Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 46, No. 2, pp. 377395 (2005) 

Spin groups over a commutative ring and the associated root data (odd rank case)Hisatoshi IkaiMathematical Institute, Tohoku University, Sendai 9808578, Japan email: ikai@math.tohoku.ac.jpAbstract: Spin and Clifford groups as group schemes of semiregular quadratic spaces of odd rank over a commutative ring are shown to be smooth and reductive. Analogously to the hyperbolic case smooth open neighborhoods of unit sections, called big cells, are constructed and examined. Jordan pairs again play a role through an imbedding into hyperbolic space whose rank is higher by one. The property reductive is now proved by constructing maximal tori and their associated root data explicitly. Keywords: Spin groups, jordan pairs, root data Classification (MSC2000): 14L15; 17C30 Full text of the article:
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