BCN-graded Lie algebras arising from fermionic representations

We use fermionic representations to obtain a class of BC N -graded Lie algebras coordinatized by quantum tori with nontrivial central extensions. 0 Introduction Lie algebras graded by the reduced finite root systems were first introduced by Berman-Moody [BM] in order to understand the generalized intersection matrix algebras of Slodowy. [BM] classified Lie algebras graded by the root systems of type Al, l ≥ 2, Dl, l ≥ 4 and E6, E7, E8 up to central extensions. Benkart-Zelmanov [BZ] classified Lie algebras graded by the root systems of type A1, Bl, l ≥ 2, Cl, l ≥ 3, F4 and G2 up to central extensions. Neher [N] gave a different approach for all reduced root systems except E8, F4 and G2. The idea of root graded Lie algebras can be traced back to Tits [T] and Seligman [S]. [ABG1] 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 (see [BGKN] and [AG]). All affine Kac-Moody Lie algebras except A (2) 2l are examples of Lie algebras graded by reduced finite root systems. To include the twisted affine Lie algebra A (2) 2l and for the purpose of the classification of the extended affine Lie algebras of non-reduced types, [ABG2] ∗Research was partially supported by NSERC of Canada and Chinese Academy of Science.