Structure and uniqueness of sums of simple Lie superalgebras

Tvalavadze,, T. (2006) Structure and uniqueness of sums of simple Lie superalgebras. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

In this thesis we consider decompositions of algebras and superalgebras into the sum of two subalgebras. The sum is understood in a sense of a vector space sum and not necessarily direct. The structure of these sums has attracted considerable attention for various types of algebras. Originally, this problem arises in the work of M. Goto (1963) where he studied the case of nilpotent Lie algebras. In 1969 A. Onishchik classified decompositions of simple complex Lie algebras into the sum of two reductive subalgebras. In 1999 Y. Bahturin and O. Kegel [1] proved that no simple associative algebra can be written as the sum of two simple subalgebras over an algebraically closed field. In the joint paper with M. Tvalavadze [24], we classify decompositions of simple Jordan algebras over an algebraically closed field of characteristic not two. -- In the case of Lie superalgebras this problem was open until now. The main result of this thesis is a classification of all such decompositions in the case of basic non-exceptional Lie superalgebras, up to conjugation, over an algebraically closed field of characteristic zero. Moreover, we construct precise matrix realizations of each decomposition. -- To prove this result we consider a Lie superalgebra as a module over its even component which is a Lie algebra. Using techniques of the representation theory of semisimple Lie algebras we present the precise description of such modules for each superalgebra in the sum. This research is significantly based on the result from [26] which extends Onishchik's Classification Theorem to an arbitrary algebraically closed field of characteristic zero.

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