Projective metric geometry and Clifford algebras (Q2051423)

Let \(V\) be a finite dimensional vector space over a field \(F\). Assume that \(V\) is endowed with a quadratic form \(Q\). As the main goal of the present note, the author interprets the Lipschitz group \(\mathrm{Lipx}(V, Q)\) in projective terms, i.e. he considers the projective metric space \(P(V,Q)\) and the projective space on the associated Clifford algebra \(\mathrm{Cl}(V,Q)\). In \(P(\mathrm{Cl}(V, Q))\) he introduces point sets \(M(V, Q)\) and \(G(V, Q)\) that arise from a quotient of the Lipschitz monoid \(\mathrm{Lip}(V, Q)\) and a quotient of the Lipschitz group \(\mathrm{Lipx}(V, Q)\). This set is made into a group that acts on the initial projective metric space \(P(V, Q)\). The author compares the Clifford algebras \(\mathrm{Cl}(V, Q)\) and \(\mathrm{Cl}(V, cQ)\).