Veränderungen über einen Satz von Timmesfeld – I. Quadratic Actions
Confluentes Mathematici, Volume 5 (2013) no. 2, p. 25-46

We classify quadratic SL 2 (𝕂)- and 𝔰𝔩 2 (𝕂)-modules by crude computation, generalising in the first case a Theorem proved independently by F.G. Timmesfeld and S. Smith. The paper is the first of a series dealing with linearisation results for abstract modules of algebraic groups and associated Lie rings.

Received : 2013-01-01
Revised : 2013-04-08
Accepted : 2013-08-21
Published online : 2017-03-27
Classification:  20G05,  20G15,  17B10,  17B45
@article{CML_2013__5_2_25_0,
     author = {Adrien Deloro},
     title = {Ver\"anderungen \"uber einen Satz von Timmesfeld -- I. Quadratic Actions},
     journal = {Confluentes Mathematici},
     publisher = {Institut Camille Jordan},
     volume = {5},
     number = {2},
     year = {2013},
     pages = {25-46},
     language = {en},
     url = {https://cml.centre-mersenne.org/item/CML_2013__5_2_25_0}
}
Deloro, Adrien. Veränderungen über einen Satz von Timmesfeld – I. Quadratic Actions. Confluentes Mathematici, Volume 5 (2013) no. 2, pp. 25-46. cml.centre-mersenne.org/item/CML_2013__5_2_25_0/

[1] Andrew Chermak Quadratic pairs, J. Algebra, Tome 277 (2004) no. 1, pp. 36-72 | Article

[2] George Glauberman A sufficient condition for p stability, Proc. London Math. Soc. (3), Tome 25 (1972), pp. 253-287

[3] Chat-Yin Ho On the quadratic pairs, J. Algebra, Tome 43 (1976) no. 1, pp. 338-358

[4] Alexander Arcadievitch Premet; Irina Dmitrievna Suprunenko Quadratic modules for Chevalley groups over fields of odd characteristics, Math. Nachr., Tome 110 (1983), pp. 65-96 | Article

[5] Stephen D. Smith Quadratic action and the natural module for SL 2 (k), J. Algebra, Tome 127 (1989) no. 1, pp. 155-162

[6] John G. Thompson Quadratic pairs, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, p. 375-376

[7] Franz Georg Timmesfeld Groups generated by k-transvections, Invent. Math., Tome 100 (1990) no. 1, pp. 167-206

[8] Franz Georg Timmesfeld Abstract root subgroups and quadratic action, Adv. Math., Tome 142 (1999) no. 1, pp. 1-150 (With an appendix by A. E. Zalesskii) | Article

[9] Franz Georg Timmesfeld Abstract root subgroups and simple groups of Lie type, Monographs in Mathematics, Tome 95, Birkhäuser Verlag, Basel, 2001, xiv+389 pages