The Fermionic-bosonic Representations of <I>A</I>(<I>M</I>-1,<I>N</I>-1)-graded Lie Superalgebras

Abstract

Lie groups, Lie algebras and their representation theories are important parts of mathematical physics. They play a crucial character in symmetries. As a generalization of Lie algebra, Lie superalgebras are from the comprehending and description for supersymmetry of mathematical physics. Unlike the semisimple Lie algebras, understanding the representation theory of Lie superalgebras is a difficult problem. Lie superalgebras graded by root supersystems are Lie superalgebras of great significance. In recent years, the representations of types <I>B</I>(<i>m,n</i>)<i>, C</i>(<i>n</i>)<i>, D</i>(<i>m,n</i>)<i>, P</i>(<i>n</i>) and <I>Q</I>(<i>n</i>)-graded Lie superalgebras coordinatized by quantum tori have been studied. In this paper, we construct fermionic-bosonic representations for a class of <I>A</I>(<I>M</I>-1,<I>N</I>-1)-graded Lie superalgebras coordinatized by quantum tori with nontrivial central extensions. At first, we introduce the background of the research on the graded Lie superalgebras and present some basics on it. Then, a set of bases for <I>A</I>(<I>M</I>-1,<I>N</I>-1)-graded Lie superalgebras and the multiplication operations among them are given specifically to present the construction of the vector space. By using the tensor product of fermionic and bosonic module, the operators and their operation relations are derived. Finally, we obtain a brief and pretty representation theorem of <I>A</I>(<I>M</I>-1,<I>N</I>-1)-graded Lie superalgebras with nontrivial central extensions.