RICAM-Logo OEAW-Logo
Johann Radon Institute for Computational and Applied Mathematics (RICAM)
Austrian Academy of Sciences (ÖAW)
Home People Research Groups and Topics Externally Funded Projects Publications Events Media/Press Jobs/Opportunities Links Internal Location/Contact Special Semesters
Search

MEGA 2007

Effective Methods in Algebraic Geometry

Strobl, Austria, June 25th - 29th

http://www.ricam.oeaw.ac.at/mega2007

 mega-logo
Electronic Proceedings

Abstract

TitleMinimal Faithful Upper-triangular Matrix Representations for Low-Dimensional Solvable Lie Algebras
Keywordssolvable Lie algebra, upper-triangular matrix representation
Abstract\begin{center}
\large\bf EXTENDED ABSTRACT
\end{center}

\noindent According Ado's Theorem, given any finite-dimensional
complex Lie algebra $\mathfrak{g}$, there exists a matrix algebra
isomorphic to $\mathfrak{g}$ (see \cite{Jac62}). In this way,
every finite-dimensional complex Lie algebra can be represented as
a Lie subalgebra of the complex general linear algebra
$\mathfrak{gl}(n,\mathbb{C})$, formed by all the complex $n\times
n$ matrices, for some $n\in\mathbb{N}$.

\vspace{0.1cm}

In this paper we consider the complex Lie algebra
$\mathfrak{b}_n$, formed by all the $n \times n$ complex
upper-triangular matrices. Obviously, this Lie algebra is a Lie
subalgebra of $\mathfrak{gl}(n,\mathbb{C})$. Besides, the Lie
algebra $\mathfrak{b}_n$ is solvable and, hence, each Lie
subalgebra $\mathfrak{s} \subset \mathfrak{b}_n$ is also solvable.
The Lie algebras $\mathfrak b_n$ and their subalgebras are
interesting because any complex finite-dimensional solvable Lie
algebra can be represented by a Lie subalgebra of some $\mathfrak
b_n$ (see \cite[Theorem 9.11]{F-H91} or \cite[Theorem
3.7.3]{Va98}).

\vspace{0.1cm}

Given $n\in\mathbb N$, the Lie algebra $\mathfrak b_n$ has
dimension $\binom{n+1}{2}$ and a basis
$\mathcal{B}=\{X_{ij}\in\mathfrak b_n \ | \ 1\le i \le j \le n \}$
of this algebra is given by the following upper-triangular
matrices:
{\small\begin{equation*} X_{ij} =\begin{pmatrix}
x_{1,1} & x_{1,2} & \cdots & x_{1,n-1} & x_{1,n} \\
0 & x_{2,2} & \cdots & x_{2,n-1} & x_{2,n} \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & x_{n-1,n-1} & x_{n-1,n} \\
0 & 0 & \cdots & 0 & x_{n,n}
\end{pmatrix}, \qquad \mathrm{with} \, \, x_{r,s}=\left\{\begin{array}{ll}
0, & \textrm{ if } (r,s)\neq(i,j) \\
1, & \textrm{ if } (r,s)=(i,j)
\end{array} \right.
\end{equation*}}
With respect to the basis $\mathcal B$, the non-zero brackets of
this algebra are the following: {\small$$ [X_{i,h}, X_{h,k}] =
X_{i,k}, \quad 1\le i\le h\le k\le n \ \wedge \ i\neq k.$$}

\vspace{-0.6cm}

Given $n\in\mathbb N$, we now wonder if every $n$-dimensional
complex solvable Lie algebra can be represented as a Lie
subalgebra of $\mathfrak{b}_n$. To answer this question, each
$n$-dimensional Lie subalgebra of $\mathfrak{b}_n$ has to be
studied and its isomorphism class has to be determined to rule out
the algebras which do not verify this property.

\vspace{0.1cm}

In this paper, we study all the complex solvable Lie algebras
whose dimension is less than 5 and we obtain minimal
representations of these algebras by using complex
upper-triangular matrices; that is, for each complex solvable Lie
algebra of such dimensions, we find the minimal $n\in \mathbb N$
such that this algebra can be represented by a subalgebra of
$\mathfrak b_n$. This is the main goal of the paper.

\vspace{0.1cm}

A Lie algebra $\mathfrak{g}$ is called {\em solvable} if there
exists a natural integer $m$ such that its {\em commutator central
series} is cancel out in the ideal $\mathcal{C}_m(\mathfrak{g})$;
that is: {\small $$\mathcal{C}_1(\mathfrak{g})= \mathfrak{g}, \
\mathcal{C}_2(\mathfrak{g})=[\mathfrak{g},\mathfrak{g}], \ \dots,
\ \mathcal{C}_k(\mathfrak{g})=[\mathcal{C}_{k-1}(\mathfrak{g}),
\mathcal{C}_{k-1}(\mathfrak{g})], \ \dots, \
\mathcal{C}_m(\mathfrak{g}) \equiv \{0\}.$$}

\vspace{-0.6cm}

In this paper, we use the following well-known result:

\vspace{0.2cm}

\noindent {\bf Proposition.} Given a Lie algebra $\mathfrak{g}$
and a Lie subalgebra $\mathfrak{h}$ of $\mathfrak{g}$, their
respective commutator central series are related as follows:
$\mathcal{C}_k(\mathfrak{h})\subseteq\mathcal{C}_k(\mathfrak{g})$,
$\forall k\in\mathbb N$.

\vspace{0.2cm}

We have used the classification given by Mubarakzyanov
\cite{AND,Mub} to deal with all the solvable Lie algebras (up to
isomorphism classes) of dimension less than 5. Nevertheless, the
classification of solvable Lie algebras is also known for
dimension 5 (see \cite{Mub1}) and 6 (see~\cite{Turi}).

\vspace{0.1cm}

With respect to minimal representations of Lie algebras, Burde
\cite{Bu98} introduced the following invariant value for an
arbitrary Lie algebra $\mathfrak{g}$:

\vspace{-0.4cm}

$${\small\mu(\mathfrak{g})={\rm min}\{{\rm dim} (M) \ | \ M \mbox{
is a faithful } \mathfrak{g}\mbox{-module}\}.}$$ This value is
also equal to the minimal value $n$ such that
$\mathfrak{gl}(\mathbb{C}^n)$ contains a subalgebra isomorphic to
$\mathfrak{g}$. However, the algebras considered in \cite{Bu98}
are nilpotent and the non-nilpotent solvable Lie algebras were not
studied.

\vspace{0.1cm}

In this paper, minimal representations of solvable Lie algebras
are found. Besides, we are interested in minimal representations
of these algebras with a particular restriction: the
representations have to be given by upper-triangular matrices. In
this way, given a solvable Lie algebra $\mathfrak{g}$, we are
going to compute the minimal value $n$ such that $\mathfrak{b}_n$
contains $\mathfrak{g}$. This value is an invariant of
$\mathfrak{g}$ and its expression is given by:
$${\small\bar{\mu}(\mathfrak{g})={\rm min}\{n\in\mathbb{N} \ |
\ \exists \mbox{ subalgebra of } \mathfrak{b}_n \mbox{ isomorphic
to } \mathfrak{g}\}.}$$

The invariant $\bar{\mu}(\mathfrak{g})$ exists for any solvable
Lie algebra $\mathfrak{g}$ because every solvable complex Lie
algebra can be represented by a Lie subalgebra of the Lie algebra
$\mathfrak{b}_n$, in virtue of Theorem 3.7.3 of \cite{Va98}.

\vspace{0.1cm}

To get minimal representations by using upper-triangular matrices
for each solvable Lie algebra of low dimension, we have used the
following method which we explain next. Given a complex solvable
Lie algebra $\mathfrak{g}$, the steps followed to obtain such a
representation are the following ones:

\begin{enumerate}
\item The commutator central series of the Lie
algebra $\mathfrak{g}$ is compared with the one of the Lie algebra
$\mathfrak{b}_n$, beginning with $n=1$. We pass to the following
step when we have found the minimal $n$ such that $\mathcal C_k(
\mathfrak g)\subset \mathcal C_k(\mathfrak b_n)$, for all
$k\in\mathbb N$.

\item Given a basis of $\mathfrak{g}$, each vector of this basis
is a linear combination of the vectors $X_{ij}$ in the basis
$\mathcal{B}$ of $\mathfrak{b}_n$. To do it, for each basic vector
of $\mathfrak g$, we consider the greatest $k\in\mathbb N$ such
that the ideal $\mathcal{C}_k(\mathfrak{b}_n)$ contains to this
basic vector. In this way, this vector is expressed as a linear
combination of the vectors $X_{ij}$ in the chosen
$\mathcal{C}_k(\mathfrak{b}_n)$.

\item All the brackets between two basic vectors of $\mathfrak{g}$
are computed by using MAPLE~9.5. By imposing the law of the Lie
algebra $\mathfrak{g}$, a system of non-linear equations are
obtained by comparing coordinate to coordinate with respect to the
basis $\mathcal B$. The unknowns of this system are the
coefficients of the vectors in the basis of $\mathfrak g$.

\item By using MATHEMATICA 5.2, this system of equations is solved and
a solution of the system will provide us a representation of the
Lie algebra $\mathfrak{g}$ in $\mathfrak{b}_n$ if such a solution
satisfies that the vectors resulting are linearly independent.
When no solution is obtained from the system, $\mathfrak{g}$
cannot be represented by a subalgebra of $\mathfrak{b}_n$. In this
case, we go back to step 2 and repeat all the steps with the Lie
algebra $\mathfrak{b}_{n+1}$.
\end{enumerate}

Let us note that the representation obtained for the Lie algebra
$\mathfrak{g}$ is minimal because we start with $n=1$ and $n$ is
increased when no representations can be obtained in
$\mathfrak{b}_n$.

\vspace{0.1cm}

Next, we write the result which summarizes the minimal
representations by using upper-triangular matrices for each
solvable Lie algebra of dimension less than 4 (up to isomorphism
classes):

\vspace{0.25cm}

\noindent{\bf Theorem.}\quad A minimal representation by
upper-triangular matrices for each solvable Lie algebra of
dimension less than 4 is given in the next table with the
dimension of such a minimal representation:

\vspace{-0.75cm}

{\small\[\setlength{\extrarowheight}{0.3cm}\begin{array}{ccc}
\mathrm{Lie \ algebra} & \mathrm{Minimal \ representation} & \mathrm{Dim.} \ (\bar{\mu}) \\
\hline\hline \mathfrak{s}_1^1: & \langle Z_1=X_{11}\rangle\subseteq\mathfrak{b}_1 & 1 \\
\mathfrak{s}_2^1: & \langle Z_1=X_{11}, Z_2=X_{22} \rangle\subseteq \mathfrak{b}_2 & 2 \\
\mathfrak{s}_2^2: & \langle Z_1=X_{11}, Z_2=X_{12}\rangle\subseteq\mathfrak{b}_2 & 2\\
\mathfrak{s}_3^1: & \langle Z_1=X_{11}, Z_2=X_{22}, Z_3=X_{33}\rangle\subseteq\mathfrak{b}_3 & 3\\
\mathfrak{s}_3^2: & \langle Z_1=X_{12}, Z_2 = X_{13}, Z_3=X_{23}\rangle\subseteq\mathfrak{b}_3 & 3\\
\mathfrak{s}_3^3: & \langle Z_1=X_{13}, Z_2=X_{12}, Z_3=-X_{1,1}\rangle\subseteq\mathfrak{b}_3 & 3\\
\mathfrak{s}_3^4: & \langle Z_1=X_{12}, Z_2=i X_{12}+ X_{13}, Z_3=iX_{22}+ X_{23}-iX_{33}\rangle\subseteq\mathfrak{b}_3 & 3\\
\mathfrak{s}_3^5: & \langle Z_1 = X_{13}, Z_2 = X_{23}, Z_3 = X_{12} - X_{33}\rangle\subseteq\mathfrak{b}_3 & 3\\
\mathfrak{s}_3^6: & \langle Z_1 = X_{12}, Z_2 = X_{11}+X_{22}, Z_3
= X_{22}\rangle \subseteq \mathfrak{b}_2 & 2
\end{array} \]}

And the following result shows the minimal representations by
using upper-triangular matrices for each 4-dimensional solvable
Lie algebras. Due to the bigger number of isomorphism classes in
dimension 4, we have only considered the non-decomposable ones
(that is, those which are not isomorphic to the direct sum of
other two Lie algebras).

\vspace{0.25cm}

\noindent{\bf Theorem.}\quad A minimal representation by
upper-triangular matrices for each non-decom\-posable solvable Lie
algebra of dimension 4 is given in the next table with the
dimension of such a minimal representation:

\vspace{0.4cm}

\hspace*{-0.6cm}\begin{tabular}{ccc}\setlength{\extrarowheight}{0.3cm}
\!\!\small{\rm L.A.} & \small{\rm Minimal representation} & \!\!\!\!\!\!\!\!\!\!\!\!\!\! \small{\rm Dim.} ($\bar{\mu}$) \\
\hline\hline $\mathfrak{s}_4^1:$ & \small $\langle Z_1\!\!=\!X_{11}, Z_2\!\!=\!X_{22}, Z_3\!\!=\!X_{33}, Z_4\!\!=\!X_{44} \rangle\!\subseteq\! \mathfrak{b}_4$ &\!\!\!\!\!\!\!\!\!\!\! \small $4$ \\
$\mathfrak{s}_4^2:$ & \small $\langle Z_1\!\!=\!-(X_{23}+X_{34}), Z_2\!\!=\!X_{14}, Z_3\!\!=\!X_{13}, Z_4\!\!=\!X_{12}\rangle\!\subseteq\!\mathfrak{b}_4$ &\!\!\!\!\!\!\!\!\!\small $4$\\
$\mathfrak{s}_4^3:$ & \small $\langle Z_1\!\!=\!X_{11}, Z_2\!\!=\!-X_{23}, Z_3\!\!=\!X_{12}, Z_4\!\!=\!X_{13}\rangle\!\subseteq\!\mathfrak{b}_3$ &\!\!\!\!\!\!\!\!\!\small $3$\\
$\mathfrak{s}_4^4:$ & \small $\langle Z_1\!\!=\!X_{11}, Z_2 \!\!=\! -\!\, i X_{22}\! +\! X_{23}\! +\! i X_{33}, Z_3 \!\!=\! X_{12}, Z_4 \!\!=\! -\!\, i X_{12}\!+\! X_{13}\rangle\!\subseteq\!\mathfrak{b}_3$ &\!\!\!\!\!\!\!\!\!\small $3$\\
$\mathfrak{s}_4^5:$ & \small $\langle Z_1\!\!=\! X_{12}\! +\! X_{33}, Z_2\!\!=\!X_{24}, Z_3\!\!=\!X_{34}, Z_4\!\!=\!X_{14}\rangle\!\subseteq\!\mathfrak{b}_4$ &\!\!\!\!\!\!\!\!\!\small $4$\\
$\mathfrak{s}_4^6:$ & \small $\langle Z_1\!\!=\!X_{14}, Z_2\!\!=\!X_{24}, Z_3\!\!=\!X_{34}, Z_4\!\!=\!(\alpha\!-\!1)X_{22}\!+\!(\beta\!-\!1)X_{33}\!-\!X_{44}\rangle\!\subseteq\!\mathfrak{b}_4$ &\!\!\!\!\!\!\!\!\!\small $4$\\
$\mathfrak{s}_4^7:$ & \small $\langle Z_1\!\!=\!X_{14}, Z_2\!\!=\!-X_{13}, Z_3\!\!=\!X_{11}\!+\!X_{23}\!+\!(1\!-\!\alpha)X_{44}, Z_4\!\!=\!X_{12}\rangle\!\subseteq\!\mathfrak{b}_4$ &\!\!\!\!\!\!\!\!\!\small $4$\\
$\mathfrak{s}_4^8:$ & \small $\langle Z_1\!\!=\!X_{12}\!-\!X_{23}\!-\!X_{44}, Z_2\!\!=\!X_{34}, Z_3\!\!=\!-X_{24}, Z_4\!\!=\!-X_{14}\rangle \!\subseteq \!\mathfrak{b}_4$ &\!\!\!\!\!\!\!\!\!\small $4$ \\
$\mathfrak{s}_4^9:$ &\!\!\!\!\!\!\!\!\! \small $\langle Z_1\!\!=\!\alpha X_{11}\!\!+\!(\alpha\!-\!\beta\!\!-\!i)X_{22}\!\!+\!(\beta\!-\!i)X_{33}, Z_2\!\!=\!i(X_{12}\!\!-\!\!X_{34}), Z_3\!\!=\!X_{12}\!\!+\!X_{34}, Z_4\!\!=\!X_{14}\rangle\! \subseteq\! \mathfrak{b}_4$ &\!\!\!\!\!\!\!\!\!\small $4$ \\
$\mathfrak{s}_4^{10}:$ & \small $\langle Z_1\!\!=\!\alpha X_{11}+X_{22}, Z_2\!\!=\!X_{12}, Z_3\!\!=\!X_{23}, Z_4 \!\!=\! X_{13}\rangle\! \subseteq\! \mathfrak{b}_3$ &\!\!\!\!\!\!\!\!\!\small $3$ \\
$\mathfrak{s}_4^{11}:$ & \small $\langle Z_1\!\!=\!X_{12}+X_{11}+X_{22}-X_{44}, Z_2\!\!=\!X_{23}-X_{34}, Z_3\!\!=\!X_{13}, Z_4\!\!=\!X_{14}\rangle\! \subseteq\! \mathfrak{b}_4$ &\!\!\!\!\!\!\!\!\!\small $4$ \\
$\mathfrak{s}_4^{12}:$ & \small $\langle Z_1\!\!=\!2 \alpha
X_{11}+(\alpha-i)X_{22}, Z_2\!\!=\!X_{23}+X_{12}, Z_3\!\!=\!i
X_{23}-i X_{12}, Z_4\!\!=\!2 i X_{13}\rangle\! \subseteq\!
\mathfrak{b}_3$ &\!\!\!\!\!\!\!\!\!\small $3$
\end{tabular}

\begin{thebibliography}{99}\small

\bibitem{AND} A. Andrada, M. L. Barberis, I. G. Dotti and G. P.
Ovando. Pro\-duct structures on four dimensional solvable Lie
algebras. {\it Homology, Homotopy and Applications}, \textbf{7}
(2005) 9--37.

\bibitem{Bu98} D. Burde. On a refinement of Ado's Theorem. {\it
Arch. Math. (Basel)} {\bf 70} (1998) 118--127.

\bibitem{F-H91} W. Fulton and J. Harris. {\it Representation Theory. A First
Course.} Springer-Verlag, New York, 1991.

\bibitem{Jac62} N. Jacobson. A note on automorphisms and
derivations of Lie algebras, {\it Proc. Amer. Math. Soc.} {\bf 6}
(1955) 281--283.

\bibitem{Mub} G. M. Mubarakzyanov. On solvable Lie algebras. {\it
Izv. Vyssh. Uchebn. Zaved. Mat.} \textbf{32} (1) (1963) 114--123
(in Russian).

\bibitem{Mub1} G. M. Mubarakzyanov. The classification of the real structure
of five-dimensional Lie algebras. {\it Izv. Vyssh. Uchebn. Zaved.
Mat.} \textbf{34} (3) (1963) 99--106 (in Russian).

\bibitem{Turi} P. Turkowski. Solvable Lie algebras of dimension
six. {\it J. Math. Phys.} \textbf{31} (1990) \linebreak
1344--1350.

\bibitem{Va98} V. S. Varadarajan. {\it Lie Groups, Lie Algebras and
their Representations}. Selected Monographies {\bf 17}, Coll\ae ge
Press, Beijing, 1998.

\end{thebibliography}

The Institute is named after the famous Austrian mathematician Johann Radon (1887-1956)

Medieninhaber:
Österreichische Akademie der Wissenschaften
Juristische Person öffentlichen Rechts (BGBl 569/1921 idF BGBl I 130/2003)
Dr. Ignaz Seipel-Platz 2, 1010 Wien
Diese Website dient zur Information über die wissenschaftlichen Aktivitäten der Österreichischen Akademie der Wissenschaften und setzt somit den gesetzlichen Auftrag um, die Wissenschaft in jeder Hinsicht zu fördern.

This RICAM page was made with 100% valid HTML & CSS - Send comments to Webmaster
Today's date and time is 02/09/12 - 10:55 CET and this file ( /mega2007/openconf/electronic/16-abs.html ) was last modified on 06/19/07 - 14:54 CEST