Centralizers of certain matrices relative to the operation related to a \(K\)-loop
It is known that the point set of the classical three-dimensional hyperbolic geometry can be described algebraically by the set \({\mathbf h}^{1+}\) consisting of all Hermitian \(2\times 2\) matrices having determinant 1 and positive trace. The latter enjoys the property that any \(X\in{\mathbf h}^{1+}\) admits a (unique) square root in \({\mathbf h}^{1+}\) and this implies that the operation \(\oplus\) defined by \(A\oplus B:=(AB^2A)^{{1\over 2}}\) makes \(({\mathbf h}^{1+},\oplus)\) a \(K\)-loop. In this note, the authors adopt a more general point of view upon considering in the vector space \(M_2(K)\) (\(2\times 2\) matrices with entries in a field \(K\)) the operation \(*\) defined by setting \(A*B:=AB^2A\). They end up (cf. Theorems 2.1 and 3.1) with a complete and exhaustive classification of the centralizers \(Z_*(A):=\{X\in M_2(K)\mid A*X=X*A\}\) as \(A\) varies in \(M_2(K)\), viewing the latter as a metric vector space with respect to the quadratic form \(q(X)=\det X\).
scientific article; zbMATH DE number 1346915
scientific article