On the Special Identities of Gelfand–Dorfman Algebras

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 Gröbner basis for the corresponding shuffle operad, we show that these two identities imply all special ones up to degree 5.

Keywords:

• Gelfand–Dorfman algebra
• Poisson algebra
• special identity
• Gröbner basis
• operad

Additional information

Funding

This work was supported by the Program of fundamental scientific researches of the Siberian Branch of Russian Academy of Sciences, I.1.1, project 0314-2019-0001. The second author was supported by grant AP08052405 of Ministry of Education and Science of Republic of Kazakhstan