On Grassmannians over *-rings
On the ring \(M_n(K)\) of all \(n\times n\) matrices there is an involution * determined by the conjugation and transpose maps, where \(K\) is the field of real numbers \(\mathbb{R}\), the field of complex numbers \(\mathbb{C}\), or the skew-field of quaternions. The tangent bundle \(TG_{n,r}(K)\) of the Grassmannian \(G_{n,r}(K)\) is diffeomorphic to the manifold \(\text{Idem}_{n,r}(K)\) of all idempotent \(n\times n\) matrices with rank \(r\) over \(K\). For any idempotent matrix \(\varphi\) in \(\text{Idem}_{n,r}(K)\) the matrix \(\varphi+\varphi^*-I_n\) is invertible and the matrix \(\psi=\varphi(\varphi+\varphi^*-I_n)^{-1}\) is Hermitian, idempotent with \(\varphi\psi=\psi\) and \(\psi\varphi=\varphi\). Then there exists an explicit regular deformation retraction \(\text{Idem}_{n,r}(K)\to G_{n,r}(K)\) of affine varieties. This paper tends to develop similar results in a wider context by replacing not only \(K\) but even \(M_n(K)\) by any ring with involution. In Section 2 the authors study the matrix ring \(M_n(R)\) over a commutative ring \(R\) with involution. Section 3 is Grassmannians over matrix rings.
scientific article; zbMATH DE number 1864739