Abstract
In this paper, first we establish the explicit relation between 3-derivations and 3- automorphisms of a Lie algebra using the differential and exponential map. More precisely, we show that the Lie algebra of 3-derivations is the Lie algebra of the Lie group of 3-automorphisms. Then we study the derivations and automorphisms of the standard embedding Lie algebra of a Lie triple system. We prove that derivations and automorphisms of a Lie triple system give rise to derivations and automorphisms of the corresponding standard embedding Lie algebra. Finally we compute the 3-derivations and 3-automorphisms of 3-dimensional real Lie algebras.
Highlights
By using the differential and exponential map, we prove that the exponential of a 3-derivation is a 3-automorphism
We prove that for any Lie algebra g, the Lie algebra of 3-derivations is the Lie algebra of the Lie group of 3-automorphisms
We study 3-derivations and 3-automorphisms on Lie algebras using the differential and exponential map
Summary
For a Lie algebra g, the space of derivations on g is a Lie algebra, which is the Lie algebra of the Lie group of automorphisms on g This is an application of Lie’s third theorem. Lie triple system to construct derivations and automorphisms on the standard embedding Lie algebra. We study the 3-derivations and 3-automorphisms on Lie algebras, and prove that the Lie algebra of 3-derivations is the Lie algebra of the Lie group of. We will prove that 3-Der(g) is the Lie algebra of 3-Aut(g). Because 3-Der(g) is the subalgebra of gl(g) and 3-Aut(g) is a closed Lie subgroup of GL(g), we only need to prove that. ∑ ∑ Cni Cn−i [[ Dn+1−i− j x, D j y]g , Di z]g j i =1 j =1 n n +1− i i =1 j =1 n n +1− i i =1 j =1 j −1. We have i.e., 3-Der(g) is the Lie algebra of 3-Aut(g)
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