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In this paper we study the BGG-categories associated to quantum groups. We prove that many properties of the ordinary BGG-category for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for and for finite dimensional -modules we are able to determine...
We study Hom-Lie superalgebras of Heisenberg type. For 3-dimensional Heisenberg Hom-Lie superalgebras we describe their Hom-Lie super structures, compute the cohomology spaces and characterize their infinitesimal deformations.
We prove that the variety of Lie algebras arising from splicing operation coincides with the variety CM of centreby-metabelian Lie algebras. Using these Lie algebras we find the minimal dimension algebras generated the variety CM and the variety of its associative envelope algebras. We study the splicing n-ary operation. We show that all n-ary (n > 2) commutator algebras arising from this operation are nilpotent of index 3. We investigate the generalization of the splicing n-ary operation, and...
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