TY - JOUR

T1 - Gröbner–Shirshov basis theory and word problems for metatrivial Poisson algebras

AU - Bokut, Leonid A.

AU - Chen, Yuqun

AU - Zhang, Zerui

N1 - Supported by the RAS Fundamental Research Program (Project FWNF–2022–0002). Supported by the NNSF of China (11571121, 12071156). Supported by the NNSF of China (12101248) and by the China Postdoctoral Science Foundation (2021M691099). Публикация для корректировки.

PY - 2023

Y1 - 2023

N2 - Metatrivial algebras (i.e., extensions of trivial algebras by trivial algebras) were studied by Agore and Militaru (J Algebra 426:1–31, 2015) for Leibniz algebras and by Militaru (Bull Malays Math Sci Soc 40:1639–1651, 2017) for associative algebras under names “metabelian" Leibniz and “metabelian" associative algebras respectively. We study metatrivial Poisson algebras. We first prove an analogy of Itô’s theorem for Poisson algebras. Then we construct a linear basis for a free metatrivial Poisson algebra. It turns out that such a linear basis depends on the characteristic of the underlying field. Finally, we elaborate the method of Gröbner–Shirshov basis for metatrivial Poisson algebras and show that the word problem for an arbitrary finitely presented metatrivial Poisson algebra is solvable. As a byproduct, we give a method of recognizing automorphisms of a finitely generated free metatrivial Poisson algebra.

AB - Metatrivial algebras (i.e., extensions of trivial algebras by trivial algebras) were studied by Agore and Militaru (J Algebra 426:1–31, 2015) for Leibniz algebras and by Militaru (Bull Malays Math Sci Soc 40:1639–1651, 2017) for associative algebras under names “metabelian" Leibniz and “metabelian" associative algebras respectively. We study metatrivial Poisson algebras. We first prove an analogy of Itô’s theorem for Poisson algebras. Then we construct a linear basis for a free metatrivial Poisson algebra. It turns out that such a linear basis depends on the characteristic of the underlying field. Finally, we elaborate the method of Gröbner–Shirshov basis for metatrivial Poisson algebras and show that the word problem for an arbitrary finitely presented metatrivial Poisson algebra is solvable. As a byproduct, we give a method of recognizing automorphisms of a finitely generated free metatrivial Poisson algebra.

KW - Gröbner–Shirshov basis

KW - Metatrivial Poisson algebra

KW - Word problem

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85176129507&origin=inward&txGid=05f9d542916ce51d1ea3e79c3f5553e1

UR - https://www.mendeley.com/catalogue/f3581cc5-cd29-3998-804f-92ccf9225635/

U2 - 10.1007/s40863-023-00386-4

DO - 10.1007/s40863-023-00386-4

M3 - Article

JO - Sao Paulo Journal of Mathematical Sciences

JF - Sao Paulo Journal of Mathematical Sciences

SN - 2316-9028

ER -