Hom-lie2-Algebras

In this thesis, we introduce the notion of hom-Lie2-algebras. We givethe definition of hom-Lie2-algebras, study the properties and give twoexamples of hom-Lie2-algebras: skeletal hom-Lie2-algebras and stricthom-Lie2-algebras.First, we introduce some basic notions about hom-Lie algebras andHL_∞-algebras, including the homotopy theory of hom-Lie algebras and2-vecter spaces.We introduce the notions of hom-Lie2-algebras, which is the categori-fication of hom-Lie algebras, HL_∞-algebras, which is the hom-analogue ofHL_∞-algebras. We prove that the category of hom-Lie2-algebras and thecategory of2-term HL_∞-algebras are equivalent.We give the classification of skeletal hom-Lie2-algebras by the3-hom cocycle and construct skeletal hom-Lie2-algebras from quadraticinvolutive hom-Lie algebras. Then, we give the hom-analogues of thestring Lie2-algebras.At last, we prove that there exists a one to one correspondence betweenstrict hom-Lie2-algebras and crossed modules of hom-Lie algebras. Wegive the construction of strict hom-Lie2-algebras from hom-left-symmetricalgebras and symplectic hom-Lie algebras.