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 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156 \documentstyle{article} \parindent0cm %\textwidth 15.5cm\textheight 22.0cm\columnwidth\textwidth %\hoffset-1.5cm\voffset-1.5cm \begin{document} %\parskip 10pt plus 1pt \parindent 0pt \title{The {LIE} Package} \author{Carsten and Franziska Sch\"obel\\ The Leipzig University, Computer Science Dept.\\ Augustusplatz 10/11, O-7010 Leipzig, Germany\\ Email: cschoeb@aix550.informatik.uni-leipzig.de} \date{22 January 1993} \maketitle {\bf LIE} is a package of functions for the classification of real n-dimensional Lie algebras. It consists of two modules: {\bf liendmc1} and {\bf lie1234}. \\[0.3cm]{\large\bf liendmc1}\\[0.1cm] With the help of the functions in this module real n-dimensional Lie algebras $L$ with a derived algebra $L^{(1)}$ of dimension 1 can be classified. $L$ has to be defined by its structure constants $c_{ij}^k$ in the basis $\{X_1,\ldots,X_n\}$ with $[X_i,X_j]=c_{ij}^k X_k$. The user must define an ARRAY LIENSTRUCIN($n,n,n$) with n being the dimension of the Lie algebra $L$. The structure constants LIENSTRUCIN($i,j,k$):=$c_{ij}^k$ for $i). \end{verbatim} {\tt } corresponds to the dimension$n$. The procedure simplifies the structure of$L$performing real linear transformations. The returned value is a list of the form \begin{verbatim} (i) {LIE_ALGEBRA(2),COMMUTATIVE(n-2)} or (ii) {HEISENBERG(k),COMMUTATIVE(n-k)} \end{verbatim} with$3\leq k\leq n$,$k$odd.\\ The concepts correspond to the following theorem ({\tt LIE\_ALGEBRA(2)}$\rightarrow L_2$, {\tt HEISENBERG(k)}$\rightarrow H_k$and {\tt COMMUTATIVE(n-k)}$\rightarrow C_{n-k}$):\\[0.2cm] {\bf Theorem.} Every real$n$-dimensional Lie algebra$L$with a 1-dimensional derived algebra can be decomposed into one of the following forms:\\[0.1cm] \hspace*{0.3cm} (i)$C(L)\cap L^{(1)}=\{0\}\, :\; L_2\oplus C_{n-2}$or\\[0.05cm] \hspace*{0.3cm} (ii)$C(L)\cap L^{(1)}=L^{(1)}\, :\; H_k\oplus C_{n-k}\quad (k=2r-1,\, r\geq 2)$, with\newpage \hspace*{0.3cm} 1.$C(L)=C_j\oplus (L^{(1)}\cap C(L))$and dim$\,C_j=j$,\\[0.05cm] \hspace*{0.3cm} 2.$L_2$is generated by$Y_1,Y_2$with$[Y_1,Y_2]=Y_1$,\\[0.05cm] \hspace*{0.3cm} 3.$H_k$is generated by$\{Y_1,\ldots,Y_k\}$with\\ \hspace*{0.7cm}$[Y_2,Y_3]=\cdots =[Y_{k-1},Y_k]=Y_1$.\\[0.2cm] (cf. \cite{cssmp92})\\[0.2cm] The returned list is also stored as LIE\_LIST. The matrix LIENTRANS gives the transformation from the given basis$\{X_1,\ldots ,X_n\}$into the standard basis$\{Y_1,\ldots ,Y_n\}$:$Y_j=($LIENTRANS$)_j^k X_k$.\\[0.1cm] A more detailed output can be obtained by turning on the switch TR\_LIE: \begin{verbatim} ON TR_LIE; \end{verbatim} before the procedure LIENDIMCOM1 is called.\\[0.1cm] The returned list could be an input for a data bank in which mathematical relevant properties of the obtained Lie algebras are stored.\\[0.3cm] {\large\bf lie1234}\\[0.1cm] This part of the package classifies real low-dimensional Lie algebras$L$of the dimension$n:=$dim$\,L=1,2,3,4$.$L$is also given by its structure constants$c_{ij}^k$in the basis$\{X_1,\ldots,X_n\}$with$[X_i,X_j]=c_{ij}^k X_k$. An ARRAY LIESTRIN($n,n,n$) has to be defined and LIESTRIN($i,j,k$):=$c_{ij}^k$for$i). \end{verbatim} {\tt } should be the dimension of the Lie algebra $L$. The procedure stepwise simplifies the commutator relations of $L$ using properties of invariance like the dimension of the centre, of the derived algebra, unimodularity etc. The returned value has the form: \begin{verbatim} {LIEALG(n),COMTAB(m)}, \end{verbatim} where $m$ corresponds to the number of the standard form (basis: $\{Y_1,\ldots,Y_n\}$) in an enumeration scheme. The corresponding enumeration schemes are listed below (cf. \cite{ntz-preprint27/92},\cite{mmpreprint1979}). In case that the standard form in the enumeration scheme depends on one (or two) parameter(s) $p_1$ (and $p_2$) the list is expanded to: \begin{verbatim} {LIEALG(n),COMTAB(m),p1,p2}. \end{verbatim} This returned value is also stored as LIE\_CLASS. The linear transformation from the basis $\{X_1,\ldots,X_n\}$ into the basis of the standard form $\{Y_1,\ldots,Y_n\}$ is given by the matrix LIEMAT: $Y_j=($LIEMAT$)_j^k X_k$.\newpage By turning on the switch TR\_LIE: \begin{verbatim} ON TR_LIE; \end{verbatim} before the procedure LIECLASS is called the output contains not only the list LIE\_CLASS but also the non-vanishing commutator relations in the standard form.\\[0.1cm] By the value $m$ and the parameters further examinations of the Lie algebra are possible, especially if in a data bank mathematical relevant properties of the enumerated standard forms are stored.\\[0.3cm] {\large\bf Enumeration schemes for lie1234}\\[0.2cm] \hspace*{0.3cm}\begin{tabular}{l|l}returned list LIE\_CLASS& the corresponding commutator relations\\[0.1cm]\hline {LIEALG(1),COMTAB(0)}&commutative case\\[0.1cm]\hline {LIEALG(2),COMTAB(0)}&commutative case\\[0.1cm] {LIEALG(2),COMTAB(1)}&$[Y_1,Y_2]=Y_2$\\[0.1cm]\hline {LIEALG(3),COMTAB(0)}&commutative case\\[0.1cm] {LIEALG(3),COMTAB(1)}&$[Y_1,Y_2]=Y_3$\\[0.1cm] {LIEALG(3),COMTAB(2)}&$[Y_1,Y_3]=Y_3$\\[0.1cm] {LIEALG(3),COMTAB(3)}&$[Y_1,Y_3]=Y_1,[Y_2,Y_3]=Y_2$\\[0.1cm] {LIEALG(3),COMTAB(4)}&$[Y_1,Y_3]=Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm] {LIEALG(3),COMTAB(5)}&$[Y_1,Y_3]=-Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm] {LIEALG(3),COMTAB(6)}&$[Y_1,Y_3]=-Y_1+p_1 Y_2,[Y_2,Y_3]=Y_1,p_1\neq 0$\\[0.1cm] {LIEALG(3),COMTAB(7)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=-Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm] {LIEALG(3),COMTAB(8)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]\hline {LIEALG(4),COMTAB(0)}&commutative case\\[0.1cm] {LIEALG(4),COMTAB(1)}&$[Y_1,Y_4]=Y_1$\\[0.1cm] {LIEALG(4),COMTAB(2)}&$[Y_2,Y_4]=Y_1$\\[0,1cm] {LIEALG(4),COMTAB(3)}&$[Y_1,Y_3]=Y_1,[Y_2,Y_4]=Y_2$\\[0.1cm] {LIEALG(4),COMTAB(4)}&$[Y_1,Y_3]=-Y_2,[Y_2,Y_4]=Y_2,$\\ &$[Y_1,Y_4]=[Y_2,Y_3]=Y_1$\\[0.1cm] {LIEALG(4),COMTAB(5)}&$[Y_2,Y_4]=Y_2,[Y_1,Y_4]=[Y_2,Y_3]=Y_1$\\[0.1cm] {LIEALG(4),COMTAB(6)}&$[Y_2,Y_4]=Y_1,[Y_3,Y_4]=Y_2$\\[0.1cm] {LIEALG(4),COMTAB(7)}&$[Y_2,Y_4]=Y_2,[Y_3,Y_4]=Y_1$\\[0.1cm] {LIEALG(4),COMTAB(8)}&$[Y_1,Y_4]=-Y_2,[Y_2,Y_4]=Y_1$\\[0.1cm] {LIEALG(4),COMTAB(9)}&$[Y_1,Y_4]=-Y_1+p_1 Y_2,[Y_2,Y_4]=Y_1,p_1\neq 0$\\[0.1cm] {LIEALG(4),COMTAB(10)}&$[Y_1,Y_4]=Y_1,[Y_2,Y_4]=Y_2$\\[0.1cm] {LIEALG(4),COMTAB(11)}&$[Y_1,Y_4]=Y_2,[Y_2,Y_4]=Y_1$ \end{tabular}\\ \hspace*{0.3cm}\begin{tabular}{l|l}returned list LIE\_CLASS& the corresponding commutator relations\\[0.1cm]\hline {LIEALG(4),COMTAB(12)}&$[Y_1,Y_4]=Y_1+Y_2,[Y_2,Y_4]=Y_2+Y_3,$\\ &$[Y_3,Y_4]=Y_3$\\[0.1cm] {LIEALG(4),COMTAB(13)}&$[Y_1,Y_4]=Y_1,[Y_2,Y_4]=p_1 Y_2,[Y_3,Y_4]=p_2 Y_3,$\\ &$p_1,p_2\neq 0$\\[0.1cm] {LIEALG(4),COMTAB(14)}&$[Y_1,Y_4]=p_1 Y_1+Y_2,[Y_2,Y_4]=-Y_1+p_1 Y_2,$\\ &$[Y_3,Y_4]=p_2 Y_3,p_2\neq 0$\\[0.1cm] {LIEALG(4),COMTAB(15)}&$[Y_1,Y_4]=p_1 Y_1+Y_2,[Y_2,Y_4]=p_1 Y_2,$\\ &$[Y_3,Y_4]=Y_3,p_1\neq 0$\\[0.1cm] {LIEALG(4),COMTAB(16)}&$[Y_1,Y_4]=2 Y_1,[Y_2,Y_3]=Y_1,$\\ &$[Y_2,Y_4]=(1+p_1) Y_2,[Y_3,Y_4]=(1-p_1) Y_3,$\\ &$p_1\geq 0$\\[0.1cm] {LIEALG(4),COMTAB(17)}&$[Y_1,Y_4]=2 Y_1,[Y_2,Y_3]=Y_1,$\\ &$[Y_2,Y_4]=Y_2-p_1 Y_3,[Y_3,Y_4]=p_1 Y_2+Y_3,$\\ &$p_1\neq 0$\\[0.1cm] {LIEALG(4),COMTAB(18)}&$[Y_1,Y_4]=2 Y_1,[Y_2,Y_3]=Y_1,$\\ &$[Y_2,Y_4]=Y_2+Y_3,[Y_3,Y_4]=Y_3$\\[0.1cm] {LIEALG(4),COMTAB(19)}&$[Y_2,Y_3]=Y_1,[Y_2,Y_4]=Y_3,[Y_3,Y_4]=Y_2$\\[0.1cm] {LIEALG(4),COMTAB(20)}&$[Y_2,Y_3]=Y_1,[Y_2,Y_4]=-Y_3,[Y_3,Y_4]=Y_2$\\[0.1cm] {LIEALG(4),COMTAB(21)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=-Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm] {LIEALG(4),COMTAB(22)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=Y_2,[Y_2,Y_3]=Y_1$ \end{tabular} \bibliography{lie} \bibliographystyle{plain} \end{document}