The QRD and SVD of matrices over a real algebra

Recent work in the field of signal processing has shown that the singular value decomposition of a matrix with entries in certain real algebras can be a powerful tool. In this article we show how to generalise the QR decomposition and SVD to a wide class of real algebras, including all finite-dimensional semi-simple algebras, (twisted) group algebras and Clifford algebras. Two approaches are described for computing the QRD/SVD: one Jacobi method with a generalised Givens rotation, and one based on the Artin-Wedderburn theorem.Comment: Uses elsarticle.cl

p-Adic Lifting Problems and Derived Equivalences

For two derived equivalent $k$-algebras $\bar\Lambda$ and $\bar\Gamma$, we introduce a correspondence between \OO-orders reducing to $\bar\Lambda$ and \OO-orders reducing to $\bar\Gamma$. We outline how this may be used to transfer properties like uniqueness (or non-existence) of a lift between $\bar\Lambda$ and $\bar\Gamma$. As an application, we look at tame algebras of dihedral type with two simple modules, where, most notably, we are able to show that among those algebras only the algebras $\mathcal D^{\kappa,0}(2A)$ and $\mathcal D^{\kappa,0}(2B)$ can actually occur as basic algebras of blocks of group rings of finite groups