Simple ternary complex Grassmann algebras

Abstract

This work introduces a new class of simple comtrans algebras obtained in the tangent space of a complex Grassmann manifold. It is shown that some of the simple algebras of this new, Grassmann type do not appear as simple algebras of any of the previously known types. In chapter one, comtrans algebras are defined and examples given for the four broad types of simple comtrans algebras currently known. Some of the prerequisite details of the general algebraic theory of comtrans algebras, particularly concerning the universal enveloping algebra of a comtrans algebra, are summarized. In chapter two we define a new class of comtrans algebras. The algebras are said to be of complex Grassmann type. We discuss their simplicity and show that each algebra in this class is an internal Thomas sum of its subalgebras E and F defined in the chapter. Chapter three is devoted to the problem of showing that the complex Grassmann comtrans algebras are not isomorphic to other types of simple comtrans algebras, and in conclusion we outline a few conjectures as open problems for further research.