The Fifth International Workshop on Analele Universita˘t¸ii din Timis¸oara Differential Geometry and Its Applications Vol XXXIX
10 pages
English

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris

The Fifth International Workshop on Analele Universita˘t¸ii din Timis¸oara Differential Geometry and Its Applications Vol XXXIX

-

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris
Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus
10 pages
English
Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

Description

The Fifth International Workshop on Analele Universita˘t¸ii din Timis¸oara Differential Geometry and Its Applications Vol. XXXIX, 2001 September 18–22, 2001, Timisoara Seria Matematica˘-Informatica˘ Romania Fascicula˘ speciala˘-Matematica˘ Generalized projective geometries by Wolfgang Bertram Abstract This is a survey paper. After reviewing some features of “ordi- nary” projective geometry over a commutative base field, generalized projective (resp. polar) geometries (over a commutative base field or ring K in which 2 is invertible) as well as the symmetric space (over K) associated to a generalized polar geometry are defined. Examples of such geometries are given. The equivalence of connected general- ized projective (resp. polar) geometries over K (with base point) with Jordan pairs(resp. triple systems) over K is described. The impor- tance of the Lie-Jordan functors is explained. Some open problems are pointed out. 1 Introduction: Projective geometry revisited Before explaining the concept of a generalized projective geometry, I would like to recall quickly some features of “ordinary” projective geome- try (over a commutative base field K) which I consider to be fundamental and which I would not like to miss in later generalizations: (1) Duality. As a matter of principle, a projective space X = KPn = P(W ) (W ?= Kn+1) should always be considered together with its dual space X ? = P(W ?) AMS Subject Classification: 51A05, 51A50, 17C37, 17C50, 53C35

  • polar geometries

  • commutative base

  • over

  • projective geometry

  • lie-jordan functors

  • can introduce

  • jordan pairs


Sujets

Informations

Publié par
Nombre de lectures 14
Langue English

Extrait

The Fifth International Workshop on Analele Universit˘at¸ii din Timi¸soara
Differential Geometry and Its Applications Vol. XXXIX, 2001
September 18–22, 2001, Timisoara Seria Matematic˘a-Informatic˘a
Romania Fascicul˘a special˘a-Matematic˘a
Generalized projective geometries
by
Wolfgang Bertram
Abstract
This is a survey paper. After reviewing some features of “ordi-
nary” projective geometry over a commutative base field, generalized
projective (resp. polar) geometries (over a commutative base field or
ringK in which 2 is invertible) as well as the symmetric space (over
K) associated to a generalized polar geometry are defined. Examples
of such geometries are given. The equivalence of connected general-
ized projective (resp. polar) geometries overK (with base point) with
Jordan pairs(resp. triple systems) over K is described. The impor-
tance of the Lie-Jordan functors is explained. Some open problems
are pointed out.
1 Introduction: Projective geometry
revisited
Before explaining the concept of a generalized projective geometry, I
would like to recall quickly some features of “ordinary” projective geome-
try (over a commutative base field K) which I consider to be fundamental
and which I would not like to miss in later generalizations:
n(1) Duality. Asamatterofprinciple,aprojectivespaceX =KP =P(W)
n+1»(W =K ) should always be considered together with its dual space
0 ⁄X =P(W )
AMS Subject Classification: 51A05, 51A50, 17C37, 17C50, 53C35
Keywords and phrases: projective geometry, generalized projective geometry, gen-
eralizedpolargeometry,symmetricspace,Jordanpair,3-gradedLiealgebra,Jordan
triple system, Jordan-Lie functor
57which can be seen as the space of hyperplanes in X (the class [‚] of
a non-zero linear form can be identified with the hyperplane [ker‚]).
0Duality defines incidence: a point [x] 2 X and an element [‚] 2 X
are incident if ‚(x) = 0. Let us say that, in general, a pair geometry
0 0is given by two sets X;X and a subset M ‰X£X , called the set of
0non-incident or remote pairs, such that, for all x2X, a2X , the sets
0 0V :=fz2Xj(z;a)2Mg; V :=fb2Xj(x;b)2Mga x
are non-empty. In the case of projective geometry, the V , respectivelya
0the V are the complements of hyperplanes, and they clearly are non-x
empty.
(2) The affine-projective relationship. It is a classical exercise in linear
algebra that the set V , i.e., the complement of a given hyperplanea
na in KP , has a canonical structure of an affine space over K. Let
us say that an affine pair geometry is a pair geometry (as defined in
0(1)) such that, for any x 2 X and a 2 X , the sets V , respectivelya
0V , have the structure of an affine space over K. In other words, forx
any (x;a) 2 M, V is a vector space over K with zero vector x, anda
for x varying in V , these vector space structures are related amonga
0each other in the usual way, and similarly for V with origin a. Thusx
(x;a) 2 M defines a vectorialization of X, and M can be seen as the
0space of vectorializations of X (and of X ).
(3) The “fundamental identities”. Let us write, for r2K and x;y2V ,a
„ (x;a;y):=r (y):=r¢y;r x;a
where the product r¢ y is the usual multiplication by scalars in the
vector space V with origin x. The map „ thus defined can be seen asa r
a ternary product map
0„ :X£X £X ?D!X; (x;a;y)7!„ (x;a;y)r r
defined on the set D given by the conditions (x;a);(y;a) 2 M (for
nX =KP , this is a Zariski-dense open subset). In the same way one
0 0 0defines a map „ : D ! X . One may ask weather the “productr
0maps” („ ;„ ) satisfy algebraic identities such as, e.g., associativityr r
or commutativity. It is fairly obvious that „ is non-associative andr
1non-commutative (however, for r = , „ is weakly commutative inr2
the sense that it is symmetric in x and y), but there are indeed other
identities: first of all, there is a set of “easy” identities which express
58just the fact that … := „ (¢;a;¢) describes the affine structure of V –r r a
nin fact, identifying V with a standard vector space V =K , the mapa
… is nothing but the binary mapr
… (x;y)=(1¡r)x+ry:r
The reader may, as an elementary exercise in linear algebra, try to find
some algebraic identities for … which in turn are sufficient to recoverr
the structure of an affine space on V – a solution can be found in
[Be01a] where a (of course non-unique) set (Af1)-(Af4) of such identi-
ties is given. Moreover, as outlined in loc. cit., this approach to affine
geometry has some advantages compared with the usual approach – in
many regards it is easier and more conceptual. Coming back to pro-
0jective geometry, the maps („ ;„ ) satisfy two other identities whichr r
call the “fundamental identities of projective geometry”, denoted by
(PG1), (PG2). Roughly, (PG1) says that, if r is invertible in K, the
map r : V ! V extends to a bijection of X which is an automor-x;a a a
0phism of all product maps („ ;„ ), s 2 K. Indeed, r is a linears x;as
bijection of V and is therefore induced by an element of the generala
projective group G = PGl(n+1;K); but clearly all elements of G act
0as automorphisms of the product maps („ ;„ ). The identity (PG2)s s
is similar in nature. Using the formalism explained in [Be01a,b], the
fundamental identities can be written in the short form
(r) t (r) (r) t (r) (r) t (r)(L ) =L ; (R ) =R ; (M ) =Mx;a a;x a;x x;a x;y y;x
wheretstandsfor“transposed”andLandM aredefinedas“operators
of left, right and middle multiplication by
(r) (r) (r)„ (x;a;y)=L (y)=R (x)=M (a):r x;a a;y x;y
(4) Scalar extension. The real projective space can be embedded into the
n ncomplex projective space: RP ‰CP – see [B87, Ch. 7] for a concep-
tual, but rather complicated construction of this inclusion. Similarly,
n nwe have inclusions likeQP ‰RP . More generally, ifK‰K is a base
field extension, then we have an “extension”
0 0(X;X )‰(X ;X )
0which is compatible with the ternary products („ ;„ ) living on theser r
spaces. Taking some care in the definitions, this carries over to the
59
KKcase of projective spaces defined over commutative base rings. Here an
important special case is given by the extension
2K=K'†K; † =0;
called the dual numbers overK and constructed and the same way as
the complex numbers from the real numbers, but replacing the con-
2 2dition i = ¡1 by the condition † = 0. In this case, the projective
0 0space (X ;X ) can be interpreted as the tangent bundle (TX;TX )
0of (X;X ). In this way we can introduce differential geometric terms
in classical projective geometry, even if the base field in question is
different from the field of real or complex numbers.
(5) Polar geometries. In geometry, one is interested in metric or pseudo-
metric structures or in their analogues. However, a projective space
P(W)does,a`priori,notcarrysuchastructure;itdependsonadditional
choices. More precisely, what one needs is a way to identify X with its
0 0 0 0dualspaceX ,usuallycalledacorrelation: if(p:X !X ;p :X !X)
0is a pair of bijections of an affine pair geometry, we say that (p;p) is
- an anti-automorphism if it is compatible with all product maps
0„ ;„ ,r r
0- a correlation if it is an anti-automorphism of order 2 (i.e. p =
¡1p ),
- a null-system if it is a correlation having only isotropic points (a
point x2X is called isotropic if (x;p(x)) is incident),
- a polarity if it is a correlation admitting some non-isotropic point.
2 Generalized projective geometries and
symmetric spaces
Now I will take the properties (1)–(5) just explained as starting point of
an axiomatic definition (see [Be01b] for the exact formulation): a generalized
projective geometry (over a commutative base field or ring K in which 2
0is invertible) is an affine pair geometry (X;X ) such that the fundamental
identities (PG1) and (PG2) hold in all scalar extensions ofK. A generalized
0polar geometry is a generalized projective geometry (X;X ) together with a
0polarity (p;p).
Apart from the case of ordinary projective geometry overK, explained in
Section1,therearemanyotherexamplesofgeneralizedprojectivegeometries
(cf. [Be01a]):
60
KK0 p+q p+q(1) Grassmanniangeometries(X;X )=(Gras (K );Gras (K )),whichp q
can be defined more generally in infinite dimensions and over rings,
0(2) Lagrangian geometries; here X = X is the space of Lagrangian sub-
spaces of some symplectic or (neutral) symmetric or Hermitian form;
correspondingly, there are two main types of such geometries, namely
symplectic and orthogonal Lagrangian geometries,
0(3) conformal geometries; here X = X is a projective quadric; the struc-
ture of generalized projective geometry is defined via a generalized
stereographic projection R

  • Univers Univers
  • Ebooks Ebooks
  • Livres audio Livres audio
  • Presse Presse
  • Podcasts Podcasts
  • BD BD
  • Documents Documents