Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-15T03:42:14.141Z Has data issue: false hasContentIssue false

A note on Clifford algebras and central division algebras with involution

Published online by Cambridge University Press:  18 May 2009

D. W. Lewis
Affiliation:
Department of Mathematics, University College, Belfif, Dublin 4Ireland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this note we consider the question as to which central division algebras occur as the Clifford algebra of a quadratic form over a field. Non-commutative ones other than quaternion division algebras can occur and it is also the case that there are certain central division algebras D which, while not themselves occurring as a Clifford algebra, are such that some matrix ring over D does occur as a Clifford algebra. We also consider the further question as to which involutions on the division algebra can occur as one of two natural involutions on the Clifford algebra.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

REFERENCES

1.Albert, A. A., Structure of algebras, revised edition (Amer. Math. Soc. Colloquium publications, 1961).Google Scholar
2.Albert, A. A., A construction of non-cyclic normal division algebras, Bull. Amer. Math. Soc. 38 (1932), 449456.CrossRefGoogle Scholar
3.Amitsur, S. A., Rowen, L. H. and Tignol, J. P., Division algebras of degree 4 and 8 with involution, Israel. J. Math. 33 (1979), 133148.CrossRefGoogle Scholar
4.Arason, J. K., A proof of Merkurjev's theorem, to appear in Proc. of conference on quadratic forms and hermitian K-theory, McMaster University, July 1983.Google Scholar
5.Elman, R., Lam, T.-Y., Tignol, J. P. and Wadsworth, A., Witt rings and Brauer groups under multiquadratic extensions I, Amer. J. Math 105 (1983), 11191170.CrossRefGoogle Scholar
6.Frohlich, A., Orthogonal and symplectic representations of groups Proc. London Math. Soc. (3) 24 (1972), 470506.CrossRefGoogle Scholar
7.Jacobson, N., Basic Algebra II (Freeman, 1980).Google Scholar
8.Lam, T.-Y., Algebraic theory of quadratic forms (Benjamin, 1973).Google Scholar
9.Lewis, D. W., Periodicity of Clifford algebras and exact octagons of Witt groups, to appear.Google Scholar
10.Merkurjev, A. S., On the norm residue symbol of degree 2, Dokl. Akad. Nauk. SSSR 261 (1981), 542547, English translation, Soviet Math. Dokl. 24 (1981), 546–551.Google Scholar
11.Racine, M. L., A simple proof of a theorem of Albert, Proc. Amer. Math. Soc. 43 (1974), 487488.Google Scholar
12.Rowen, L. H., Central simple algebras, Israel J. Math. 29 (1978), 285301.CrossRefGoogle Scholar
13.Tignol, J. P., Corps a Involution de Rang Fini sur leur centre et de caracteristique differente de 2, memoire de licence, Bruxelles (1975).Google Scholar
14.Tignol, J. P., Central simple algebras with involution, in Ring Theory, Proc. of 1978 Antwerp conference (Marcel Dekker, 1979).Google Scholar
15.Tignol, J. P., Cyclic algebras of small exponent, Proc. Amer. Math. Soc. 89 (1983), 587588.CrossRefGoogle Scholar