K-loops from classical subgroups of \(\operatorname{GL} (\mathcal H)\), \(\mathcal H\) being a separable Hilbert space (Q2798960)

A loop \((Q,\oplus)\) is called a K-loop if it obeys the left Bol identity \(a\oplus (b\oplus (a\oplus c))=(a\oplus (b\oplus a))\oplus c\) and the automorphic inverse property \((a\oplus b)^{-1}=a^{-1}\oplus b^{-1}\) for all \(a,b,c\in Q\). One of the most interesting examples of a K-loop is the non-associative and non-commutative structure \((\mathbb{R}^3,\oplus)\) where \(\mathbb{R}_c^3\) is the set of relativistically admissible velocities that are vectors in \(\mathbb{R}^3\) whose norms are strictly less than \(c\) (the vacuum speed of light) and \(\oplus\) is the Einstein's velocity addition binary operation over it. In the past, efforts have been made to construct more examples of K-loops using various mathematical systems. For instance, \textit{H. Kiechle} [J. Geom. 61, No. 1--2, 105--127 (1998; Zbl 0904.20051)] remarked that the construction of K-loops from classical groups over ordered fields can be generalized to K-loops from \(\operatorname{GL}(\mathcal{H})\) using the polar decomposition theorem, where \(\operatorname{GL}(\mathcal{H})\) is the unit group of bounded linear operators over the Hilbert space \(\mathcal{H}\). The article under review is divided into three sections. The first section is dedicated to an introduction on K-loops where an example of an K-loop derived from classical groups over ordered fields by Kiechle [loc. cit.] is cited. The second section summarized the method of \textit{A. Kreuzer} and \textit{H. Wefelscheid} [Result. Math. 25, No. 1--2, 79--102 (1994; Zbl 0803.20052)] of forming a K-loop from group transversals which would enable the author to be able to extend the examples of K-loops to algebraic groups with additional structures such as groups with manifolds or topological groups and not only purely algebraic groups. Section 3 contains the main result (Theorem 3.1) of the work where they formed infinite dimensional K-loops from some subgroups of \(\operatorname{GL}(\mathcal{H})\) such as symplectic and orthogonal classical Banach Lie groups.NEWLINENEWLINE Some typographical errors: `Einsten' in the first line of the third paragraph of the introduction should be `Einstein'. `Kerby' in the last paragraph of the introduction should be `Kreuzer'. In reference [7], the volume number should be \(61\) and not \(113\).