Research ArticleNo Access

Defining relations and Gröbner–Shirshov bases of Poisson algebras as of conformal modules

P. S. Kolesnikov

https://orcid.org/0000-0002-7534-1534

Sobolev Institute of Mathematics, Akad. Koptyug prosp., 4, Novosibirsk 630090, Russia

E-mail Address: pavelsk@math.nsc.ru

Corresponding author.

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and

A. S. Panasenko

Novosibirsk State University, Pirogova str., 1, Novosibirsk 630090, Russia

E-mail Address: a.panasenko@g.nsu.ru

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https://doi.org/10.1142/S0219498822501389Cited by:0 (Source: Crossref)

Abstract

We study the relation between Poisson algebras and representations of Lie conformal algebras. We establish a Gröbner–Shirshov basis theory framework for modules over associative conformal algebras and apply this technique to Poisson algebras considered as conformal modules over appropriate associative conformal envelopes of current Lie conformal algebras. As a result, we obtain a setting for the calculation of a Gröbner–Shirshov basis in a Poisson algebra.

Communicated by S.-J. Cheng

Keywords:

AMSC: 17B63, 17B69, 17A61

References

1. B. Bakalov and V. G. Kac , Field algebras, Int. Math. Res. Not. 3 (2003) 123–159. Crossref, Google Scholar
2. L. A. Bokut, Y. Q. Chen and Z. Zhang , Some algorithmic problems for Poisson algebras, J. Algebra 525 (2019) 562–588. Crossref, Web of Science, Google Scholar
3. Y. Chen and L. Ni , Gröbner–Shirshov bases for associative conformal modules, in Proc. 3rd Int. Congress in Algebra and Combinatorics, eds. K. P. Shum et al., New Trends in Algebra and Combinatorics (World Scientific, River Edge, NJ, 2020), pp. 408–446. Link, Google Scholar
4. S.-J. Cheng and V. G. Kac , Conformal modules, Asian J. Math. 1 (1997) 181–193. Crossref, Google Scholar
5. V. G. Kac , Vertex Algebras for Beginners, 2nd edn., University Lecture Series, Vol. 10 (American Mathematical Society, Providence, RI, 1998). Crossref, Google Scholar
6. S.-J. Kang and K.-H. Lee , Gröbner–Shirshov bases for representation theory, J. Korean Math. Soc. 37 (2000) 55–72. Google Scholar
7. P. S. Kolesnikov , Gröbner–Shirshov bases for associative conformal algebras with arbitrary locality function, in Proc. 3rd Int. Congress in Algebra and Combinatorics, eds. K. P. Shum et al., New Trends in Algebra and Combinatorics (World Scientific, River Edge, NJ, 2020), pp. 255–267. Link, Google Scholar
8. P. S. Kolesnikov , Universal enveloping Poisson conformal algebras, Internat. J. Algebra Comput. 30(5) (2020) 1015–1034. Link, Web of Science, Google Scholar
9. P. S. Kolesnikov and R. A. Kozlov, Standard bases for the universal associative conformal envelopes of Kac–Moody conformal algebras, preprint (2020), arXiv:2009.12062. Google Scholar
10. M. Roitman , On free conformal and vertex algebras, J. Algebra 217 (1999) 496–527. Crossref, Web of Science, Google Scholar
11. M. Roitman , Universal enveloping conformal algebras, Sel. Math. (N.S.) 6 (2000) 319–345. Crossref, Google Scholar