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Keywords:
group of automorphisms; monomorphism; Lie algebra; Witt algebra; Virasoro algebra; automorphism; locally nilpotent derivation
group of automorphisms; monomorphism; Lie algebra; Witt algebra; Virasoro algebra; automorphism; locally nilpotent derivation
Summary:
Let $L_n=K[x_1^{\pm 1} , \ldots , x_n^{\pm 1}]$ be a Laurent polynomial algebra over a field $K$ of characteristic zero, $W_n:= {\rm Der}_K(L_n)$ the Lie algebra of $K$-derivations of the algebra $L_n$, the so-called Witt Lie algebra, and let ${\rm Vir}$ be the Virasoro Lie algebra which is a $1$-dimensional central extension of the Witt Lie algebra. The Lie algebras $W_n$ and ${\rm Vir}$ are infinite dimensional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: ${\rm Aut}_{{\rm Lie}} ({\rm Vir}) \simeq {\rm Aut}_{{\rm Lie}} (W_1) \simeq \{\pm 1\} \ltimes K^*$, and give a short proof that ${\rm Aut}_{{\rm Lie}} (W_n) \simeq {\rm Aut_{K-{\rm alg}}} (L_n)\simeq {\rm GL}_n(\mathbb {Z}) \ltimes K^{*n}$.
References:
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