# On the structure of graded transitive Lie algebras

Research output: Book/ReportReportOther research output

### Abstract

We study finite-dimensional Lie algebras ${\mathfrak L}$ of polynomial vector fields in $n$ variables that contain the vector fields $\dfrac{\partial}{\partial x_i} \; (i=1,\ldots, n)$ and $x_1\dfrac{\partial}{\partial x_1}+ \dots + x_n\dfrac{\partial}{\partial x_n}$. We show that the maximal ones always contain a semi-simple subalgebra $\bar{{\mathfrak g}}$, such that $\dfrac{\partial}{\partial x_i}\in \bar{{\mathfrak g}} \; (i=1,\ldots, m)$ for an $m$ with $1 \leq m \leq n$. Moreover a maximal algebra has no trivial $\bar{{\mathfrak g}}$-module in the space spanned by $\dfrac{\partial}{\partial x_i} (i=m+1,\ldots, n)$. The possible algebras $\bar{{\mathfrak g}}$ are described in detail, as well as all $\bar{{\mathfrak g}}$-modules that constitute such maximal ${\mathfrak L}$. All maximal ${\mathfrak L}$ are described explicitly for $n\leq 3$.
Original language Undefined Enschede University of Twente, Department of Applied Mathematics Published - 2000

### Publication series

Name Department of Applied Mathematics, University of Twente 1557 0169-2690

• MSC-17B05
• MSC-17B66
• IR-65744
• EWI-3377
• MSC-17B70

### Cite this

Post, G. F. (2000). On the structure of graded transitive Lie algebras. Enschede: University of Twente, Department of Applied Mathematics.
Post, Gerhard F. / On the structure of graded transitive Lie algebras. Enschede : University of Twente, Department of Applied Mathematics, 2000.
@book{9a4d12942dc54d9fbbe9e8f2e46e3a4d,
title = "On the structure of graded transitive Lie algebras",
abstract = "We study finite-dimensional Lie algebras ${\mathfrak L}$ of polynomial vector fields in $n$ variables that contain the vector fields $\dfrac{\partial}{\partial x_i} \; (i=1,\ldots, n)$ and $x_1\dfrac{\partial}{\partial x_1}+ \dots + x_n\dfrac{\partial}{\partial x_n}$. We show that the maximal ones always contain a semi-simple subalgebra $\bar{{\mathfrak g}}$, such that $\dfrac{\partial}{\partial x_i}\in \bar{{\mathfrak g}} \; (i=1,\ldots, m)$ for an $m$ with $1 \leq m \leq n$. Moreover a maximal algebra has no trivial $\bar{{\mathfrak g}}$-module in the space spanned by $\dfrac{\partial}{\partial x_i} (i=m+1,\ldots, n)$. The possible algebras $\bar{{\mathfrak g}}$ are described in detail, as well as all $\bar{{\mathfrak g}}$-modules that constitute such maximal ${\mathfrak L}$. All maximal ${\mathfrak L}$ are described explicitly for $n\leq 3$.",
keywords = "MSC-17B05, MSC-17B66, IR-65744, EWI-3377, MSC-17B70",
author = "Post, {Gerhard F.}",
note = "Imported from MEMORANDA",
year = "2000",
language = "Undefined",
publisher = "University of Twente, Department of Applied Mathematics",
number = "1557",

}

Post, GF 2000, On the structure of graded transitive Lie algebras. University of Twente, Department of Applied Mathematics, Enschede.
Enschede : University of Twente, Department of Applied Mathematics, 2000.

Research output: Book/ReportReportOther research output

TY - BOOK

T1 - On the structure of graded transitive Lie algebras

AU - Post, Gerhard F.

N1 - Imported from MEMORANDA

PY - 2000

Y1 - 2000

N2 - We study finite-dimensional Lie algebras ${\mathfrak L}$ of polynomial vector fields in $n$ variables that contain the vector fields $\dfrac{\partial}{\partial x_i} \; (i=1,\ldots, n)$ and $x_1\dfrac{\partial}{\partial x_1}+ \dots + x_n\dfrac{\partial}{\partial x_n}$. We show that the maximal ones always contain a semi-simple subalgebra $\bar{{\mathfrak g}}$, such that $\dfrac{\partial}{\partial x_i}\in \bar{{\mathfrak g}} \; (i=1,\ldots, m)$ for an $m$ with $1 \leq m \leq n$. Moreover a maximal algebra has no trivial $\bar{{\mathfrak g}}$-module in the space spanned by $\dfrac{\partial}{\partial x_i} (i=m+1,\ldots, n)$. The possible algebras $\bar{{\mathfrak g}}$ are described in detail, as well as all $\bar{{\mathfrak g}}$-modules that constitute such maximal ${\mathfrak L}$. All maximal ${\mathfrak L}$ are described explicitly for $n\leq 3$.

AB - We study finite-dimensional Lie algebras ${\mathfrak L}$ of polynomial vector fields in $n$ variables that contain the vector fields $\dfrac{\partial}{\partial x_i} \; (i=1,\ldots, n)$ and $x_1\dfrac{\partial}{\partial x_1}+ \dots + x_n\dfrac{\partial}{\partial x_n}$. We show that the maximal ones always contain a semi-simple subalgebra $\bar{{\mathfrak g}}$, such that $\dfrac{\partial}{\partial x_i}\in \bar{{\mathfrak g}} \; (i=1,\ldots, m)$ for an $m$ with $1 \leq m \leq n$. Moreover a maximal algebra has no trivial $\bar{{\mathfrak g}}$-module in the space spanned by $\dfrac{\partial}{\partial x_i} (i=m+1,\ldots, n)$. The possible algebras $\bar{{\mathfrak g}}$ are described in detail, as well as all $\bar{{\mathfrak g}}$-modules that constitute such maximal ${\mathfrak L}$. All maximal ${\mathfrak L}$ are described explicitly for $n\leq 3$.

KW - MSC-17B05

KW - MSC-17B66

KW - IR-65744

KW - EWI-3377

KW - MSC-17B70

M3 - Report

BT - On the structure of graded transitive Lie algebras

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -

Post GF. On the structure of graded transitive Lie algebras. Enschede: University of Twente, Department of Applied Mathematics, 2000.