Jordan- and Lie geometries

Bertram, Wolfgang

Archivum mathematicum, Tome 49 (2013) no. 5, pp. 275-293

Abstract (VO)

In these lecture notes we report on research aiming at understanding the relation beween algebras and geometries, by focusing on the classes of Jordan algebraic and of associative structures and comparing them with Lie structures. The geometric object sought for, called a generalized projective, resp. an associative geometry, can be seen as a combination of the structure of a symmetric space, resp. of a Lie group, with the one of a projective geometry. The text is designed for readers having basic knowledge of Lie theory – we give complete definitions and explain the results by presenting examples, such as Grassmannian geometries.

Détail

DOI : 10.5817/AM2013-5-275

Classification : 16-02, 16W10, 17C37, 20N10, 22A30, 51B25, 51P05, 81P05
Keywords: Jordan algebra (triple system, pair); associative algebra (triple systems, pair); Lie algebra (triple system); graded Lie algebra; symmetric space; torsor (heap, groud, principal homogeneous space); homotopy and isotopy; Grassmannian; generalized projective geometry

Bibliographie

[1] Berger, M.: Les espaces symétriques non–compacts. Ann. Sci. ENS (1957). | Zbl

[2] Bertram, W.: Jordan geometries by inversions. Preprint, 2013, | arXiv

[3] Bertram, W.: The projective geometry of a group. arXiv: math.GR/1201.6201.

[4] Bertram, W.: The Geometry of Jordan and Lie Structures. Lecture Notes in Math., vol. 1754, Springer, Berlin, 2000. | DOI | Zbl

[5] Bertram, W.: Generalized projective geometries: General theory and equivalence with Jordan structures. Advances in Geometry 3 (2002), 329–369. | MR

[6] Bertram, W.: The geometry of null systems, Jordan algebras and von Staudt’s Theorem. Ann. Inst. Fourier 53 fasc. 1 (2003), 193–225. | DOI | MR | Zbl

[7] Bertram, W.: Differential geometry, Lie groups and symmetric spaces over general base fields and rings. Mem. Amer. Math. Soc. 192 (900) (2008), x+202, arXiv: math.DG/0502168. | MR | Zbl

[8] Bertram, W.: Homotopes and conformal deformations of symmetric spaces. J. Lie Theory 18 (2008), 301–333, math.RA/0606449. | MR | Zbl

[9] Bertram, W.: Is there a Jordan geometry underlying quantum physics?. Int. J. Theor. Phys. 47 (2) (2008), 2754—2782, arXiv: math-ph/0801.3069. | DOI | MR | Zbl

[10] Bertram, W.: On the Hermitian projective line as a home for the geometry of Quantum Theory. AIP Conference Proceedings 1079, p. 14–25 (XXVII Workshop on Geometrical Methods in Physics, Bialowieza 2008), American Institute of Physics, New York, 2008. | MR | Zbl

[11] Bertram, W., Bieliavsky, P.: Homotopes of symmetric spaces. I : Construction by algebras with two involutions. arXiv: math.DG/1011.2923.

[12] Bertram, W., Bieliavsky, P.: Homotopes of symmetric spaces. II : Structure Variety and Classification. arXiv: math.DG/1011.3161.

[13] Bertram, W., Glöckner, H., Neeb, K.–H.: Differential calculus over general base fields and rings. Exposition. Math. 22 (2004), 213–282, arXiv: math.GM/030330. | DOI | MR | Zbl

[14] Bertram, W., Kinyon, M.: Associative geometries. I: Torsors, linear relations and grassmannians. J. Lie Theory 20 (2) (2010), 215–252, arXiv: math.RA/0903.5441. | MR | Zbl

[15] Bertram, W., Kinyon, M.: Associative geometries. II: Involutions, the classical torsors, and their homotopes. J. Lie Theory 20 (2) (2010), 253–282, arXiv: math.RA/0909.4438. | MR | Zbl

[16] Bertram, W., Neeb, K.–H.: Projective completions of Jordan pairs. I: The generalized projective geometry of a Lie algebra. J. Algebra 277 (2) (2004), 193–225, arXiv: math.RA/0306272. | MR | Zbl

[17] Bertram, W., Neeb, K.–H.: Projective completions of Jordan pairs. Part II: Manifold structures and symmetric spaces (avec K.-H. Neeb). vec K.-H. Neeb), Geom. Dedicata 112 (1) (2005), 73–113, arXiv: math.GR/0401236. | DOI | MR

[18] Chenal, J.: Generalized flag geometries and manifolds associated to short Z-graded Lie algebras in arbitrary dimension. C. R. Math. Acad. Sci. Paris 347 (2009), 21–25, arXiv: 1007.4076v1 [math.RA]. | DOI | MR

[19] Chu, Ch.–H.: Jordan Structures in Geometry and Analysis. Cambridge University Press, 2012. | MR | Zbl

[20] Connes, A.: Non–commutative Geometry. Academic Press, 1994.

[21] Emch, G.: Mathematical and Conceptual Foundations of 20th Century Physics. North Holland, 1985.

[22] Faraut, J., Koranyi, A.: Analysis on Symmetric Cones. Clarendon Press, Oxford, 1994. | Zbl

[23] Grgin, E., Petersen, A.: Algebraic implications of composability of physical systems. Comm. Math. Phys. 50 (1976), 177–188. | DOI | Zbl

[24] Jordan archive. Jordan preprint server) http://molle.fernuni-hagen.de/~loos/jordan/index.html

[25] Kaneyuki, S.: On classification of parahermitian symmetric spaces. Tokyo J. Math. 8 (1985), 473–482. | DOI | Zbl

[26] Koecher, M.: The Minnesota Notes on Jordan Algebras and Their Applications (reprint). eprint), Lecture Notes in Mat., vol. 1710, Springer, Berlin, 1999.

[27] Koufany, K.: Réalisation des espaces symétriques de type Cayley. C. R. Math. Acad. Sci. Paris 318 (1994), 425–428. | Zbl

[28] Loos, O.: Symmetric Spaces I. Benjamin, New York, 1969. | Zbl

[29] Loos, O.: Jordan triple systems, R-spaces and bounded symmetric domains. Bull. Amer. Math. Soc. 77 (1971), 558–561. | DOI | Zbl

[30] Loos, O.: Jordan Pairs. Lecture Notes in Math., vol. 460, Springer, Berlin, 1975. | Zbl

[31] Loos, O.: Bounded Symmetric Domains and Jordan Pairs. University of California, Irvine, 1977, [See http://molle.fernuni-hagen.de/~loos/jordan/index.html]

[32] Loos, O.: Charakterisierung symmetrischer $R$–Räume durch ihre Einheitsgitter. Math. Z. 189 (1985), 211–226. | DOI | Zbl

[33] McCrimmon, K.: A Taste of Jordan Algebras. Springer-Verlag, New York, 2004. | MR | Zbl

[34] Nagano, Tadashi: Transformation groups on compact symmetric spaces. Trans. Amer. Math. Soc. 118 (1965), 428–453. | DOI

[35] Springer, T. A.: Jordan Algebras and Algebraic Groups. Classics in Mathematics, Springer-Verlag, 1998, Reprint of the 1973 edition. | Zbl

[36] Takeuchi, Masaru: Cell decompositions and Morse equalities on certain symmetric spaces. J. Fac. Sci. Univ. Tokyo Sect. I 12 (1965), 81–192.

[37] Upmeier, H.: Symmetric Banach Manifolds and Jordan $C^*$-algebras. North-Holland Math. Stud., North-Holland Publishing Co., Amsterdam, 1985.

[38] Upmeier, H.: Jordan algebras in analysis, operator theory, and quantum mechanics. CBMS Regional Conference Series in Mathematics, 67. Published for the Conference Board of the Mathematical Sciences, Washington, DC, American Mathematical Society, Providence, RI, 1987.