A Survey of Lipschitz Monoids

Projective Metric Geometry and Clifford Algebras

“…see (17). By our construction, ker θ is just the kernel of the group action described in (25). This means…”

“…The following exposition runs along the lines of the survey [25] and the summary in [31, 5.10]. Historical remarks and additional results may be retrieved from [26][27][28]30] and [44, pp.…”

“…3, we collect from various sources those results about the Clifford algebra Cl(V , Q) which are needed later on. Section 4 is based on the work of Helmstetter as summarised in [25]. We recall from there the Lipschitz monoid Lip(V , Q) and the twisted adjoint 0123456789().…”

Minimal Algorithms for Lipschitz Monoids and Vahlen Monoids

“…In this way we obtain all orthogonal transformations g of V such that firstly, q(g(z)) = q(z) for all z ∈ V, and secondly (when q is degenerate), g(z) = z for all z ∈ ker(b q ). When dim(V) is infinite, these conditions on g must be strengthened (Helmstetter, 2005;Helmstetter & Micali, 2008;Helmstetter, 2012).…”

“…My first versions of the invariance theorem are in the references (Helmstetter, 1977(Helmstetter, , 1985(Helmstetter, , 1987(Helmstetter, , 1992, where the name "Clifford monoid" is used instead of "Lipschitz monoid". But to-day, I recommend (Helmstetter, 2005) for the proofs of the theorems about Lipschitz monoids, (Helmstetter, 2011) for more information, and (Helmstetter, 2012) for a survey of the subject. A quite different version of the invariance theorem was published by Sato, Miwa and Jimbo (1978); it was preceded in 1977 by preliminary publications in Japanese.…”