ar X iv : m at h - ph / 0 11 20 22 v 1 1 2 D ec 2 00 1 COMPLEXES OF MODULES OVER EXCEPTIONAL LIE

In this paper complexes of generalized Verma modules over the infinitedimensional exceptional Lie superalgebras E(3, 8) and E(5, 10) are constructed and studied. 0. Introduction. In our papers [KR1]–[KR3] we constructed all degenerate irreducible modules over the exceptional Lie superalgebra E(3, 6). In the present paper we apply the same method to the exceptional Lie superalgebras E(3, 8) and E(5, 10). The Lie superalgebra E(3, 8) is strikingly similar to E(3, 6). In particular, as in the case of E(3, 6), the maximal compact subgroup of the group of automorphisms of E(3, 8) is isomorphic to the group of symmetries of the Standard Model. However, as the computer calculations by Joris van Jeugt show, the fundamental particle contents in the E(3, 8) case is completely different from that in the E(3, 6) case [KR2]. All the nice features of the latter case, like the CPT symmetry, completely disappear in the former case. We believe that the main reason behind this is that, unlike E(3, 6), E(3, 8) cannot be embedded in E(5, 10), which, we believe, is the algebra of symmetries of the SU5 Grand Unified Model (the maximal compact subgroup of the automorphism group of E(5, 10) is SU5). However, similarity with E(3, 6) allows us to apply to E(3, 8) all the arguments from [KR2] almost verbatim, and Figure 1 of the present paper that depicts all degenerate E(3, 8)modules is almost the same as Figure 3 from [KR2] for E(3, 6). The picture in the E(5, 10) case is quite different (see Figure 2). We believe that it depicts all degenerate irreducible E(5, 10)-modules, but we still do not have a proof. 1. Morphisms between generalized Verma modules. Let L = ⊕j∈Zgj be a Z-graded Lie superalgebra by finite-dimensional vector spaces. Let L− = ⊕j<0 gj , L+ = ⊕j>0 gj , L0 = g0 + L+ . Given a g0-module V , we extend it to a L0-module by letting L+ act trivially, and define the induced L-module M(V ) = U(L)⊗U(L0) V . If V is a finite-dimensional irreducible g-module, the L-module M(V ) is called a generalized Verma module (associated to V ), and it is called degenerate if it is not irreducible. ∗ Supported in part by NSF grant DMS-9970007.