On the Special Identities of Gelfand-Dorfman Algebras

Kolesnikov P. S., Sartayev B. K.

EXPERIMENTAL MATHEMATICS, 2022 (SCI-Expanded) identifier identifier identifier

  • Publication Type: Article / Article
  • Publication Date: 2022
  • Doi Number: 10.1080/10586458.2022.2041134
  • Journal Name: EXPERIMENTAL MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Keywords: Gelfand-Dorfman algebra, Poisson algebra, special identity, Grobner basis, operad, BASES
  • Süleyman Demirel University Affiliated: No

Abstract

A Gelfand-Dorfman algebra (GD-algebra) is said to be special if it can be embedded into a differential Poisson algebra. In this paper, we prove that the class of all special GD-algebras is closed with respect to homomorphisms and thus forms a variety. We describe a technique for finding potentially all special identities of GD-algebras and derive two known special identities of degree 4 in this way. By computing the Grobner basis for the corresponding shuffle operad, we show that these two identities imply all special ones up to degree 5.