Abstract
A Lie algebra \(\mathfrak{g}\) is considered generalized reductive if it is a direct sum of a semisimple Lie algebra and a commutative radical. This paper extends the BGG category \({\cal O}\) over complex semisimple Lie algebras to the category \({{\cal O}^\prime }\) over complex generalized reductive Lie algebras. Then, we preliminarily research the highest weight modules and the projective modules in this new category \({{\cal O}^\prime }\), and generalize some conclusions for the classical case. Also, we investigate the associated varieties with respect to the irreducible modules in \({{\cal O}^\prime }\) and obtain a result that extends Joseph’s result on the associated varieties for reductive Lie algebras. Finally, we study the center of the universal envelo** algebra U(\(\mathfrak{g}\)) and independently provide a new proof of a theorem by Ou–Shu–Yao for the center in the case of enhanced reductive Lie algebras.
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Acknowledgements
The author would like to thank Professors B. Shu and V. Mazorchuk for their helpful comments and suggestions. Also my deep thanks are given to the referee for helpful comments. This work is partially supported by NFSC (No. 12071136).
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Ren, Y. The BGG Category for Generalized Reductive Lie Algebras. Front. Math 19, 127–142 (2024). https://doi.org/10.1007/s11464-021-0352-8
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DOI: https://doi.org/10.1007/s11464-021-0352-8
Keywords
- Generalized reductive Lie algebras
- BGG category
- irreducible modules
- projective modules
- associated varieties
- center