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Metric and Involution Scores of Clifford Algebras

Research paper by Greg Trayling

Indexed on: 09 Nov '16Published on: 01 Dec '16Published in: Advances in Applied Clifford Algebras



Abstract

Abstract The notion of a metric score is introduced as a separate tally of the total number of standard (Grassmann) basis elements spanning a Clifford algebra \({C\ell_{p,q}}\) that square to + 1 and −1. A closed-form expression is derived for any given vector space dimension n = p+q. This is then generalized to reversion and Clifford-conjugation. A central application is that two real Clifford algebras are isomorphic if and only if their metric scores are identical.AbstractThe notion of a metric score is introduced as a separate tally of the total number of standard (Grassmann) basis elements spanning a Clifford algebra \({C\ell_{p,q}}\) that square to + 1 and −1. A closed-form expression is derived for any given vector space dimension n = p+q. This is then generalized to reversion and Clifford-conjugation. A central application is that two real Clifford algebras are isomorphic if and only if their metric scores are identical. \({C\ell_{p,q}}\) \({C\ell_{p,q}}\)npq