Geodesic
The groups of automorphisms of the Witt $W_n$ and Virasoro Lie algebras
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 4, pp. 1129-1141
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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 ${\rm Aut}_{{\rm Lie}} (W_n) \simeq {\rm Aut_{K-{\rm alg}}} (L_n)\simeq {\rm GL}_n(\mathbb {Z}) \ltimes K^{*n}$.
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 ${\rm Aut}_{{\rm Lie}} (W_n) \simeq {\rm Aut_{K-{\rm alg}}} (L_n)\simeq {\rm GL}_n(\mathbb {Z}) \ltimes K^{*n}$.
DOI : 10.1007/s10587-016-0314-6
Classification : 17B20, 17B30, 17B40, 17B65, 17B66
Keywords: group of automorphisms; monomorphism; Lie algebra; Witt algebra; Virasoro algebra; automorphism; locally nilpotent derivation 
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     title = {The groups of automorphisms of the {Witt} $W_n$ and {Virasoro} {Lie} algebras},
     journal = {Czechoslovak Mathematical Journal},
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     year = {2016},
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Bavula, Vladimir V. The groups of automorphisms of the Witt $W_n$ and Virasoro Lie algebras. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 4, pp. 1129-1141. doi: 10.1007/s10587-016-0314-6

[1] Bavula, V. V.: Every monomorphism of the Lie algebra of triangular polynomial derivations is an automorphism. C. R., Math., Acad. Sci. Paris 350 (2012), 553-556. | DOI | MR | Zbl

[2] Bavula, V. V.: Lie algebras of triangular polynomial derivations and an isomorphism criterion for their Lie factor algebras. Izv. Math. 77 (2013), 1067-1104. | DOI | MR | Zbl

[3] Bavula, V. V.: The groups of automorphisms of the Lie algebras of triangular polynomial derivations. J. Pure Appl. Algebra 218 (2014), 829-851. | DOI | MR | Zbl

[4] Bavula, V. V.: The group of automorphisms of the Lie algebra of derivations of a polynomial algebra. Algebra Appl. 16 (2017), 175-183 DOI: | DOI | MR

[5] Djoković, D. Ž., Zhao, K.: Derivations, isomorphisms, and second cohomology of generalized Witt algebras. Trans. Am. Math. Soc. 350 (1998), 643-664. | DOI | MR | Zbl

[6] Grabowski, J.: Isomorphisms and ideals of the Lie algebras of vector fields. Invent. Math. 50 (1978), 13-33. | DOI | MR | Zbl

[7] Grabowski, J., Poncin, N.: Automorphisms of quantum and classical Poisson algebras. Compos. Math. 140 (2004), 511-527. | DOI | MR | Zbl

[8] Osborn, J. M.: Automorphisms of the Lie algebras $W^*$ in characteristic $0$. Can. J. Math. 49 (1997), 119-132. | DOI | MR | Zbl

[9] Rudakov, A. N.: Subalgebras and automorphisms of Lie algebras of Cartan type. Funct. Anal. Appl. 20 (1986), 72-73. | DOI | MR | Zbl

[10] Shanks, M. E., Pursell, L. E.: The Lie algebra of a smooth manifold. Proc. Am. Math. Soc. 5 (1954), 468-472. | DOI | MR | Zbl

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