Some Root Graded Lie Algebras And Their Representations

In this thesis, we will construct representations of some BC_N -graded Lie algebras and B(0, N) -graded Lie superalgebras.In 1992, Lie algebras graded by the reduced finite root systems were first introducedby Berman-Moody in order to understand the generalized intersection matrix algebras of Slodowy. Berman-Moody classified Lie algebras graded by the root systems of type A_l ,l≥2, D_l,l≥4 and E₆, E₇, E₈ up to central extensions. BenkartZelmanov classified Lie algebras graded by the root systems of type A₁, B_l, l≥2, C_l,l≥3, F4 and G₂ up to central extensions. Neher gave a different approach for all the (reduced) root systems except E₈,F₄ and G₂. The main idea of root graded Lie algebras can be traced back to Tits and Seligman.In 2000, Allison-Benkart-Gao completed the classification of the above root graded Lie algebras by figuring out explicitly the centers of the universal coverings of those root graded Lie algebras. It turns out that the classification of those root graded Lie algebras played a crucial role in classifying the newly developed extended affine Lie algebras. All affine Kac-Moody Lie algebras except A_2l⁽²⁾ are important examples of Lie algebras graded by reduced finite root systems. To include the twisted affine Lie algebra A_2l⁽²⁾ and for the purpose of the classification of the extended affine Lie algebras of non-reduced types, Allison-Benkart-Gao introduced Lie algebras graded by the non-reduced root system BC_N. BC_N -graded Lie algebras do appear not only in the extended affine Lie algebras (see [1]) including the twisted affine Lie algebra A_2l⁽²⁾ but also in the finite-dimensional isotropic simple Lie algebras studied by Seligman. The other important examples include the "odd symplectic" Lie algebras studied by Gelfand-Zelevinsky, Maliakas and Proctor.In 2002, Benkart and Elduque extended the theory of root-graded Lie algebra to Lie superalgebras and completely determined root-graded Lie superalgebras for all finite dimensional split simple classical superalgebras except P(n), Q(n) up to central extensions. In 2003, Martinez and Zelmanov considered Lie superalgebrs graded by P(n) and Q(n). Those root graded Lie superalgebras are a super-analog of Lie algebras graded by the reduced root system.So far, the structure of root graded Lie algebras and superalgebras is clear, however there is not yet an extensive development of the representation theory of these algebras except in some special cases (notably affine Kac-Moody Lie algebras). In the Chapter 2 and Chapter 3 of this thesis, we shall construct fermions and bosons depending on the parameter q which will lead to representations for some BC_N -graded Lie algebras and 5(0, N)-graded Lie superalgebras coordinatized by quantum tori with nontrivial central extensions. The representations are given in Theorem 2.3, 2.4 and Theorem 3.3.