Hom-Lie Algebra Structures of the n-th Schr?dinger Algebra
A Hom-structure on a Lie algebra(L,[·])is a linear map φ:L → L which satisfies the Hom-Jacobi identity[[x,y],φ(z)]+[[z,x],φ(y)]+[[y,z],φ(x)]=0 for any x,y,z ∈ L.A Hom-structure is called regular(respectively,a derivation double Lie algebra)if φ is also a Lie algebra isomorphism(respectively,derivation).The n-th Schrödinger algebra is the semi-direct product of the simple Lie algebra(s)(i)2 with the n-th Heisenberg Lie algebra(o)n.In this paper,we prove that any Hom-Lie algebra structure is a sum of a scalar multiplication and a central Hom-structure.Furthermore,any regular Hom-structure is an identity mapping,and any derivation double Lie algebra is a zero mapping.
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