| Title:
|
The groups of automorphisms of the Witt $W_n$ and Virasoro Lie algebras (English) |
| Author:
|
Bavula, Vladimir V. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
4 |
| Year:
|
2016 |
| Pages:
|
1129-1141 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 ${\rm Aut}_{{\rm Lie}} (W_n) \simeq {\rm Aut_{K-{\rm alg}}} (L_n)\simeq {\rm GL}_n(\mathbb {Z}) \ltimes K^{*n}$. (English) |
| Keyword:
|
group of automorphisms |
| Keyword:
|
monomorphism |
| Keyword:
|
Lie algebra |
| Keyword:
|
Witt algebra |
| Keyword:
|
Virasoro algebra |
| Keyword:
|
automorphism |
| Keyword:
|
locally nilpotent derivation |
| MSC:
|
17B20 |
| MSC:
|
17B30 |
| MSC:
|
17B40 |
| MSC:
|
17B65 |
| MSC:
|
17B66 |
| idZBL:
|
Zbl 06674866 |
| idMR:
|
MR3572927 |
| DOI:
|
10.1007/s10587-016-0314-6 |
| . |
| Date available:
|
2016-11-26T20:54:24Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145923 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
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| . |
