Four Z + -bigraded complexes with the action of the exceptional infinitedimensional Lie superalgebra E(3, 6) are constructed. We show that all the images and cokernels and all but three kernels of the differentials are irreducible E(3, 6)-modules. This is based on the list of singular vectors and the calculation of homology of these complexes. As a result, we obtain an explicit construction of all degenerate irreducible E(3, 6)-modules and compute their characters and sizes. Since the group of symmetries of the Standard Model SU (3) × SU (2) × U(1) (divided by a central subgroup of order six) is a maximal compact subgroup of the group of automorphisms of E(3, 6), our results may have applications to particle physics.
IntroductionIt has been established by A. Rudakov [R] some 25 years ago that all degenerate irreducible continuous modules over the Lie algebra W n of all formal vector fields in n indeterminates occur as kernels and as cokernels of the differential of the Z + -graded de Rham complex n of all formal differential forms in n indeterminates.The main objective of the present paper is to obtain a similar result for the exceptional infinite-dimensional Lie superalgebra E(3, 6) from the list of simple linearly compact Lie superalgebras classified by V. Kac [K]. It turned out that the situation is much more interesting: we have constructed four Z + -bigraded complexes with the action of E(3, 6) and certain connecting homomorphisms between these complexes. All images and cokernels and all but three kernels of differentials turn out to be irreducible, and all degenerate irreducible E(3, 6)-modules occur among them. The failure of irreducibility is connected to non-triviality of homology of these complexes, which we compute as well. At the end of the paper we compute the characters and sizes of all degenerate irreducible E(3, 6)-modules and speculate on their relation to the Standard Model.
We propose the notion of stability on a triangulated category that is a generalization of the T. Bridgeland's stability data. We establish connections between stabilities and t-structures on a category and as application we get the classification of bounded tstructures on the category D b (Coh P 1 ).
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