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

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 B(m,n), C(n), D(m,n), P(n) and Q(n)-graded Lie superalgebras coordinatized by quantum tori have been studied. In this paper, we construct fermionic-bosonic representations for a class of A(M-1,N-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 A(M-1,N-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 A(M-1,N-1)-graded Lie superalgebras with nontrivial central extensions.


Introduction
At the end of the 19th century, Lie proposed the concept of continuous transformation group when he studied the invariance of differential equations. In memory of his contribution, futurity call it Lie group. Later, mathematicians considered its linearization at the infinitesimal level, and thus obtained a structure of infinitesimal group, which is called Lie algebra. In the 1970s, the development of bosons and fermions theory in physics prompted people to propose supersymmetry. Therefore, people introduced Lie superalgebra structure to characterize it.
Lie superalgebras are Lie algebras with 2 Z -graded structure [1], which are natural generalizations of Lie algebras. Lie superalgebra theory is more complex than Lie algebra theory. On the one hand, Lie superalgebra theory is closely related to quantum field theory, statistical mechanics and string theory, so scholars in various fields need to have a profound physical background to study Lie superalgebra; on the other hand, close contact between Lie superalgebra and many branches of mathematics, such as quantum supergroup, differential geometry, topology and algebraic groups, etc.
In 1992, Berman and Moody [2] gave the concept of Lie algebras graded by finite root systems so as to comprehend the notion of generalized intersection matrix algebras proposed by Slodowy. Since then, under the research of many scholars, complete classification of all graded Lie algebras has been realized [3][4][5]. Graded Lie superalgebras are natural extension of graded Lie algebras. Unlike Lie algebras, even for finite dimensional Lie superalgebras has no strict root theory. Since entering the 21st century, Benkart and Elduque have given some concepts of classical finite dimensional graded Lie superalgebras based on the root supersystems of classical Lie superalgebras given by Kac [6], and have classified types A(m,n), B(m,n), C(n), D(m,n), D (2,1; α), F(4) and G(3)-graded Lie superalgebras [7][8][9]. Gao [10] gave the fermionic and bosonic representations of the high dimensional affine Lie algebra ( ) N q gl ℂ with quantum torus q ℂ of two variables as coordinate algebra. Later, Chen-Gao [12,13] structured representations for N BC -graded Lie algebras and (0, ) B N -graded Lie superalgebras coordinatized by quantum tori. Cheng established the bosonic and fermionic representations for a kind of generalized B(m,n), C(n), D(m,n), P(N), Q(N)-graded Lie superalgebras with the quantum tori as the coordinate algebra [14][15][16]. At present, the research of Lie superalgebra is still limited, and many works are far from complete.
In the letter, by constructing fermionic-bosonic representations, one obtains Notations: In this paper, we assume that F is a field of characteristic zero. One denotes ℤ as the set of integers and * ℕ as the set of positive integers.

Preliminaries
In this section, we recall some basics. First, we review the basic concepts of Lie superalgebras. Then we introduce Lie superalgebras ( 1, 1) A M N − − and the definition of its graded Lie superalgebras. Finally, we introduce the quantum tori and the matrix Lie superalgebras with the quantum tori as the coordinate algebra.

Notions of Lie Superalgebras
Definition 1. Vector spaceV is a 2 Z -graded vector space if V is the direct sum of two vector spaces, i.e. Remark 2. Let ( , )( ) gl m n F (or simply as ( , ) gl m n ) be a set of all block matrices over field F with the following form: where A and D are matrices over F with order m m × and n n × , respectively.
x y x y xy yx x y gl m n = − − ∈ is a Lie superalgebra. And we usually call it the general linear Lie superalgebra.

Lie Superalgebras
Now we present some basics on Lie superalgebras sl m n is a subsuperalgebra of ( , ) gl m n , which is ordinarily named the special linear Lie superalgebra.
− is as follows: Next, we present an important concept in this paper. Definition 3. (see [8]) A Lie superalgebra L over F is graded by the root system ∆ of ( 1,

The Matrix Lie Superalgebras with the Quantum Tori
From [17], we are aware that quantum tori related to x y ± ± given by x y x y x y q We have a Lie superalgebra Then following the central extensions of general linear Lie algebras ( ) where , , , m n s t ∈ ℤ ; 2 , ( , ) , ;

A(M-1,N-1)-Graded Lie Superalgebras
In this section, we first present the construction of ( 1, − -graded Lie superalgebras with the quantum tori as the coordinate algebra. And then we construct the representation of ( 1, − -graded Lie superalgebras by using bosons and fermions.

The Construction of A(M-1,N-1)-Graded Lie Superalgebras
In this subsection, we study the structure of ( 1, − -graded Lie superalgebras with coordinate algebra as quantum tori. We set We have the following root space decomposition: with the Lie superbracket as (1).
Now we give the multiplication of Lie superalgebra G specifically.
two cases will be discussed as follows.   (6) and (11)

Conclusion
In this paper, we studied the representation theory of ( 1, 1) A M N − − -graded Lie superalgebras with coordinate algebra as quantum tori. We first gave the structure of ( 1, 1) A M N − − -graded Lie superalgebras concretely, i.e., a set of bases and their multiplication operations. And finally, we constructed the representations by using the tensor product module which are extremely concise and pretty results.