A commutative algebra approach to multiplicative Hom-Lie algebras. Issue 7 (3rd May 2023) - Chen, Yin; Zhang, Runxuan
Abstract : Let g be a finite-dimensional complex Lie algebra and HLie m ( g ) be the affine variety of all multiplicative Hom-Lie algebras on g . We use a method of computational ideal theory to describe HLie m ( g l n ( C ) ), showing that HLie m ( g l 2 ( C ) ) consists of two 1-dimensional and one 3-dimensional irreducible components and HLie m ( g l n ( C ) ) = { diag { δ, …, δ, a } ∣ δ = 1 or 0, a ∈ C } for n ⩾ 3 . We construct a new family of multiplicative Hom-Lie algebras on the Heisenberg Lie algebra h 2 n + 1 ( C ) and characterize the affine varieties HLie m ( u 2 ( C ) ) and HLie m ( u 3 ( C ) ) . We also study the derivation algebra Der D ( g ) of a multiplicative Hom-Lie algebra D on g and, under some hypotheses on D, we prove that the Hilbert series H ( Der D ( g ), t ) is a rational function.