Abstract
The paper gives an introduction to Steinberg groups for Jordan pairs, a theory developed in the book recent book by Ottmar Loos and the author.
Dedicated to Vyjayanthi Chari on the occasion of her 60th birthday
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References
A. Bak, K-theory of forms, Ann. of Math. Studies, vol. 98, Princeton University Press, 1981.
N. Bourbaki, Groupes et algèbres de Lie, chapitres 4–6 , Masson, Paris, 1981.
C. Chevalley and R. D. Schafer, The exceptional simple Lie algebras F4 and E6, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 137–141.
D. L. Costa and G. E. Keller, The E(2, A) sections of SL(2, A), Ann. of Math. (2) 134 (1991), no. 1, 159–188.
A. J. Hahn and O. T. OMeara, The classical groups and K-theory , Grundlehren, vol. 291, Springer-Verlag, 1989.
R. Hazrat, N. Vavilov, and Z. Zhang, The commutators of classical groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 443 (2016), no. Voprosy Teorii Predstavleniı̆ Algebr i Grupp. 29, 151–221.
J. Humphreys, Introduction to Lie algebras and Representation Theory , Graduate Texts in Mathematics vol. 9, Springer-Verlag, New York, 1972.
N. Jacobson, Some groups of transformations defined by Jordan algebras. I , J. Reine Angew. Math. 201 (1959), 178–195.
——, Some groups of transformations defined by Jordan algebras. II. Groups of type F4 , J. Reine Angew. Math. 204 (1960), 74–98.
——, Some groups of transformations defined by Jordan algebras. III, J. Reine Angew. Math. 207 (1961), 61–85.
——, Lectures on quadratic Jordan algebras , Tata Institute of Fundamental Research, 1969.
——, Structure theory of Jordan algebras , Lecture Notes in Math., vol. 5, The University of Arkansas, 1981.
I. L. Kantor, Classification of irreducible transitive differential groups , Dokl. Akad. Nauk SSSR 158 (1964), 1271–1274.
——, Non-linear groups of transformations defined by general norms of Jordan algebras, Dokl. Akad. Nauk SSSR 172 (1967), 779–782.
——, Certain generalizations of Jordan algebras, Trudy Sem. Vektor. Tenzor. Anal. 16 (1972), 407–499.
M. Kervaire, Multiplicateurs de Schur et K-théorie , Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 212–225.
M. Koecher, Imbedding of Jordan algebras into Lie algebras. I, Amer. J. Math. 89 (1967), 787–816.
——, Über eine Gruppe von rationalen Abbildungen , Invent. Math. 3 (1967), 136–171.
——, Imbedding of Jordan algebras into Lie algebras. II , Amer. J. Math. 90 (1968), 476–510.
——, Gruppen und Lie-Algebren von rationalen Funktionen , Math. Z. 109 (1969), 349–392.
A. Lavrenov, Another presentation for symplectic Steinberg groups, J. Pure Appl. Algebra 219 (2015), no. 9, 3755–3780.
A. Lavrenov and S. Sinchuk, On centrality of even orthogonal K2, J. Pure Appl. Algebra 221 (2017), no. 5, 1134–1145.
O. Loos, Jordan pairs , Springer-Verlag, Berline, 1975, Lecture Notes in Mathematics, vol 460.
——, Homogeneous algebraic varieties defined by Jordan pairs, Mh. Math. 86 (1978), 107–129.
——, On algebraic groups defined by Jordan pairs, Nagoya Math. J. 74 (1979), 23–66.
——, Elementary groups and stability for Jordan pairs, K-Theory 9 (1995), 77–116.
——, Steinberg groups and simplicity of elementary groups defined by Jordan pairs, J. Algebra 186 (1996), no. 1, 207–234.
——, Rank one groups and division pairs, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 3, 489–521.
O. Loos and E. Neher, Locally finite root systems , Mem. Amer. Math. Soc. 171 (2004), no. 811.
——, Steinberg groups for Jordan pairs , Progress in Mathematics, vol. 332, Birkhäuser Springer, 2019
O. Loos, H. P. Petersson and M. L. Racine, Inner derivations of alternative algebras over commutative rings , Algebra & Number Theory, 2, (2008), 927–968.
B. Magurn, An algebraic introduction to K-Theory , Encyclopedia of Mathematics and Its Applications 87, Cambridge University Press 2002.
J. Milnor, Introduction to algebraic K-theory , Ann. of Math. Studies, vol. 72, Princeton University Press, 1971.
E. Neher, Jordan triple systems by the grid approach, Lecture Notes in Math., vol. 1280, Springer-Verlag, 1987.
——, Systèmes de racines 3-gradués , C. R. Acad. Sci. Paris Sér. I 310 (1990), 687–690.
——, Lie algebras graded by 3-graded root systems and Jordan pairs covered by a grid, Amer. J. Math 118 (1996), 439–491.
——, Polynomial identities and nonidentities of split Jordan pairs , J. Algebra 211 (1999, 206–224.
J. Rosenberg, Algebraic K-theory and its applications , Graduate Texts in Mathematics, vol. 147, Springer-Verlag, New York, 1994.
S. Sinchuk, On centrality of K2 for Chevalley groups of type Eℓ, J. Pure Appl. Algebra 220 (2016), 857–875.
T. A. Springer, Jordan algebras and algebraic groups , Springer-Verlag, New York, 1973, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 75.
T. A. Springer and F. D. Veldkamp, Octonions, Jordan algebras and exceptional groups . Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000
R. Steinberg, Générateurs, relations, et revêtements de groupes algébriques , Colloq. Théorie des Groupes Algébriques (Bruxelles), 1962, 113–127.
——, Lectures on Chevalley groups , Yale University Lecture Notes, New Haven, Conn., 1967.
——, Generators, relations and coverings of algebraic groups II , J. Algebra 71 (1981), 527–543.
J. R. Strooker, The fundamental group of the general linear group, J. Algebra 48 (1977), 477–508.
J. Tits, Une classe d’algèbres de Lie en relation avec les algèbres de Jordan , Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indag. Math. 24 (1962), 530–535.
J. Tits, Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles. I. Construction, Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28 (1966), 223–237.
W. van der Kallen, Another presentation for Steinberg groups, Nederl. Akad. Wetensch. Proc. Ser. A 80 = Indag. Math. 39 (1977), 304–312.
C. Weibel, The K-book: an introduction to algebraic K-theory , Graduate Studies in Mathematics 145, AMS 2013.
Acknowledgements
The author thanks Ottmar Loos for many helpful comments on an earlier version of this paper.
The author acknowledges with thanks partial support by the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery grant.
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Neher, E. (2021). Steinberg Groups for Jordan Pairs: An Introduction with Open Problems. In: Greenstein, J., Hernandez, D., Misra, K.C., Senesi, P. (eds) Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification. Progress in Mathematics, vol 337. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-63849-8_4
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