Various characteristic properties of Lipschitzian elements in Clifford algebras (Q6169969)

This work can be of interest for algebraists who want to know more about Clifford algebras and the properties of their Lipschitzian elements, a topic that is important for example for the representation of orthogonal groups. The multiplicative Lipschitz monoid is generated by the Clifford product of vectors. The present work offers a number of characteristic properties, which can also serve as alternative, but equivalent definitions. The introduction presents the definition as multiplicative monoid in a Clifford algebra over a vectors space with given quadratic form: it is generated by the elements of the field, the elements of the vector space and elements of the form \(1+uv\), \(u\) and \(v\) spanning a totally isotropic vector space plane. Then a theorem characterizes homogeneous multivectors as Liptschitzian, using a \textit{derivative} in the twisted tensor product of the Clifford algebra with itself. A second characterization theorem is based on a Clifford algebra as an irreducible module over the Clifford algebra arising from the vector space and its dual space. Section 2 reviews Clifford algebra properties. Section 3 presents polynomials with two arguments form the set \(P(N)\) of all subsets of\( N=\{1,2, \dots, n\}\), related to the above mentioned \textit{derivative} in the twisted tensor product of the Clifford algebra with itself, and Section 4 presents more information on this derivative operator. Section 5 relates the derivative operator to simple (pure) spinors in the Clifford algebra. Section 6 goes deeper into twisted algebras. Section 7 presents a further definition arising from the characterization of homogeneous multivectors based on twisted subalgebras. In his final comments (Section 8), the author gives an overview of the developments during the last 45 years, starting with his work [J. Algebra 111, 14--48 (1987; Zbl 0667.15024)].