Simple subalgebras of simple Jordan algebras and simple decompositions of simple Jordan superalgebras

Tvalavadze, M. (2006) Simple subalgebras of simple Jordan algebras and simple decompositions of simple Jordan superalgebras. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

In 1952 E. Dynkin classified semisimple subalgebras of semisimple Lie algebras over an algebraically closed field F of zero characteristic. Until now there was no classification of simple (semisimple) subalgebras of simple finite-dimensional Jordan algebras. As a consequence the first problem of this thesis is a description of simple subalgebras in finite-dimensional special simple Jordan algebras over an algebraically closed field F of characteristic not 2. Using a slightly generalized version of Malcev's Theorem, Racine's classification of maximal subalgebras and other techniques developed in the thesis we can show that each simple subalgebra of a simple Jordan algebra can be reduced to an appropriate canonical form. Besides we formulate necessary and sufficient conditions for conjugacy of simple subalgebras of simple special Jordan algebra J. Therefore, in Jacobson's terminology we describe orbits of the algebra of symmetric matrices under O (n ) (the orthogonal group), orbits of the algebra of symplectic matrices under Sp (n ) (the symplectic group) and orbits of full matrix algebra under GL (n ) (the general linear group). -- The other problem considered in this thesis is the classification of simple decompositions that occur in simple Jordan superalgebras with semisimple even part over an algebraically closed field F of characteristic not 2. By a simple (semisimple ) decomposition of any algebra J (not necessarily simple) we understand any representation of J as vector sum space of two proper simple (semisimple) subalgebras. In general, the sum in this decomposition is not necessarily direct, and the subalgebras may not be ideals. The problem of finding simple decompositions has drawn researchers' interest in late 60's after the pioneering works of O. Kegel, A. Onishchik and others. Given J = A + B , the sum of two proper simple subalgebras A and B , what abstract properties of A and B does J inherit? In addition, information about the structure of simple subalgebras can be used to describe the lattice properties of simple algebras. In this thesis we determined the conjugacy classes of simple decompositions of simple matrix Jordan superalgebras with semisimple even part over an algebraically closed field F of characteristic not 2.

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