ cos θ − 3 sin θ)s+

12θ

(sin θ)s2 − 12θ3 (θ cos θ − sin θ)s3,

which is similar to the well known Rodrigues formula for rotations.

Another map from the Lie algebra to the group of proper Euclidean transforma-tions is given by the Cayley map. This is defined as g = (1 − s)(1 + s)−1. In asimilar manner this map can also be expressed as a cubic polynomial in the twists. This map is useful for numerical methods since it does not involve trigonometricfunctions.

These two maps will be compared and also compared to the Cayley maps derivedfrom different matrix representations of the group, see [3]. The geometry of theseCayley map in relation to the Study quadric will be explored. Next relations for thederivatives of these maps are found, following Hausdorff [2]. Using the system ofidempotents and nilpotents these relations are easily inverted to give relations forthe derivative of the twist.

Finally, the problem of finding all possible analytic maps from the Lie algebra tothe groups is studied.

[1] W.K. Clifford, Preliminary Sketch of the biquaternions. Proc. London Math. Soc. s1-4(1):381.395,1871.

[2] F. Hausdorff. Die Symbolische exponential formel in den gruppen theorie. Berichte de SlachicenAkademie de Wissenschaften (Math Phys Klasse) vol. 58, pp. 19.48, 1906.

[3] J.M. Selig, Cayley Maps for SE(3), .The International Fedaration of Theory of Machines and Mech-anisms 12th World Congress, Besancÿon 2007.

[4] G. Sobczyk. The generalized spectral decomposition of a linear operator. The College Mathematics

Journal pp. 27.38, 1997.

64

Polynomial Invariants

for Rarita-Schwinger Representations

Dalibor Smid

Rarita-Schwinger operators are generalizations of Dirac operators to higher spinrepresentations. We describe the space of invariant polynomial endomorphisms ofsuch representations, as a first step for establishing the Fischer decomposition ofpolynomials with values in Rarita-Schwinger representations.

(Joint work with David Eelbode.)

65

Unification of Space-Time-Matter-Energy

Garret Sobczyk and Tolga Yarman

A complete description of space-time, matter and energy is given in terms ofthe conservation of energy-momentum in Einstein’s special theory of relativity. Wederive explicit equations of motion for two falling bodies, based upon the principlethat each body must subtract the mass-equivalent for any change in its kinetic energythat is incurred during the fall. In this theory, we find that there are no singularitiesand consequently no blackholes.

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Graphical Calculi and

Categories with Additional Structure

Fernando Souza

Graphical calculi representing categorical structures have been used extensivelyin various branches of mathematics and other areas, particularly topology, algebra,category theory, logic, classical and quantum information processing, and physics.They provide tools for faithfully visualizing abstract generalizations of tensor prod-ucts, duality, traces, braidings, twists, formal summations, and irreducible represen-tations, among other structures. Through this, they have comprised a powerful wayto calculate, proof, and generalize in a variety of contexts. They have also played akey role in the typical application of categories to the establishment of connectionsbetween seemingly diverse areas.

There are various approaches to those graphical calculi, developed at differentlevels of rigor. In this survey, we will review some of the major approaches froma categorical viewpoint, revising the relationship between them. After a short in-troduction to the underlying notions from knot theory (diagrams for links, braids,tangles, and rigid-vertex graphs), we will cover the following treatments: The well-studied case of the categories with additional structure that are freely generated bya given category; categories that have additional structure themselves; Penrose’s ar-row notation and its variations (including operadic ones and non-aligned diagrams),as well as its correspondence to approaches based on words and incidence relationsvia Penrose’s tensor notation; and the construction of algebraic objects (includingsome universal ones) in the presence of categorical structures, including a brief dis-cussion of planar algebras.

Our presentation will be as self-contained as possible, emphasizing the usage ofthese techniques. Important examples from several sources shall be mentioned. Em-phasis will be given to two overlapping, interdisciplinary subjects: Quantum algebra,particularly Hopf-algebra objects and bialgebra objects in various kinds of categorieswith additional structure; and combinatorial representation theory.

67

Hydrodynamics with Thermal

and Magnetic Effects

in Complex Quaternionic Setting

Wolfgang Sprößig

The talk is considered with Poisson–Stokes equations where thermal effects aredescribed by Boussinesq approximations. The methods based on the Bergman–Hodge decomposition of the corresponding complex quaternionic Hilbert space.Rothe’ s method is applied for solutions of the time-dependent Boussinesq problem.The same ideas can be transfered for the consideration of problems of magneto-hydrodynamics. Representation formulae can be presented.

68

Clifford Algebras, Graph Problems,

and Computational Complexity

G. Stacey Staples

Extending Clifford-algebraic methods to graph theory opens the door to applica-tions in theoretical computer science, symbolic dynamics, and coding theory. Forexample, defining the adjacency matrix A of a graph whose vertices and edges arelabeled with basis vectors of C`p,q allows one to enumerate the graph’s k-cyclesby examining the trace of Ak, provided k is odd. More general results are possi-ble by constructing commutative subalgebras of Clifford algebras. Three general-purpose algebras can be constructed within a Clifford algebra of appropriate sig-nature. The algebras are generated by the unit scalar along with elements xisatisfying xixj = xjxi and one of the following: (i) xi

2 = 0 (null-square); (ii) xi2 = 1

(unipotent); (iii) xi2 = xi (idempotent).

Combinatorial properties of these algebras make them useful for studying a widevariety of graph problems. In addition, they illustrate a potential reduction in com-putational complexity. The problem of enumerating a graph’s k-cycles is known tobe NP-complete. By considering entries of Λk, where Λ is an appropriate “nilpo-tent adjacency matrix” associated with a finite graph on n vertices, the k-cycles inthe graph are recovered by performing O(n3 log k) Clifford operations or “C` ops”.While the number of geometric products required is not a natural measure of com-plexity in classical computing, it is natural if one assumes the existence of a Cliffordcomputer that can perform such operations in either constant or polynomial time. Anumber of applications will be discussed, including processes on geometric randomgraphs. These processes are related to addition-deletion networks used to modelwireless networks.

(Joint work with René Schotty – yIECN and LORIA Universié Henri Poincaré-NancyI, BP 239, 54506 Vandoeuvre-ltes- Nancy, France, email: [email protected])

69

Oriented Projective Geometry

Jorge Stolfi

Oriented Projective Geometry (OPG) [1] is a variant of Real Projective Geometry(RPG) that supports the concepts of orientation, sideness, handedness, convexity,etc., in any dimension. Many algorithms of computational geometry depend onthese concepts, and therefore can be formulated in OPG but not in RPG.

On the other hand, OPG retains most of the advantages that RPG has over Eu-clidean Geometry, such as the smooth handling of points at infinity and parallelism,geometric duality, and projective transformation. Thus, geometric algorithms thatwere developed for Euclidean Geometry often become simpler, more elegant, andmore general when recast in OPG.

Mathematically, the n-dimensional oriented projective space Tn is a double coverof the real projective n-space Pn — which is simply the sphere Sn. The lines of Tn

are the great circles of Sn, and are endowed with an internal orientation (a sense oftravel along the line) and an external orientation (a sense of turning around the line).Planes and higher-dimensional subspaces are also oriented. The fundamental RPGoperations, join (the smallest projective subspace enclosing two given subspaces) andmeet (the intersection of two subspaces) are replaced by orientation-sensitive and an-ticommutative versions.

Computationally, points in Tn can be represented by n + 1 homogeneous coordi-nates, as in Pn; except that reversing the signs of all coordinates, which is a no-op inPn, yields a distinct point of Tn. Lines and higher-dimensional subspaces can be ho-mogeneously represented by Plücker coordinates; or, more compactly, by projectiveframes.

OPG can be viewed as an algebraic structure embeedded in a larger Clifford alge-bra. The elementary entities of OPG (points, lines, planes, etc.) can be identified with

70

certain “pure” elements of a Clifford algebra (products of points), and the join andmeet operations of OPG are essentially components of the Clifford algebra’s prod-uct. However, representing an element of a Clifford algebra requires 2n+1 numbers,whereas a point or hyperplane of OPG requires only n + 1 homogeneous coordi-nates, and a line or hyperline requires only n(n + 1)/2. Thus, while Clifford algebraoften provides more succint and general formulas than OPG, the latter usually yieldsmore efficient algorithms — and is easier to interpret in geometric terms.

[1] Stolfi, J., Oriented Projective Geometry: A Framework for Geometric Computations, AcademicPress (1991).

71

New Insights in the Physics of Spacetime

Using Quaternions

Douglas Sweetser

A real 4 × 4 matrix representation of quaternions is shown to be a commutingdivision algebra so long as zero and quaternions where one element equals the sumof the other three are excluded, as happens in physics for light-like events. A twolimit derivative is defined, where first the 3-vector goes to zero, then the scalar, ashappens in physics for time-like events. This directional derivative along the scalaror time axis is the well behaved domain of classical physics where events are orderedin time. If the limit process is reversed as happens for space-like events, then onlythe normed derivative can be calculated. This is the domain of quantum mechanicswhere one can know all the possible states. Finally, events in spacetime will beanimated directly from quaternion equations. Analytic animations of trig functionssuch as a sine can be surprising. The first visualization of all the groups that makeup the standard model - U(1), SU(2), and SU(3) - will be shown.

72

Twisted Clifford Bundles, Gravity

and the Mass of the Higgs Boson

Jürgen Tolksdorf

We discuss the basic ideas of “gauge theories of Dirac type” and summarize someof its basic features. In particular, we discuss general relativity from the point ofview of (twisted) Clifford bundles and Dirac type first order differential operators.This geometrical description of general relativity allows to relate gravity to sponta-neous symmetry breaking without the use of a Higgs potential. In fact, from thepoint of view of Dirac type gauge theories the Higgs potential is regarded as thesum of two terms with a rather different geometrical origin. One of these terms isshown to be related to gravity, whereas the other term is related to Yang-Mills gaugetheory. In particular, the Higgs potential is not needed to provide the gauge bosonswith a non-trivial mass spectrum. The latter can be expressed in terms of a (certainclass of) Dirac type first order operators on a twisted Clifford bundle. The basicreason for the occurrence of the Higgs potential in the Standard Model of ParticlePhysics is to provide the Higgs boson itself with an appropriate mass. We brieflydiscuss how the Higgs potential of the Standard Model can be derived from Diractype operators on a twisted Clifford bundle and how this allows to make a predic-tion for the mass of the yet to be find Higgs boson. The predicted value turns out tobe compatible with all the known data derived from the Standard Model of ParticlePhysics. Hence, this predicted value for the mass of the Higgs boson may be verifiedin the nearby future, for instance, by the LHC accelerator.

Eventually, we shall conclude our discussion with some remarks on possible rela-tions between Dirac type gauge theories and a natural generalization of Maxwell’sequations in Dirac form: 6∂F = 0

73

Clifford Algebras and the

Supersymmetric Quantum Mechanics

Francesco Toppan

The algebra of the one-dimensional N-Extended Supersymmetry (the superalge-bra of the Supersymmetric Quantum Mechanics) induces Clifford algebras whenformulating the eigenvalue problem. The classification of its irreducible represen-tations realized by linear derivative operators acting on a finite number of bosonicand fermionic fields is presented. The presentation of the irreducible representa-tions in terms of N-colored oriented graphs is discussed. Off-shell invariant actionsfor 1D sigma models are constructed. For N=1,2,4,8, the division algebra structureconstants enter the invariant actions as coupling constants.

74

Hyperbolic Pseudoanalytic Function Theory

and the Klein-Gordon Equation

Sébastien Tremblay

Elliptic pseudoanalytic function theory was considered independently by Bers andVekua decades ago. In this talk we develop a hyperbolic analogue of pseudoanalyticfunction theory using the algebra of hyperbolic numbers. We consider the Klein-Gordon equation with a potential:

( − ν(x, t)

)ϕ(x, t) = 0. With the aid of one

particular solution we factorize the Klein-Gordon operator in terms of two Vekua-type operators. We show that real parts of the solutions of one of these Vekua-typeoperators are solutions of the Klein-Gordon equation. Using hyperbolic pseudo-analytic function theory, we then obtain explicit construction of infinite systems ofsolutions of the Klein-Gordon equation with potential.

(This work was done in collaboration with V.V. Kravchenko and D. Rochon)

75

An Algebraic Remark on Clifford Algebras

and Differential Operators

Alexandre Trovon

The purpose of this talk is to study, from an algebraic viewpoint, the rule playedby differential operators on abstract Clifford algebras. Within this framework it ispossible to find a relationship among linear differential operators and a kind of gen-eralized Dirac operator, which points to an investigation on algebraic spin structures.

76

Noncommutative Field Theory

and Twisted Symmetries

J. David Vergara

Within the context of quantum field theory, a considerable amount of work hasbeen recently done dealing with quantum field theories in noncommutative space-times (NCQFT). One of the most relevant issues in this area is related to the sym-metries under which these noncommutative systems are invariant. The most recentcontention being that NCQFT are invariant under global “twisted symmetries”. Thiscriterion has been extended to the case of the twisting of local symmetries, such asdiffeomorphisms by Wess, et al, and this has been used to propose some noncom-mutative theories of gravity. Another possible extension of this idea is to considerthe construction of noncommutative gauges theories with an arbitrary gauge group.Regarding this latter line of research there is, however, some level of controversy asto whether it is possible to construct twisted gauge symmetries. In this lecture weaddress this issue from the point of view of canonically reparametrized noncommu-tative field theories.

77

Cohomological Approach to Diagonal

Noncommutative

Nonassociative Graded Algebras

Luis A. Wills-Toro, Juan D. Velez and Thomas Craven

We study finite abelian group graded algebras with no zero divisors on monomialin the generators. The algebra is generated by a set of algebra elements graded bya minimal set of generators of the grading group, and as many as them. There arefunctions q and r coding the noncommutativity and nonassociativity of the algebra.We study the cohomology of such q- and r-functions. We discover that the r-functioncoding nonassociativity has always trivial cohomology. Quaternions, octonions andsedenions are constructed in this manner and we study their noncommutativityand nonassociativity using cohomological tools. For deformed graded Lie algebraswhose noncommutative nonassociative transformation parameters are governed bysuch algebras, even if the cohomology of the q-function is not trivial, there is a oneto one correspondence with a plain (non-deformed) Lie algebra while maintainingtheir grading. We show then that in the case of colored or epsilon-Lie algebras, thereis a transformation to plain Lie algebras that transform self-adjoint generators intoself-adjoint generators, complementing the result of Scheunert.

78

Historical Talk

Did Copernicus Incorporate

some Arab Innovations

for Describing Planetary Motion

into his Work Without Acknowledgement?

John Snygg

In 1957, a paper written by ibn al-Shatir (circa 1304 - 1375AD) was discovered inthe Vatican archives which presented a variation of Ptolemy’s methods for describ-ing planetary motion. When this paper was shown to Copernican scholars, it wasrecognized that Copernicus used the same approaches without mention of any Arabsource. Since then, it has become conventional wisdom among most historians ofIslamic science that Copernicus did not reinvent these methods. In this talk, I willdiscuss the merits of these accusations against Copernicus.

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List of Participants

Rafal Ablamowicz

Department of MathematicsTennessee Technological UniversityCookeville, TN, [email protected]

Thomas Batard

Mathématiques, Image et ApplicationsUniversité de La RochelleLa Rochelle, [email protected]

William Baylis

Department of PhysicsUniversity of WindsorWindsor, ON, [email protected]

Eduardo Bayro-Corrochano

Dept. of Electrical Engineering and CSCINVESTAV, Unidad GuadalajaraGuadalajara, JAL, Mé[email protected]

Troy Brachey

Department of MathematicsTennessee Technological UniversityCookeville, TN, [email protected]

Fred Brackx

Department of Mathematical AnalysisGhent UniversityGhent, [email protected]

Oliver Conradt

Section for Mathematics and AstronomyGoetheanumDornach, [email protected]

Marek Czachor

Theo. Phys. and Quantum InformaticsPolitechnika GdanskaGdansky, [email protected]

Cássius De Melo

Instituto de Física TeóricaUNESPSão Paulo, SP, [email protected]

Hennie De Schepper

Department of Mathematical AnalysisGhent UniversityGhent, [email protected]

80

Nedim Degirmenci

Department of MathematicsAnadolu UniversityEskisehir, [email protected]

Suleyman Demir

Department of PhysicsAnadolu UniversityEskisehir, [email protected]

Sirkka-Liisa Eriksson

Department of MathematicsTampere University of TechnologyTampere, [email protected]

Rita Fioresi

Dipartimento di MatematicaUniversità degli Studi di BolognaBologna, [email protected]

Ghislain Franssens

Belgian Institutefor Space AeronomyBrussels, [email protected]

Klaus Gürlebeck

Institute of Mathematics and PhysicsBauhaus University WeimarWeimar, [email protected]

Jacques Helmstetter

Institut FourierUniversité Grenoble IGrenoble, [email protected]

David Hestenes

Department of PhysicsArizona State UniversityTempe, AZ, [email protected]

Eckhard S. M. Hitzer

Department of Applied PhysicsUniversity of FukuiFukui, [email protected]

Eduardo Hoefel

Departamento de MatemáticaUniversidade Federal do ParanáCuritiba, PR, [email protected]

81

Bernard Jancewicz

Institute of Theoretical PhysicsWroclaw UniversityWroclaw, [email protected]

Marcos Jardim

Departamento de MatemáticaIMECC-UNICAMPCampinas, SP, [email protected]

Lukas Krump

Mathematical InstituteCharles University in PraguePrague, Czech [email protected]

Zhanna Kuznetsova

Departamento de MatemáticaUniversidade Federal de Juíz de ForaJuíz de Fora, ES, [email protected]

Anthony Lasenby

Department of PhysicsUniversity of CambridgeCambridge, England, [email protected]

Carlile Lavor

Departmento de Matemática AplicadaIMECC-UNICAMPCampinas, SP, [email protected]

Julian Ławrynowicz

Department of Solid State PhysicsUniversity of LodzLodz, [email protected]

Rafael Leão

Departamento de MatemáticaUniversidade Federal do ParanáCuritiba, PR, [email protected]

Murat Limoncu

Department of MathematicsAnadolu UniversityEskisehir, [email protected]

Paul Loya

Department of MathematicsBinghamton University - SUNYBinghamton, NY, [email protected]

82

Douglas Lundholm

Department of MathematicsKTH - Royal Institute of TechnologyStockholm, [email protected]

Marco A. Macías-Cedeño

Tecnológico de MonterreyCampus Santa FéCiudad de México, DF, Mé[email protected]

Emilio Marmolejo-Olea

Instituto de MatematicasUnidad Cuernavaca – UNAMCuernavaca, MOR, [email protected]

Mircea Martin

Department of MathematicsBaker UniversityBaldwin City, KS, [email protected]

Nolmar Melo

Departamento de Matemática AplicadaIMECC-UNICAMPCampinas, SP, [email protected]

Artibano Micali

Dép. des Sciences MathématiquesUniversité Montpellier IIMontpellier, [email protected]

Igor Monteiro

Departamento de MatemáticaUniversidade Federal do Rio GrandeRio Grande, RS, [email protected]

Ricardo Mosna

Departamento de Matemática AplicadaIMECC-UNICAMPCampinas, SP, [email protected]

Eduardo Notte-Cuello

Departamento de MatemáticasUniversidad de La SerenaLa Serena, [email protected]

Matej Pavšic

Department of Theoretical PhysicsJosef Stefan InstituteLjubljana, [email protected]

83

Alessandro Perotti

Dipartimento di MatematicaUniversità degli Studi di TrentoPovo, Trento, [email protected]

Bruto Max Pimentel

Instituto de Física TeóricaUNESPSão Paulo, [email protected]

Dimitrios Pinotsis

Department of MathematicsUniversity of ReadingReading, England, [email protected]

Rafael Reséndiz

Departamento de MatematicaUniversidad Autónoma MetropolitanaCiudad de México, DF, Mé[email protected]

Roldão da Rocha Jr.CMCCUniversidade Federal do ABCSanto André, SP, [email protected]

Dominic Rochon

Dép. mathématiques et d’informatiqueUniversité du QuébecTrois-Rivières, QC, [email protected]

Waldyr A. Rodrigues Jr.Departamento de Matemática AplicadaIMECC-UNICAMPCampinas, SP, [email protected]

Thalanayar Santhanam

Department of PhysicsSaint Louis UniversitySt. Louis, MO, [email protected]

Bernd Schmeikal

Am Platzl 1A-4451 [email protected]

Jon Selig

MSFS DepartmentLondon South Bank UniversityLondon, [email protected]

84

Dalibor Smid

Mathematical InstituteCharles University in PraguePrague, Czech [email protected]

John Snygg

433 Prospect StreetEast Orange, NJ 07017-3101, [email protected]

[email protected]

Garret Sobczyk

Dpto. de Actuaría, Física y MatemáticasUniversidad de Las Américas PueblaPuebla, [email protected]

Fernando J. O. Souza

Departamento de MatemáticaUniversidade Federal de PernambucoRecife, PE, [email protected]

Wolfgang Sprößig

Institute of Applied AnalysisTU Bergakademie FreibergFreiberg, [email protected]

George Stacey Staples

Dept. of Mathematics and StatisticsSouthern Illinois Univ. EdwardsvilleEdwardsville, IL, [email protected]

Jorge Stolfi

Instituto de ComputaçãoUniversidade Estadual de CampinasCampinas, SP, [email protected]

Douglas Sweetser

Quaternions.com39 Drummer RoadActon, MA 01720, [email protected]

Jürgen Tolksdorf

Max Planck Institutfor Mathematics in the SciencesLeipzig, [email protected]

Francesco Toppan

Física TeóricaCBPFRio de Janeiro, RJ, [email protected]

85

Sébastien Tremblay

Dép. mathématiques et d’informatiqueUniversité du QuébecTrois-Rivières, QC, [email protected]

Alexandre Trovon

Departamento de MatemáticaUniversidade Federal do ParanáCuritiba, PR, [email protected]

Jayme Vaz Jr.Departamento de Matemática AplicadaIMECC-UNICAMPCampinas, SP, [email protected]

José David Vergara

Instituto de Ciencias NuclearesUNAMCiudad de México, DF, Mé[email protected]

Georges Weill

Department of MathematicsCooper Union School of EngineeringNew York, NY, [email protected]

Luis A. Wills-Toro

Dept. of Mathematics and StatisticsAmerican University of SharjahSharjah, United Arab [email protected]

Photography on page 3 by Antoninho Perri.Backcover photography captured by IKONOS-2 Satellite.

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May 4, 1845Exeter, England

March 3, 1879Madeira, Portugal