The groups of automorphisms of the Witt $W_n$ and Virasoro Lie algebras

Bavula, Vladimir V.. "The groups of automorphisms of the Witt $W_n$ and Virasoro Lie algebras." Czechoslovak Mathematical Journal 66.4 (2016): 1129-1141. <http://eudml.org/doc/287585>.

@article{Bavula2016,
abstract = {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: $\{\rm Aut\}_\{\{\rm Lie\}\} (\{\rm Vir\}) \simeq \{\rm Aut\}_\{\{\rm Lie\}\} (W_1) \simeq \lbrace \pm 1\rbrace \ltimes K^*$, and give a short proof that $\{\rm Aut\}_\{\{\rm Lie\}\} (W_n) \simeq \{\rm Aut_\{K-\{\rm alg\}\}\} (L_n)\simeq \{\rm GL\}_n(\mathbb \{Z\}) \ltimes K^\{*n\}$.},
author = {Bavula, Vladimir V.},
journal = {Czechoslovak Mathematical Journal},
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},
language = {eng},
number = {4},
pages = {1129-1141},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The groups of automorphisms of the Witt $W_n$ and Virasoro Lie algebras},
url = {http://eudml.org/doc/287585},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Bavula, Vladimir V.
TI - The groups of automorphisms of the Witt $W_n$ and Virasoro Lie algebras
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 4
SP - 1129
EP - 1141
AB - 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: ${\rm Aut}_{{\rm Lie}} ({\rm Vir}) \simeq {\rm Aut}_{{\rm Lie}} (W_1) \simeq \lbrace \pm 1\rbrace \ltimes K^*$, and give a short proof that ${\rm Aut}_{{\rm Lie}} (W_n) \simeq {\rm Aut_{K-{\rm alg}}} (L_n)\simeq {\rm GL}_n(\mathbb {Z}) \ltimes K^{*n}$.
LA - eng
KW - 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
UR - http://eudml.org/doc/287585
ER -