We start in §2 by reviewing the quaternions and defining the concepts of AH-modules and AH-morphisms. Section 3 defines hypercomplex manifolds and q-holomorphic functions, and shows that the q-holomorphic functions on a hypercomplex manifold are an AH-module. The key to the algebraic side of this paper is the definition in §4 of the quaternionic tensor product, and an exploration of some of its properties. Section 5 defines H-algebras, the quaternionic analogues of commutative algebras, and

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... s that the q-holomorphic functions on a hypercomplex manifold form an H-algebra. We briefly discuss the problem of recovering a hypercomplex manifold from its H-algebra, which leads to the idea of an algebraic geometry of hypercomplex manifolds. In §6 hyperkähler manifolds are defined, and we explain how the H-algebra of a hyperkähler manifold acquires an additional algebraic structure, making it into an HP-algebra. Sections 7-9 study the quaternionic algebra of finite-dimensional AH-modules in greater depth. In §7 a series of examples are given which illustrate differences between real and quaternionic algebra. Sections 8 and 9 concern two special classes of AH-modules, stable and semistable AH-modules. Amongst other things, we show that the quaternionic tensor product U ⊗ H V of two stable AH-modules is stable, and give a formula for dim U ⊗ H V . By restricting to stable AH-modules, some of the differences between real and quaternionic algebra are resolved, and the analogy between real and quaternionic algebra becomes more complete. Finally, § §10-12 give geometrical applications of the theory to hypercomplex and hyperkähler manifolds. Section 10 studies q-holomorphic functions on H , and § §11 and 12 prove some new results on hyperkähler manifolds with large symmetry groups, including an algebraic construction of Kronheimer's hyperkähler metrics on coadjoint orbits [10], [11] . Acknowledgements. I would like to thank John Cernes, Peter Kronheimer, Peter Neumann, and Simon Salamon for interesting conversations.