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ON SUBGROUPS OF CLIFFORD GROUPS DEFINED BY JORDAN PAIRS OF RECTANGULAR MATRICES

  • HISATOSHI IKAI (a1)

Abstract

Some embeddings of general linear groups into hyperbolic Clifford groups are constructed generically by using Jordan pairs of rectangular and alternating matrices over a ring. In low rank cases through exceptional isomorphisms, their direct description and relationships to some automorphisms of Clifford groups are given. Generic norms are calculated in detail, and equivariant embeddings of representation spaces are constructed.

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References

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[1]Bourbaki, N., Elements of Mathematics: Algebra, Part I (Addison-Wesley, Reading, MA, 1974), Chapters 1–3.
[2]Chevalley, C., The Constructions and Study of Certain Important Algebras (The Mathematical Society of Japan, Tokyo, 1955), reprinted in: Collected Works, Vol. 2, Springer, Berlin, Heiderberg, New York.
[3]Demazure, M. and Gabriel, P., Groupes Algébriques, Tome I (Masson, Paris, 1970).
[4]Igusa, J.-I., ‘A classification of spinors up to dimension twelve’, Amer. J. Math. 92 (1970), 9971028.
[5]Ikai, H., ‘Spin groups over a commutative ring and the associated root data’, Monatsh. Math. 139 (2003), 3360.
[6]Ikai, H., ‘On Lipschitz’ lifting of the Cayley transform’, J. Indian Math. Soc. 72 (2005), 112.
[7]Ikai, H., ‘An approach through big cells to Clifford groups of low rank’, Ark. Mat. 47 (2009), 313330.
[8]Ikai, H., ‘An explicit formula for Cayley–Lipschitz transformations’, Preprint.
[9]Ikai, H., ‘Big cells of Clifford groups attached to weak Witt decompositions’, Preprint.
[10]Knus, M.-A., Quadratic and Hermitian Forms Over Rings, Grundlehren der Mathematischen Wissenschaften, 294 (Springer, Berlin, 1991).
[11]Loos, O., Jordan Pairs, Lecture Notes in Mathematics, 460 (Springer, Berlin, 1975).
[12]Loos, O., ‘On algebraic groups defined by Jordan pairs’, Nagoya Math. J. 74 (1979), 2366.
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ON SUBGROUPS OF CLIFFORD GROUPS DEFINED BY JORDAN PAIRS OF RECTANGULAR MATRICES

  • HISATOSHI IKAI (a1)

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