Abstract
In this paper we develop a structure theory of algebraic right distributive quasigroups which correspond to closed and connected conjugacy classes\(\mathfrak{D}\) generating algebraic Fischer groups (in the sense of [6]) such that the mappingx ↦x −1 ax, fora ε\(\mathfrak{D}\), is an automorphism of\(\mathfrak{D}\) (as variety). We also give examples of algebraic Fischer groups where this does not happen. It becomes clear that the class of algebraic right distributive quasigroups has nice properties concerning subquasigroups, normal subquasigroups and direct product.
We give a complete classification of one- and two-dimensional as well as of minimal algebraic right distributive quasigroups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Birkenhake, Ch. and Lange, H.: Fixed-point free automorphisms of Abelian varieties,Geom. Dedicata 51 (1994), 201–213.
Borel, A.:Linear Algebraic Groups, Springer, 1991.
Burstin, C. and Mayer, W.: Distributive Gruppen von endlicher Ordnung,J. reine angew. Math. 160 (1929), 111–130.
Chein, O., Pflugfelder, H. O., Smith, J. D. H. (eds): Quasigroups and loops. Theory and applications,Sigma Series in Pure Mathematics, Vol. 8, Heldermann Verlag, Berlin.
Demazure, M. and Gabriel, P.:Groupes Algebriques, Tome I, 1970.
Enea, M. R.: Algebraic groups with a distinguished conjugacy class (to appear inForum Math.).
Fischer, B.: Distributive Quasigruppen endlicher Ordnung,Math. Z. 83 (1964), 267–303.
Hertzig, D.: ‘Fixed-point-free automorphisms of algebraic tori’,Amer. J. Math. 90 (1968), 1041–1047.
Hertzig, D.: The structure of Frobenius algebraic groups,Amer. J. Math. 83 (1961), 421–431.
Hochschild, G. and Mostow, G. D.: Automorphisms of affine groups,J. Algebra 13 (1969), 535–543.
Humphreys, J. E.:Linear Algebraic Groups, Springer, 1975.
Husemöller, D.:Elliptic Curves, Springer, 1987.
Kambayashi, T. and Russel, P.: On linearizing algebraic torus actions,J. Pure Appl. Algebra 23 (1982), 243–250.
Matsumura, H. and Oort, F.: Representability of group functors and automorphisms of algebraic schemes,Invent. Math. 4 (1967), 1–25.
Mumford, D.:Abelian Varieties, Oxford University Press, Oxford, Bombay, 1974.
Plaumann, P. und Strambach, K.: Partitionen Liescher und algebraischer Gruppen,Forum Math. 2 (1990), 523–578.
Ramanujan, C. P.: A note on automorphism groups of algebraic varieties,Math. Ann. 156 (1964), 25–33.
Rosenlicht, M.: Some basic theorems on algebraic groups,Amer. J. Math. 78 (1956), 401–443.
Rosenlicht, M.: Extension of vector groups by Abelian varieties,Amer. J. Math. 80 (1958), 685–714.
Serre, J. P.:Groupes Algébriques et Corps de Classe, Hermann, Paris, 1959.
Smith, J. H. W. D.: Finite distributive quasigroups,Math. Proc. Cambridge Phil. Soc. 80, No. 1 (1976), 37–41.
Springer, T. A.:Linear Algebraic Groups, Birkhäuser, Boston, Basel, Stuttgart, 1981.
Strambach, K.: Rechtsdistributive Multiplikationen auf homogenen Räumen,J. reine angew. Math. 373 (1987), 185–217.
Tate, J. T.: The arithmetic of elliptic curves,Invent. Math. 23 (1974), 170–206.