  Motivated 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.
