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Abstract
The paper is devoted to the study of graded-simple modules and gradings on simple modules over finite-dimensional simple Lie algebras. In general, a connection between these two objects is given by the so-called loop construction. We review the main features of this construction as well as necessary and sufficient conditions under which finite-dimensional simple modules can be graded. Over the Lie algebra sl2(C), we consider specific gradings on simple modules of arbitrary dimension.
Highlights
We start this paper by reviewing the criteria of [12, 13, 8] for the existence of a compatible grading on a finite-dimensional simple module V
We focus on the case L = sl2(C), where we give explicit gradings for those V that admit them. After this we switch to infinite-dimensional simple sl2(C)-modules
We turn our attention to reviewing the main results of [14]. Therein, it is described how the so-called loop construction could be used for the classification of graded-simple modules of arbitrary dimension
Summary
We restrict ourselves to the case where L is a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero and focus on simple L-modules. We start this paper by reviewing the criteria of [12, 13, 8] for the existence of a compatible grading on a finite-dimensional simple module V. We focus on the case L = sl2(C), where we give explicit gradings for those V that admit them After this we switch to infinite-dimensional simple sl2(C)-modules. We turn our attention to reviewing the main results of [14] Therein, it is described how the so-called loop construction could be used for the classification of graded-simple modules of arbitrary dimension.
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