Abstract
We describe the centers of the universal enveloping algebras of nilpotent Lie algebras of dimension at most six over fields of prime characteristic. If the characteristic is not smaller than the nilpontency class, then the center is the integral closure of the algebra generated over the p-center by the same generators that also occur in characteristic zero. Except for three examples, two of which are standard filiform, this algebra is already integrally closed and hence it coincides with the center. In the case of these three exceptional algebras, the center has further generators. Then we show that the center of the universal enveloping algebra of the algebras investigated in this paper is isomorphic to the Poisson center (the algebra of invariants under the adjoint representation). This shows that Braun’s conjecture is valid for this class of Lie algebras.
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Acknowledgements
Vanderlei Lopes de Jesus was financially supported by a PhD scholarship awarded by CNPq (Brazil). Csaba Schneider acknowledges the financial support of the CNPq projects Produtividade em Pesquisa (project no.: 308212/2019-3) and Universal (project no.: 421624/2018-3). We thank Lucas Calixto, Letterio Gatto, Tiago Macedo, and Renato Vidal Martins for their valuable comments. We also thank the anonymous referee for his or her useful suggestions.
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de Jesus, V.L., Schneider, C. The center of the universal enveloping algebras of small-dimensional nilpotent Lie algebras in prime characteristic. Beitr Algebra Geom 64, 243–266 (2023). https://doi.org/10.1007/s13366-022-00631-5
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DOI: https://doi.org/10.1007/s13366-022-00631-5
Keywords
- Lie algebras
- Nilpotent Lie algebras
- Universal enveloping algebras
- Center
- Poisson center
- Invariant ring
- Poincaré–Birkhoff–Witt Theorem