Monoids of Compatible Bilinear Forms in Relation to Lipschitz Monoids

Abstract

Let V be a vector space of finite dimension over a field K,  and Q a quadratic form on V. A bilinear form compatible with Q is a bilinear form \(\varphi \) defined on any subspace S of V such that \(\varphi (s,s)=Q(s)\) for all \(s\in S\). The bilinear forms compatible with Q,  together with an exceptional empty element, constitute an associative and unital monoid \(\textrm{Cbf}(V,Q)\). In the first part of this work, the main purpose is a surjective homomorphism from the Lipschitz monoid \(\textrm{Lip}(V,Q)\) onto this monoid \(\textrm{Cbf}(V,Q)\). In the second part, V is provided with an alternating bilinear form \(\Omega ,\) and some analogous properties are established for the monoid of bilinear forms compatible with \(\Omega \). When K is the field of real numbers, the controversy about an eventual Lipschitz monoid for \(\Omega \) is recalled.