Real projective quadrics, conformal structures and conformal spin structures

Pierre Anglès, Institut de Mathématiques de Toulouse, Université Paul Sabatier, France

This lecture is dedicated to the memories of Pertti Lounesto, Jaime Keller and Artibano Micali.

Abstract

This lecture wants to emphasize the links between some fundamental real projective quadrics, conformal structures and conformal spin structures.

It is divided into three sections.

• First, basics of general linear groups and of Clifford algebras will be recalled.

• The following section deals with the study of the real projective quadric $\widetilde{Q_{r,s}}$ associated with a standard regular pseudo-Euclidean space $E_{r,s},r+s=m=\dim _{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion }E_{r,s}$ , with the corresponding conformal pseudo-riemannian structures and the real conformal spin structures.

Let $S^{r}$ be the unit sphere of $E_{r+1,0}$. One recalls how $\widetilde{% Q_{r,s}}$ is naturally provided with a pseudo-riemannian conformal structure of type $(r-1,s-1).$

Consider $E_{r+1,s+1}.$The corresponding $m-$dimensional projective quadric $% M=\widetilde{Q_{r+1,s+1}}$ is called the compactified of $E_{r,s}.$One shows how $M$ is identical to the homogeneous space $PO(E_{r+1,s+1})/\mathrm{Sim}% (E_{p,q}),$ quotient group of $PO(E_{r+1,s+1})=O(r+1,s+1)/% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2\text{ }}$by the group $\mathrm{Sim}E_{r,s}$ of similarities of $E_{r,s}.$ One shows how $M$ is identical to the quotient of the manifold $S^{r}\times S^{s\text{ }}$by the equivalence relation $(a,b)\sim (-a,-b).$One can study the compactified $M=\widetilde{Q_{r+1,s+1}}$ of $E_{r,s}$ by three different ways.

The first one is a geometrical one that shows how the compactified $M$ is got, talking abusively, by adjoining to $E_{r,s}$ a projective cone at infinity. The second one is algebraic and gives a method for building up covering groups for the conformal group of $E_{r,s},$using a theorem found out by J.Haantjes that extends Liouville theorem to pseudo-Euclidean spaces.The third one needs the study of some Lie algebras.

The topology of real projective quadrics is studied.The embedding of projective quadrics is given. In particular, one finds that the conformal compactified $\widetilde{Q_{2,4}}$ of the Minkowski space $E_{1,3}$ can be identified with the group $U(2).$The study of conformal spin structures on riemannian or pseudo-riemannian manifolds is given. Groups called conformal spinoriality groups play an essential part.The links between classical spin structures and conformal spin structures is given. Indications concerning the study of the corresponding conformal connections are given.

• The third section deals with the study of the properties of some real projective quadrics associated with a standard pseudo-Hermitian space $% \mathbf{H}_{p,q}\mathbf{,}$with $p+q=n=\dim _{% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion }\mathbf{H}_{p,q}$, namely: $\widetilde{Q}_{p,q}$, $\widetilde{Q}_{2p+1,1}$, $\widetilde{Q}_{1,2q+1}$, $\widetilde{H_{p,q}}$. $\widetilde{\text{ }Q}_{p,q}$ is the $(2n-2)$ real projective quadric diffeomorphic to $(S^{2p-1}\times S^{2q-1})/% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2}$, inside the real projective space $P(E_{1})$, where $E_{1}$ is the real $% (2n)-$dimensional space subordinate to $\mathbf{H}_{p,q}.$ The properties of $% \widetilde{Q}_{p,q}$ are investigated.

$\widetilde{H_{p,q}}$ is the real $(2n-3)-$dimensional compact manifold-projective quadric- associated with $\mathbf{H}_{p,q}$, inside the complex projective space $P(\mathbf{H}_{p,q})$. $\widetilde{\text{ }H_{p,q}}$ is diffeomorphic to $(S^{2p-1}\times S^{2q-1})/S^{1}.$The properties of $% \widetilde{H_{p,q}}$ are studied. $\widetilde{Q}_{2p+1,1}$ is a $(2p)$-dimensional standard real projective quadric and $\widetilde{Q}_{1,2q+1}$ is another standard real $(2q)-$dimensional projective quadric. $\widetilde{Q}% _{2p+1,1}\cup \widetilde{Q}_{1,2q+1},$ union of two compact quadrics plays a part in the understanding of the "special pseudo-unitary conformal compactified" of $\mathbf{H}_{p,q}.$ It is shown how a distribution $% y\rightarrow D_{y},$where $y\in H\backslash \{0\},H$ being the isotropic cone of $\mathbf{H}_{p,q},$allows to consider $\widetilde{H_{p+1,q+1}}$ as a "special pseudo-unitary conformal compactified" of $\mathbf{H}_{p,q}\times %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion .$

The study of pseudo-unitary spin structures and conformal pseudo-unitary spin structures over an almost complex $(2n)-$dimensional manifold $V$ is sketched after a recall of the building up of a Clifford algebra $Cl^{p,q}$ associated with $\mathbf{H}_{p,q}.$

The Odyssey of Geometric Algebras

Pierre Anglès, Institut de Mathématiques de Toulouse, Université Paul Sabatier, France

This lecture is dedicated to the memories of Pertti Lounesto, Jaime Keller and Artibano Micali.

“In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.” Hermann Weyl, “Invariants”, 1939, “Gesammelte Werke”, Band III, page 681.

Abstract

During the 20th century, two symmetrical processes of unification appear in Mathematics. The first one is, in effect, the outcome of the long way starting from the Greeks, which wanted to understand and to solve the opposition first between Geometry and Arithmetic and then between Geometry and Algebra. The second process of unification focuses on Langlands correspondences.This lecture wants to study thoroughly the first process.We will show how, in the course of time, algebra became geometric algebra and how appears progressively a coherent mode fundamental for the development of the physics of the 20th century.

Historically, the notion of Clifford algebra appeared in many different ways. Its destiny is closely joined to the development of generalized complex numbers and the success of quadratic forms. A first try to solve the opposition between algebra and geometry was realized by the French mathematician and philosopher René Descartes who developed in 1637 the first Mathesis universalis, the first model of a unique science, by building up analytic geometry. Objectively, two criticisms can be made: first, the analysis of R. Descartes was relative to scalars; secondly, such an analysis established only correspondences between geometry and the only theory of real numbers. The story of complex imaginary numbers starts in the 16th century when Italian mathematicians Girolamo Cardano (1501-1576), Raphaele Bombelli ( born in 1530, whose algebra was published in 1572), and Niccolo Fontana, called "Tartaglia"-which means stammerer- realizing that a negative real number cannot have a square root, began to use a symbol for its representation. Thus came into the world the symbol $i$ such as $i^{2}=-1,$ a very real mathematical oxymoron. We recall that starting from the classical field $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$ or real numbers we can define the three following generalized numbers of order two: $a+ib;a,b\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,$with $i^{2}=-1,$ classical complex numbers or elliptic numbers ; $% a+\varepsilon b,a,b\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,$with $\varepsilon ^{2}=0,$dual numbers or parabolic numbers ;$a+eb,a,b\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,$with $e^{2}=1,$double numbers or hyperbolic numbers.

The introduction of generalized numbers of order more than $2$ appears to be linked to the attempts made in the 18th century by Caspar Wessel, J.R.Argand, J.F. Français, F.G. Servois to extend the geometrical theory of imaginary numbers of the plane to the usual space.W.R. Hamilton was the first to introduce in 1842 a system of numbers of order $2^{2}=4,$ with a noncommutative multiplicative law: the sfield $\mathbf{H.}$The study of the group of rotations in the classical $3-$dimensional space led W.R. Hamilton to his discovery. The discovery of Hamilton gave a simple "spinorial representation" of rotations.If $q=ai+bj+ck$ is a "pure" quaternion and $u$ is a unit quaternion, then $q\rightarrow uqu^{-1}$is a rotation and every rotation can be so obtained. Dual and double numbers were studied by two mathematicians: Eug\`{e}ne Study and W.K. Clifford.The German mathematician Hermann Günther Grassmann developed in 1844 the idea of an algebra where the corresponding symbols stood for quantities or geometrical objects, (points, lines, planes), subjected to some rules, allowing the construction of a structure both algebraic and geometric found upon an axiomatic conception of a $n-$dimensional space. He defined notions such as: linearly independence of vectors, dimension of a space, subspace of a given space. Many problems of linear algebra and of Euclidean geometry got solved by use of different "products": scalar product, outer product, exterior product.

In his paper from 1878, Clifford defined an algebra generated by $1$ and elements $i_{1},..,i_{n},$with the conditions $i_{a}^{2}=-1$ and $% i_{a}i_{b}=-i_{b}i_{a}$ when $a\neq b.$

As in the case of Grassmann algebra, the following basis for this new algebra appeared: $\{1,i_{a},i_{a}i_{b},i_{a}i_{b}i_{c},....\mid a < b < c...\}.$However, in this paper Clifford was only considering the cases when $% i_{a}^{2}=-1.$If $n=1,$one gets the complex numbers and for $n=2,$the quaternions. The part played by the skew field $\mathbf{H}$ of quaternions for the description of rotations of $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}$will be generalized to the part played by Clifford algebras $Cl(q),$for the description of elements of $O(q),$where $(E,q)$ is a standard $n$-dimensional Euclidean or pseudo-Euclidean space.The Clifford idea of adding a formal square root of a quadratic quantity works marvellously in any dimension.Very soon it was, the Clifford construction is clearly explained in terms of the spin group, which is the group counter part of the Clifford algebra. Physicists also quickly recognized the importance of the spin group and of its spin representation both in Euclidean and Minkowski signature. In 1929, the physicist P.A.M. Dirac wanted to solve the problem of the relativistic equation of the electron and sought to linearize the Klein-Gordon operator, which is the restricted relativistic form of the Schrödinger equation and found complex matrices of order $4$, which constitutes a Clifford algebra.

The works of Cayley in 1854 about algebras of matrices, the theory of determinants by Sylvester in 1878 led naturally to the spinorial algebras of Pauli-Dirac. Such algebras, together with the new theory of Clifford algebras improved by the theory of differential forms and the theory of spinors found by Elie Cartan, the vectorial calculus of Gibbs, beginning in 1881, the tensor calculus created by Ricci in 1890 were the foundation stones of geometric algebras.

Thus, by an algebraic way , there appeared the realization of the dream of unification of two fundamental domains of Mathematics, a dream expected for almost two thousand and five hundred years.