Some Research On Linear Poisson Lie2-groups

Lie2-groups and Lie2-algebras are the results of“categorification”respectively from Lie groups and Lie algebras. A strict Lie2-group is astrict monoidal category, in which every object and every morphism has itsinverse. And there is a1-1correspondence between Lie2-groups and Liegroup crossed modules. A Lie2-algebra has a2-vector space which is aninternal category in Vect, as its underlying structure. To get a Lie2-algebra,we equip a2-vector space with a skew-symmetric bilinear functor, and thisfunctor satisfies the Jacobi identity up to a natural transformation satisfyingcertain conditions. A strict Lie2-algebra is the case when the Jacobi identityholds strictly. According to the equivalence between Lie2-algebras and2-term DGLA, we could also describe the structure of a strict Lie2-algebrain detail. In the end we give a proof for the correspondence between Lie2-algebra structures and Lie Poisson Structures, and thus reach the conclusionthat there is a1-1correspondence between Lie2-algebras and linear PoissonLie2-groups.