Classifications And Representations Of Some Generalized Root Systems Graded Lie Superalgebras

In this thesis, we will classify the generalized P(n)-graded Lie superalgebras, and then we construct representations of some Q(n)-graded and generalized P(n)-graded Lie superalgebras.In 1992, S. Berman and R. V. Moody[11] gave the strict definition of Lie algebras graded by the reduced finite root systems in order to understand the generalized inter-section matrix algebras of P. Slodowy. And then they classified Lie algebras graded by the root systems of type Al, l≥ 2, Dl,l≥ 4 and E6,E7, E8 up to central exten-sions. In 1996, G. Benkart and E. Zelmanov[17] gave the classification of Al, Bl,l≥ 2, Cl, l≥ 3-graded and F4, G2-graded Lie algebras up to central extensions. E. Ne-her[67,68] gave the classifications of Lie algebras graded by the root systems of type Al, Bl,l≥ 2, Cl,l> 3, Dl, l≥ 4, E6, E7 up to central extensions through Jordan alge-bras.In 2000, B. N. Allison, G. Benkart and Y. Gao[2] completed the classification of the above root graded Lie algebras by determined explicitly the centers of the universal coverings of those root graded Lie algebras. The classification of these root graded Lie algebras played a very important role in classifying the extended affine Lie algebras.G. Benkart and A. Elduque[5-7] extended the theory of root graded Lie algebras to Lie superalgebras and classified Lie superalgebras graded by the root systems of type A(m,n), B(m,n),C(n), D(m,n) and D(2,1;α), F4,G(3) up to central extensions. C. Martinez and E. Zelmanov[64] investigated the P(n), Q(n)-graded Lie superalgebras.In the chapter 3 of this thesis, we gave the concept of the generalized P(n)-graded Lie superalgebras, and then classified it. The P(n)-graded Lie superalgebras is a spe-cial case of it. Then we construct fermions and bosons depending on the parameter q which lead to representations for a class of the generalized P(n)-graded Lie superal-gebras. In the chapter 4 of this thesis, we used fermions and bosons to construct the representations for the Q(n)-graded Lie superalgebras.