**Jacques Helmstetter ^{[1]}**

[1] : Institut Fourier (Mathématiques), B.P. 74, 38400 Saint-Martin d'Hères, France

**Abstract:**
Although much attention has already been paid to
graded central simple algebras and their involutions, one piece of
information is not yet well known: their involutions are classified
by a cyclic group of order 8. The properties of this classification,
and several problems in which it is useful, are here explained. Graded
modules are the natural domain of application, but modules without
gradation are also dealt with. A precise example (coming from quantum
mechanics) shows which progresses this theory can achieve.

**References**

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[MR0279159]

[2] J. Helmstetter, *The group of classes of involutions of graded central simple algebras*, in "Clifford algebras" (Cookeville, TN, 2002), R. Ablamowicz,
ed., Prog. Math. Phys., 34, Birkhäuser, Boston MA, 2004, 335-341.
[MR2025989]

[3] J. Helmstetter and A. Micali, *Quadratic mappings and Clifford algebras*, Birkhauser, Verlag, Basel 2008.
[MR2408410]

[4] M. A. Knus, A. Merkurjev, M. Rost and J.-P. Tignol, *The book of involutions*, Amer. Math. Soc. Colloq. Publ., 44, American Mathematical Society, Providence, RI 1998.
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[8] A. Weil, *Algebras with involutions and the classical groups*, J. Indian Math. Soc. (N.S.) 24 (1960), 589-623. Reprinted in "Oeuvres Scientifiques,
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[MR0136682]

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