Generalized Virasoro algebra: left-symmetry and related algebraic and hydrodynamic properties

TL;DR

This paper explores the algebraic structure and properties of a generalized Virasoro algebra, focusing on left-symmetry, geometric connections, and hydrodynamic applications, including derivation of key operators and compatibility conditions.

Contribution

It introduces a generalized Virasoro algebra with conditions for quasi-associativity, links to geometry and hydrodynamics, and develops operators and equations relevant to its structure.

Findings

Identifies conditions for left-symmetry and quasi-associativity.

Establishes links between algebraic properties and geometric/hydrodynamic systems.

Derives operators and compatibility equations for the generalized algebra.

Abstract

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.