M.Sc Thesis


M.Sc StudentLevit David
SubjectFinite Dimensional Graded Division Algebras over the Field
of Real Numbers
DepartmentDepartment of Mathematics
Supervisor PROF. Eli Aljadeff


Abstract

The theory of G-graded division algebras is a generalization of the theory of division algebras. Indeed, division algebras are G-graded division algebras for the trivial group G = {e}. Every finite dimensional division algebra over the field of real numbers is isomorphic to one of the following three algebras: (a) the field of real numbers R (b) the field of complex numbers C (c) the algebra of real quaternions H. When we let G be any finite group, we discover infinitely many more G-graded division algebras of finite dimension over the field of real numbers. Real finite dimensional G-graded division algebras, where G is a finite abelian group, were classified up to (a) G-graded isomorphism (b) G-graded equivalence in [3] by Bahturin and Zaicev. In this thesis we provide such a classification, up to isomorphism, which is relatively simpler and may be generalized to nonabelian groups. This classification requires calculating

certain cohomology groups, one of which is the Schur multiplier. We develop tools for calculating these cohomologies in some cases. In particular we show how to build G-graded division algebras when G = N H is a semidirect product, from N-graded and H-graded division algebras. We count all different, up to isomorphism, G-graded real finite dimensional division algebras for a given finite abelian group. We discuss the classification in case G is nonabelian. In particular we classify such algebras in case G is a dihedral group.