×

Algebras with scalar involution revisited. (English) Zbl 1229.14002

In the present paper, the author continues McCrimmon’s work in a different direction. In Section 1, “Conic algebras”, he establishes several basic facts on exterior products and formal directions of multilinear maps. Then, the author deal with constructions of conic algebras. Section 2 is the canonical 3-form of a conic algebra, Section 3 is coordinates for conic algebras, Section 4 is algebras with scalar involution, and the final section 5 is extending the theory to an arbitrary base scheme.

MSC:

14A15 Schemes and morphisms
14L30 Group actions on varieties or schemes (quotients)
17A45 Quadratic algebras (but not quadratic Jordan algebras)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Becker, E., Über eine Klasse flexibler quadratischer Divisionsalgebren, J. Reine Angew. Math., 256, 25-57 (1972) · Zbl 0222.17001
[2] Bourbaki, N., Elements of mathematics, (Algebra, Part I: Chapters 1-3 (1974), Hermann: Hermann Paris), Translated from the French · Zbl 0175.27001
[3] Bourbaki, N., (Éléments de mathématique. Algèbre Chapitre 10. Algèbre homologique (2007), Springer-Verlag: Springer-Verlag Berlin), Reprint of the 1980 original [Masson, Paris; MR0610795]
[4] Demazure, M.; Gabriel, P., Groupes algébriques, vol. 1 (1970), Masson: Masson Paris
[5] Elduque, A., Quadratic alternative algebras, J. Math. Phys., 31, 1, 1-5 (1990) · Zbl 0716.17002
[6] Hahn, A. J.; O’Meara, O. T., The classical groups and K-theory, (Grundlehren der mathematischen Wissenschaften, vol. 291 (1989), Springer-Verlag)
[7] Knus, M.-A., Quadratic and Hermitian forms over rings, (Grundlehren der Mathematischen Wissenschaften, vol. 294 (1991), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0656.10015
[8] Kunze, R. A.; Scheinberg, S., Alternative algebras having scalar involutions, Pacific J. Math., 124, 1, 159-172 (1986) · Zbl 0603.17013
[9] Loos, O., Tensor products and discriminants of unital quadratic forms over commutative rings, Mh. Math., 122, 45-98 (1996) · Zbl 0872.11022
[10] Loos, O., Discriminant algebras of finite rank algebras and quadratic trace modules, Math. Z., 257, 467-523 (2007) · Zbl 1136.11031
[11] Manin, Yu. I., Some remarks on Koszul algebras and quantum groups, Ann. Inst. Fourier, 37, 4, 191-205 (1987) · Zbl 0625.58040
[12] McCrimmon, K., Nonassociative algebras with scalar involution, Pacific J. Math., 116, 1, 85-109 (1985) · Zbl 0558.17002
[13] Osborn, J. M., Quadratic division algebras, Trans. Amer. Math. Soc., 105, 202-221 (1962) · Zbl 0136.30303
[14] Petersson, H. P., Composition algebras over algebraic curves of genus zero, Trans. Amer. Math. Soc., 337, 473-491 (1993) · Zbl 0778.17001
[15] H.P. Petersson, Polar decomposition of quaternion algebras over arbitrary rings, Mathematikberichte der FernUniversität in Hagen, 80, 2008, pp. 83-102.; H.P. Petersson, Polar decomposition of quaternion algebras over arbitrary rings, Mathematikberichte der FernUniversität in Hagen, 80, 2008, pp. 83-102.
[16] M. Rost, The discriminant algebra of a cubic algebra, preprint, 2002, 3 pp..; M. Rost, The discriminant algebra of a cubic algebra, preprint, 2002, 3 pp..
[17] L. Zagler, Ausgeartete, alternative, quadratische Algebren, Dissertation, University of Munich, 1968.; L. Zagler, Ausgeartete, alternative, quadratische Algebren, Dissertation, University of Munich, 1968.
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.