Homotopy methods in absolute valued structures theory
A real or complex \(2\)-graded (\(\mathbb{Z}/2\mathbb{Z}\)-graded) (non-associative) algebra \(A\) with homogeneous components \(A_0\), \(A_1\) is said to be absolute valued if each \(i=0,1\) admits a norm \(|\,\,\,|: A_i\rightarrow \mathbb{R}_+\) such that \(|xy|=|x|\,|y|\) for all homogeneous elements in \(A\). In the paper under review, finite-dimensional \(2\)-graded absolute valued algebras over the base field of real numbers are studied using homotopical techniques. The authors show that every \(2\)-graded absolute valued algebra \(A\) as above up to isomorphism can be brought into normal form where \(A_0=A_1=\mathbb{R}^n\), \(n=2,4,6\), the norms are derived from the canonical scalar product on \(\mathbb{R}^n\), and the multiplication of \(A\) is determined by four principal Albert isotopes [see \textit{A. A. Albert}, Ann. Math. 43, No. 2, 685--707 (1942; Zbl 0061.04807)] of the unique real \(n\)-dimensional composition division algebra via eight proper isometries affected by the conjugation of the composition algebra. Given \(2\)-graded absolute valued algebras \(A\) and \(B\) in normal form, the authors define a homotopy from \(A\) to \(B\) as a continuous map \(H:(\mathbb{R}^n\times \mathbb{R}^n)\times (\mathbb{R}^n\times \mathbb{R}^n)\times [0,1] \rightarrow \mathbb{R}^n\times \mathbb{R}^n\) satisfying \(H(-,-,0)=A\), \(H(-,-,1)=B\) and for each \(t \in [0,1]\), \(H(-,-,t)\) is a \(2\)-graded absolute valued algebra in normal form. They prove that isomorphic \(2\)-graded absolute valued algebras in normal form are homotopic but not conversely. The main result in the paper provides a complete classification of \(2\)-graded absolute valued algebras. Moreover, \(2\)-graded absolute triple systems are studied.
scientific article; zbMATH DE number 5706257