Ternary q-Virasoro-Witt Hom-Nambu-Lie algebras

TL;DR

This paper introduces a new family of ternary q-Virasoro-Witt algebras that generalize previous structures, demonstrating their Hom-Nambu-Lie properties across various parameter values using algebraic deformation techniques.

Contribution

It constructs ternary q-Virasoro-Witt algebras as Hom-Nambu-Lie algebras for all parameters, extending prior work on ternary Virasoro-Witt algebras.

Findings

The constructed algebras are Hom-Nambu-Lie for all parameter values.

The q-deformation generalizes previous ternary Virasoro-Witt algebras.

Different construction methods show the universal Hom-Nambu-Lie structure.

Abstract

In this paper we construct ternary $q$-Virasoro-Witt algebras which $q$-deform the ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos using $su(1,1)$ enveloping algebra techniques. The ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos depend on a parameter and are not Nambu-Lie algebras for all but finitely many values of this parameter. For the parameter values for which the ternary Virasoro-Witt algebras are Nambu-Lie, the corresponding ternary $q$-Virasoro-Witt algebras constructed in this article are also Hom-Nambu-Lie because they are obtained from the ternary Nambu-Lie algebras using the composition method. For other parameter values this composition method does not yield Hom-Nambu Lie algebra structure for $q$-Virasoro-Witt algebras. We show however, using a different construction, that the ternary Virasoro-Witt algebras of Curtright, Fairlie and Zachos, as well as the general ternary $q$-Virasoro-Witt algebras we construct, carry a structure of ternary Hom-Nambu-Lie algebra for all values of the involved parameters.