×

Steinberg groups for Jordan pairs: an introduction with open problems. (English) Zbl 1523.20090

Greenstein, Jacob (ed.) et al., Interactions of quantum affine algebras with cluster algebras, current algebras and categorification. In honor of Vyjayanthi Chari on the occasion of her 60th birthday. Based on the summer school and the conference, Washington, DC, USA, June 2018. Cham: Birkhäuser. Prog. Math. 337, 77-114 (2021).
Summary: 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 [New York, NY: Birkhäuser (2019; Zbl 1521.20001)].
For the entire collection see [Zbl 1481.17001].

MSC:

20H25 Other matrix groups over rings
17C27 Idempotents, Peirce decompositions
17C30 Associated groups, automorphisms of Jordan algebras
19Cxx Steinberg groups and \(K_2\)
20E42 Groups with a \(BN\)-pair; buildings
17B22 Root systems

Citations:

Zbl 1521.20001

References:

[1] A. Bak, K-theory of forms, Ann. of Math. Studies, vol. 98, Princeton University Press, 1981. · Zbl 0465.10013
[2] Bourbaki, N., Groupes et algèbres de Lie, chapitres 4-6 (1981), Paris: Masson, Paris · Zbl 0483.22001
[3] Chevalley, C.; Schafer, R. D., The exceptional simple Lie algebras F_4 and E_6, Proc. Nat. Acad. Sci. U.S.A., 36, 137-141 (1950) · Zbl 0037.02003 · doi:10.1073/pnas.36.2.137
[4] D. L. Costa and G. E. Keller, TheE(2, A) sections of SL(2, A), Ann. of Math. (2) 134 (1991), no. 1, 159-188. · Zbl 0743.20048
[5] A. J. Hahn and O. T. OMeara, The classical groups and K-theory , Grundlehren, vol. 291, Springer-Verlag, 1989. · Zbl 0683.20033
[6] 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.
[7] J. Humphreys, Introduction to Lie algebras and Representation Theory , Graduate Texts in Mathematics vol. 9, Springer-Verlag, New York, 1972. · Zbl 0254.17004
[8] N. Jacobson, Some groups of transformations defined by Jordan algebras. I , J. Reine Angew. Math. 201 (1959), 178-195. · Zbl 0084.03601
[9] ——, Some groups of transformations defined by Jordan algebras. II. Groups of type F4 , J. Reine Angew. Math. 204 (1960), 74-98. · Zbl 0142.26401
[10] ——, Some groups of transformations defined by Jordan algebras. III, J. Reine Angew. Math. 207 (1961), 61-85. · Zbl 0142.26501
[11] ——, Lectures on quadratic Jordan algebras , Tata Institute of Fundamental Research, 1969. · Zbl 0253.17013
[12] ——, Structure theory of Jordan algebras , Lecture Notes in Math., vol. 5, The University of Arkansas, 1981. · Zbl 0492.17009
[13] Kantor, I. L., Classification of irreducible transitive differential groups, Dokl. Akad. Nauk SSSR, 158, 1271-1274 (1964) · Zbl 0286.17011
[14] Kantor, I. L., Non-linear groups of transformations defined by general norms of Jordan algebras, Dokl. Akad. Nauk SSSR, 172, 779-782 (1967) · Zbl 0167.30601
[15] Kantor, I. L., Certain generalizations of Jordan algebras, Trudy Sem. Vektor. Tenzor. Anal., 16, 407-499 (1972) · Zbl 0272.17001
[16] Kervaire, M., Multiplicateurs de Schur et K-théorie, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), 212-225 (1970), New York: Springer, New York · Zbl 0211.32903
[17] Koecher, M., Imbedding of Jordan algebras into Lie algebras. I, Amer. J. Math., 89, 787-816 (1967) · Zbl 0209.06801 · doi:10.2307/2373242
[18] Koecher, M., Über eine Gruppe von rationalen Abbildungen, Invent. Math., 3, 136-171 (1967) · Zbl 0163.03002 · doi:10.1007/BF01389742
[19] ——, Imbedding of Jordan algebras into Lie algebras. II , Amer. J. Math. 90 (1968), 476-510. · Zbl 0311.17005
[20] ——, Gruppen und Lie-Algebren von rationalen Funktionen , Math. Z. 109 (1969), 349-392. · Zbl 0181.04503
[21] Lavrenov, A., Another presentation for symplectic Steinberg groups, J. Pure Appl. Algebra, 219, 9, 3755-3780 (2015) · Zbl 1319.19001 · doi:10.1016/j.jpaa.2014.12.021
[22] Lavrenov, A.; Sinchuk, S., On centrality of even orthogonal K_2, J. Pure Appl. Algebra, 221, 5, 1134-1145 (2017) · Zbl 1360.19002 · doi:10.1016/j.jpaa.2016.09.004
[23] O. Loos, Jordan pairs , Springer-Verlag, Berline, 1975, Lecture Notes in Mathematics, vol 460. · Zbl 0301.17003
[24] ——, Homogeneous algebraic varieties defined by Jordan pairs, Mh. Math. 86 (1978), 107-129. · Zbl 0404.14020
[25] ——, On algebraic groups defined by Jordan pairs, Nagoya Math. J. 74 (1979), 23-66. · Zbl 0424.17001
[26] ——, Elementary groups and stability for Jordan pairs, K-Theory 9 (1995), 77-116. · Zbl 0835.17021
[27] ——, Steinberg groups and simplicity of elementary groups defined by Jordan pairs, J. Algebra 186 (1996), no. 1, 207-234. · Zbl 0869.17023
[28] ——, Rank one groups and division pairs, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 3, 489-521. · Zbl 1330.20046
[29] O. Loos and E. Neher, Locally finite root systems , Mem. Amer. Math. Soc. 171 (2004), no. 811. · Zbl 1195.17007
[30] ——, Steinberg groups for Jordan pairs , Progress in Mathematics, vol. 332, Birkhäuser Springer, 2019 · Zbl 1521.20001
[31] Loos, O.; Petersson, H. P.; Racine, M. L., Inner derivations of alternative algebras over commutative rings, Algebra & Number Theory, 2, 927-968 (2008) · Zbl 1191.17011 · doi:10.2140/ant.2008.2.927
[32] B. Magurn, An algebraic introduction to K-Theory , Encyclopedia of Mathematics and Its Applications 87, Cambridge University Press 2002. · Zbl 1002.19001
[33] J. Milnor, Introduction to algebraic K-theory , Ann. of Math. Studies, vol. 72, Princeton University Press, 1971. · Zbl 0237.18005
[34] E. Neher, Jordan triple systems by the grid approach, Lecture Notes in Math., vol. 1280, Springer-Verlag, 1987. · Zbl 0621.17001
[35] ——, Systèmes de racines 3-gradués , C. R. Acad. Sci. Paris Sér. I 310 (1990), 687-690. · Zbl 0719.17014
[36] ——, Lie algebras graded by 3-graded root systems and Jordan pairs covered by a grid, Amer. J. Math 118 (1996), 439-491. · Zbl 0857.17019
[37] ——, Polynomial identities and nonidentities of split Jordan pairs , J. Algebra 211 (1999, 206-224. · Zbl 0920.17014
[38] Rosenberg, J., Algebraic K-theory and its applications, Graduate Texts in Mathematics (1994), New York: Springer-Verlag, New York · Zbl 0801.19001 · doi:10.1007/978-1-4612-4314-4
[39] Sinchuk, S., On centrality of K_2for Chevalley groups of type E_ℓ, J. Pure Appl. Algebra, 220, 857-875 (2016) · Zbl 1327.19004 · doi:10.1016/j.jpaa.2015.08.003
[40] T. A. Springer, Jordan algebras and algebraic groups , Springer-Verlag, New York, 1973, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 75. · Zbl 0259.17003
[41] T. A. Springer and F. D. Veldkamp, Octonions, Jordan algebras and exceptional groups . Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000 · Zbl 1087.17001
[42] R. Steinberg, Générateurs, relations, et revêtements de groupes algébriques , Colloq. Théorie des Groupes Algébriques (Bruxelles), 1962, 113-127. · Zbl 0272.20036
[43] ——, Lectures on Chevalley groups , Yale University Lecture Notes, New Haven, Conn., 1967.
[44] ——, Generators, relations and coverings of algebraic groups II , J. Algebra 71 (1981), 527-543. · Zbl 0468.20038
[45] Strooker, J. R., The fundamental group of the general linear group, J. Algebra, 48, 477-508 (1977) · Zbl 0409.20039 · doi:10.1016/0021-8693(77)90323-4
[46] 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. · Zbl 0104.26002
[47] 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. · Zbl 0139.03204
[48] W. van der Kallen, Another presentation for Steinberg groups, Nederl. Akad. Wetensch. Proc. Ser. A 80 = Indag. Math. 39 (1977), 304-312. · Zbl 0375.20034
[49] C. Weibel, The K-book: an introduction to algebraic K-theory , Graduate Studies in Mathematics 145, AMS 2013. · Zbl 1273.19001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.