Kupershmidt-Nijenhuis structures on pre-Malcev algebras

TL;DR

This paper explores advanced algebraic structures called Kupershmidt-Nijenhuis structures on pre-Malcev algebras, introducing new algebra families and analyzing their geometric properties using computational methods.

Contribution

It introduces new families of complex pre-Malcev algebras beyond pre-Lie algebras and connects various operators through compatibility conditions.

Findings

Constructed new complex pre-Malcev algebra families in low dimensions.

Established relationships between Kupershmidt, Nijenhuis, and Kupershmidt-Nijenhuis structures.

Characterized geometric structures of operator varieties using computational ideal theory.

Abstract

We study Kupershmidt operators, Nijenhuis operators, and Kupershmidt-Nijenhuis structures on finite-dimensional pre-Malcev algebras over a field of characteristic zero. We construct several new families of complex pre-Malcev algebras that are not pre-Lie algebras in dimensions two, three and four. We use the compatibility of linear operators to establish connections between Kupershmidt operators, Nijenhuis operators, and Kupershmidt-Nijenhuis structures on pre-Malcev algebras. Moreover, we use a method from computational ideal theory to characterize the geometric structures of the varieties of Kupershmidt operators and Nijenhuis operators on a three-dimensional complex pre-Malcev algebra.