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

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 paper 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 a 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 the ternary Hom–Nambu–Lie algebra for all values of the involved parameters.