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

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 ρ–compatibility equation and perform a phase-space extension. Finally, concrete relevant particular cases are investigated.
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Mahouton Norbert Hounkonnou, Partha Guha, Tudor Ratiu. Generalized Virasoro algebra: left-symmetry and related algebraic and hydrodynamic properties. Journal of Nonlinear Mathematical Physics, Taylor & Francis, 2016, 23 (1), pp.47-73 ⟨10.1080/14029251.2016.1135642⟩. ⟨hal-01391404⟩

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