Abstract
Split Poisson algebras are one of the most known examples of graded Poisson algebras. Since an important category in the class of graded algebras is the one of strongly graded algebras, we introduce in a natural way the category of strongly split Poisson algebras and show that if \((\mathfrak{P},\{\cdot,\cdot \})\) is a centerless strongly split Poisson algebra, then \(\mathfrak{P}\) is the direct sum of split-ideals, each one being a split-simple strongly split Poisson algebra. In case of being \(\mathfrak{P}\) infinite dimensional and locally finite, we also show that if \((\mathfrak{P},\{\cdot,\cdot \})\) is furthermore simple then it is the direct limit of finite dimensional simple (strongly) split Poisson algebras.
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Acknowledgements
The authors would like to thank the referees for their reviews of the paper as well as for the suggestions which have helped to improve the work. A.J. Calderón Martín was supported by the PCI of the UCA ‘Teoría de Lie y Teoría de Espacios de Banach’, by the PAI with project numbers FQM298, FQM7156 and by the project of the Spanish Ministerio de Educación y Ciencia MTM2010-15223.
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Martı́n, A.J.C., Cheikh, D.M. (2016). Strongly Split Poisson Algebras. In: Gueye, C., Molina, M. (eds) Non-Associative and Non-Commutative Algebra and Operator Theory. Springer Proceedings in Mathematics & Statistics, vol 160. Springer, Cham. https://doi.org/10.1007/978-3-319-32902-4_11
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DOI: https://doi.org/10.1007/978-3-319-32902-4_11
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