Hounkonnou, Mahouton NorbertGuha, ParthaRatiu, Tudor2016-07-192016-07-192016-07-19201610.1080/14029251.2016.1135642https://infoscience.epfl.ch/handle/20.500.14299/128014WOS:000373068200004Motivated by the work of Kupershmidt (J. Nonlin. Math. Phys. 6 (1998), 222 -245) we discuss the occurrence of left symmetry in a generalized Virasoro algebra. The multiplication rule is defined, which is necessary and sufficient for this algebra to be quasi-associative. Its link to geometry and nonlinear systems of hydrodynamic type is also recalled. Further, the criteria of skew-symmetry, derivation and Jacobi identity making this algebra into a Lie algebra are derived. The coboundary operators are defined and discussed. We deduce the hereditary operator and its generalization to the corresponding 3 ary bracket. Further, we derive the so-called rho compatibility equation and perform a phase-space extension. Finally, concrete relevant particular cases are investigated.Virasoro algebraLeft-symmetric algebrasQuasi-associativityCoboundary operatorsNonlinear systems of hydrodynamic typetext::journal::journal article::research article