Norms and gauges on Clifford Algebra

ABSTRACT

From a normed quadratic space (V,q), we construct a norm on the Clifford algebra C(V,q). We describe the associated graded form of this norm and give a condition for this norm to be a gauge. Then, we apply our results to prove that for a complete discrete valued field, an anisotropic quadratic form q with dimq = 0 mod 8 and nonsplit Clifford algebra cannot be at the same time a transfer of a K-hermitian form with K∕F an inertial quadratic field extension and a transfer of a T-hermitian form with T∕F a ramified quadratic field extension.

KEYWORDS:

• Clifford algebras
• gauge
• norm
• quadratic and bilinear forms
• valuation

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W60
• 16W50
• 11E88
• 15A66
• 12J20
• 13A18
• 20E42

Acknowledgment

This paper is based on the first part of my PhD thesis. I am particularly grateful to my thesis advisor Jean-Pierre Tignol for his kindness and more particularly for all his advice and suggestions. I would like to thank the members of the jury of my thesis for all the comments they give to improve the text of my PHD thesis. My thanks also go to my husband Corentin, my children Benoît, François, Sophie, Marine and baby for their encouragement and support to finish this article.

Additional information

Funding

This work was supported by the FRS-FNRS (fonds national de la recherche scientifique - Belgique).