Graded modules over the simple Lie algebra sl2(C)

Shihadeh, Abdallah A. (2019) Graded modules over the simple Lie algebra sl2(C). Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

This thesis provides a contribution to the area of group gradings on the simple modules over simple Lie algebras. A complete classification of gradings on finite-dimensional simple modules over arbitrary finite-dimensional simple Lie algebras over algebraically closed fields of characteristic zero, in terms of graded Brauer groups, has been recently given in the papers of A. Elduque and M. Kochetov. Here we concentrate on infinite-dimensional modules. A complete classification by R. Block of all simple modules over a simple Lie algebra is known only in the case of sl2(C). Thus, we restrict ourselves to the gradings on simple sl2(C)-modules. We first give a full description for the Z- and Z₂² -gradings of all weight modules over sl₂(C). Then we show that Z-gradings do not exist on any torsion-free sl₂(C)-modules of finite rank. After this, we treat Z₂² -gradings on torsion-free modules of various ranks. A construction for these modules was given by V. Bavula, and J. Nilson gave a classification of the torsion-free sl₂(C)-modules of rank 1. After giving some, mostly negative, results about the gradings on these latter modules, we construct the first family of simple Z₂² -graded sl₂(C)-modules (of rank 2). We also construct a family of graded-simple torsion-free modules of rank 2. For each of the modules in these families, we give a complete description of their tensor products with simple graded finite-dimensional modules. we restrict ourselves to the gradings on simple sl₂(C)-modules. We first give a full description for the Z- and Z₂² -gradings of all weight modules over sl₂(C). Then we show that Z-gradings do not exist on any torsion-free sl₂(C)-modules of finite rank. After this, we treat Z₂² -gradings on torsion-free modules of various ranks. A construction for these modules was given by V. Bavula, and J. Nilson gave a classification of the torsion-free sl₂(C)-modules of rank 1. After giving some, mostly negative, results about the gradings on these latter modules, we construct the first family of simple Z₂² -graded sl₂(C)-modules (of rank 2). We also construct a family of graded-simple torsion-free modules of rank 2. For each of the modules in these families, we give a complete description of their tensor products with simple graded finite-dimensional modules.

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