Lie Bialgebra Structures And Classification Of Simple Weight Modules With Finite-dimensional Weight Spaces Over The Planar Galilean Conformal Algebra

In this paper,we investigate the structure of Lie bialgebra of planar Galilean con-formal algebra and classification of irreducible weight module with a finite-dimensional weight space over the planar Galilean conformal algebra.First we determine the deriva-tion and the first cohomology group of the planar Galilean conformal algebra,and prove that Lie bialgebra(g,[·,·],?)with ?=?r+?0,where r?Im(1-?)and D0 is an outer derivation.Furthermore,A Lie bialgebra over the planar Galilean conformal alge-bra is triangular coboundary if and only if the coproduct is an inner derivation.Second,we prove that the support of an irreducible weight module with an infinite dimensional weight space coincides with the weight lattices and all nontrivial weight spaces of such a module are infinite dimensional.As a byproduct,we obtain that every simple weight planar Galilean conformal algebra module with a finite dimensional weight space,is a Harish-Chandra module.