A theory of quaternionic algebra, with applications to hypercomplex geometry (Q2715738)

This paper has four chapters. The first one explains \(A\mathbb{H}\)-modules and the quaternionic tensor product. This part is wholly algebraic; the philosophy is that much algebra that works over a commutative field such as \(\mathbb{R}\) or \(\mathbb{C}\) also has a close analogue over \(\mathbb{H}\), when we replace vector space over \(\mathbb{R}\) on \(\mathbb{C}\) by \(A\mathbb{H}\)-modules, and tensor products of vector spaces by the quaternionic tensor product. Essential properties of the usual tensor product also hold for the quaternionic tensor product. However, the quaternionic tensor product also has properties very unlike to the usual tensor product, which come from the noncommutativity of the quaternions.NEWLINENEWLINENEWLINEIn chapter 2 the author gives quaternionic analogues of various algebraic structures. The most important idea is that of an \(\mathbb{H}\)-algebra, the quaternionic version of a commutative algebra. Chapter 3 is concerned with hypercomplex geometry. \(Q\)-holomorphic functions are defined and it is shown that the \(q\)-holomorphic functions on a hypercomplex manifold form an \(\mathbb{H}\)-algebra. A similar result is proved for hyper-Kähler manifolds. Chapter 4 is a collection of examples and applications of the theory.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00032].