Lie Structure and Finite-Dimensional Simple Poisson Modules of two Poisson Algebras
Article Sidebar
Main Article Content
Abstract
This research classifies all finite-dimensional simple Poisson modules for two distinct Poisson algebras, and , defined over the complex field . The Poisson brackets are explicitly defined as: and , . Applying Jordan’s classification framework (2010), we identify the Poisson maximal ideals, , for both and , and analyze the structure of the associated Lie algebras, We establish that Poisson algebras, and , admit exactly two Poisson maximal ideals, which are and . Crucially, we prove that for all four cases, the associated Lie algebras are three-dimensional and isomorphic to the special linear algebra . Consequently, utilizing the well-known representation theory of we show that the finite-dimensional simple Poisson modules of both and consist of two infinite families. For every dimension there exists exactly one dimensional simple Poisson module (up to isomorphism) annihilated by and one by .
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
The Journal of Science and Science Education (JSSE) retain the right of all articles published in JSSE. The coresponding author or the authorized person on behalf of the authors must send the complete Copyright Transfer Form to JSSE before any article get published in JSSE.
Copyright Transfer Form
The JSSE request the coresponding author or the authorized person on behalf of the authors upload the manuscript under the together with the Copyright Transfer Form under the supplementary data. The guidline for uploading both manuscript and Copyright Transfer Form is shown below:
1. Upload the manuscript in the sub-menu, Article Component > Article Text.
2. Upload the the Copyright Transfer Form in the sub-menu, Article Component > Other.
Download Copyright Transfer Form
References
Changtong, K. and Sasom, N. (2018). Poisson Maximal ideals and the Finite-dimensional Simple Poisson Modules of a Certain Poisson Algebra. KKU Science Journal, 46(3), 606-613.
Chansuriya, N. and Fongchanta, S. (2022). Finite Dimensional Simple Poisson Modules of a Poisson Algebra A. The Journal of Applied Science and Emerging Technology, 21(1), 244580.
Chansuriya, N. and Sasom, N. (2021). Simple Poisson modules over a Poisson Algebra S_q.Thai Journal of Mathematics, 19(3), 1134-1143.
Chen, W. (2019). Simple Poisson Modules over Certain Polynomial Poisson Algebras. International Journal of Algebra, 13(2), 83-96.
Dixmier, J. (1996). Enveloping Algebras. American Mathematical Society.
Erdmann, K. and Wildon, M. J. (2006). Introduction to Lie Algebras. London: Springer.
Farkas, D. R. (2000). Modules for Poisson algebras, Communications in Algebra, 28(7), 3293-3306.
Hall, B. (2015). Lie groups, Lie Algebras and Lie Algebras and Representations: An Elementary Introduction (2nd Ed). Cham: Springer.
Humphreys, J. E. (1972). Introduction to Lie algebras and Representation Theory. New York: Springer.
Jordan D. A. (2010). Finite-dimensional simple Poisson modules. Algebras and Representation Theory, 13(1), 79-101.
Oh, S. O. (1999). Poisson enveloping algebras. Communications in Algebra, 27(5), 2181–2186.
Sasom, N. and Changtong, K. (2023). On finite-dimensional Simple Poisson module on a certain Poisson algebra by using the specific Lie structure method on Poisson maximal ideals, Journal of Science and Science Education, 6(1), 40-49.
Sasom, N. (2006). Reversible skew Laurent polynomial rings, rings of invariants and related rings. Doctoral Dissertation, Sheffield: University of Sheffield.
Sharp, R. Y. (2000). Steps in Commutative Algebra (2nd Ed). Cambridge University Press.
Xia, L. and Lie, D. (2022). Finite Dimensional Simple Modules over Some GIM Lie Algebras, Mathematics, 10(15), 2658
