Second cohomology groups and left-symmetric algebraic structures of the generalized loop Heisenberg-Virasoro algebra
Qingyan Ren, Liming Tang
TL;DR
The paper determines the second cohomology $H^2(\mathcal{L}(\Gamma),\mathbb{C})$ of the generalized loop Heisenberg-Virasoro algebra and analyzes compatible left-symmetric (pre-Lie) algebra structures under natural grading. Using a standard cocycle reduction and analyses based on the Witt subalgebra, it classifies $H^2$ into three families of cocycles with explicit representatives $\phi_{k,1},\ \phi_{k,2},\ \phi_{k,3}$ and provides the full product decomposition $H^2(\mathcal{L}(\Gamma),\mathbb{C})=\prod_{k\in\mathbb{Z},x\in\{1,2,3\}}\mathbb{C}\overline{\phi}_{k,x}$. For left-symmetric structures, the authors derive a graded, compatible pre-Lie product on $\mathcal{L}(\Gamma)$ completely determined by structure functions, culminating in explicit formulas for $a,b,c,d,e$—with $d=e=0$—and parameterized by $(\epsilon,m)$, producing the graded generalized loop Heisenberg-Virasoro left-symmetric algebra. The results provide a complete description of central extensions and graded pre-Lie structures on this infinite-dimensional family, facilitating representations and cohomological applications in related loop algebras.
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.