Second cohomology groups and left-symmetric algebraic structures of the generalized loop Heisenberg-Virasoro algebra

Abstract

This is the second paper in our series of papers dedicated to the study of the generalized loop Heisenberg-Virasoro algebra. The first paper is dedicated to the study of maps on the generalized loop Heisenberg-Virasoro algebra, including derivations, $2$-local derivations, biderivations the automorphism groups. The present paper is dedicated to the study of the second cohomology groups and left-symmetric algebraic structures on the generalized loop Heisenberg-Virasoro algebra. We describe the second cohomology group and left-symmetric algebra structures on the generalized loop Heisenberg-Virasoro algebra by use of the Witt algebras being Lie subalgebra of the generalized loop Heisenberg-Virasoro algebra up to isomorphism.