lie.tex 7.8 KB

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  7. \title{The {LIE} Package}
  8. \author{Carsten and Franziska Sch\"obel\\
  9. The Leipzig University, Computer Science Dept.\\
  10. Augustusplatz 10/11, O-7010 Leipzig, Germany\\
  11. Email:}
  12. \date{22 January 1993}
  13. \maketitle
  14. {\bf LIE} is a package of functions for the classification of real n-dimensional
  15. Lie algebras. It consists of two modules: {\bf liendmc1} and {\bf lie1234}.
  16. \\[0.3cm]{\large\bf liendmc1}\\[0.1cm]
  17. With the help of the functions in this module real n-dimensional Lie algebras
  18. $L$ with a derived algebra $L^{(1)}$ of dimension 1 can be classified. $L$ has
  19. to be defined by its structure constants $c_{ij}^k$ in the basis
  20. $\{X_1,\ldots,X_n\}$ with $[X_i,X_j]=c_{ij}^k X_k$. The user must define an
  21. ARRAY LIENSTRUCIN($n,n,n$) with n being the dimension of the Lie algebra $L$.
  22. The structure constants LIENSTRUCIN($i,j,k$):=$c_{ij}^k$ for $i<j$ should be
  23. given. Then the procedure LIENDIMCOM1 can be called. Its syntax is:
  24. \begin{verbatim}
  25. LIENDIMCOM1(<number>).
  26. \end{verbatim}
  27. {\tt <number>} corresponds to the dimension $n$. The procedure simplifies
  28. the structure of $L$ performing real linear transformations. The returned
  29. value is a list of the form
  30. \begin{verbatim}
  31. (i) {LIE_ALGEBRA(2),COMMUTATIVE(n-2)} or
  32. (ii) {HEISENBERG(k),COMMUTATIVE(n-k)}
  33. \end{verbatim}
  34. with $3\leq k\leq n$, $k$ odd.\\
  35. The concepts correspond to the following theorem ({\tt LIE\_ALGEBRA(2)}
  36. $\rightarrow L_2$, {\tt HEISENBERG(k)} $\rightarrow H_k$ and
  37. {\tt COMMUTATIVE(n-k)} $\rightarrow C_{n-k}$):\\[0.2cm]
  38. {\bf Theorem.} Every real $n$-dimensional Lie algebra $L$ with a 1-dimensional
  39. derived algebra can be decomposed into one of the following forms:\\[0.1cm]
  40. \hspace*{0.3cm} (i) $C(L)\cap L^{(1)}=\{0\}\, :\; L_2\oplus C_{n-2}$
  41. or\\[0.05cm]
  42. \hspace*{0.3cm} (ii) $C(L)\cap L^{(1)}=L^{(1)}\, :\; H_k\oplus C_{n-k}\quad
  43. (k=2r-1,\, r\geq 2)$, with\newpage
  44. \hspace*{0.3cm} 1. $C(L)=C_j\oplus (L^{(1)}\cap C(L))$
  45. and dim$\,C_j=j$ ,\\[0.05cm]
  46. \hspace*{0.3cm} 2. $L_2$ is generated by
  47. $Y_1,Y_2$ with $[Y_1,Y_2]=Y_1$ ,\\[0.05cm]
  48. \hspace*{0.3cm} 3. $H_k$ is generated by $\{Y_1,\ldots,Y_k\}$ with\\
  49. \hspace*{0.7cm} $[Y_2,Y_3]=\cdots =[Y_{k-1},Y_k]=Y_1$.\\[0.2cm]
  50. (cf. \cite{cssmp92})\\[0.2cm]
  51. The returned list is also stored as LIE\_LIST. The matrix LIENTRANS gives the
  52. transformation from the given basis $\{X_1,\ldots ,X_n\}$ into the standard
  53. basis $\{Y_1,\ldots ,Y_n\}$: $Y_j=($LIENTRANS$)_j^k X_k$.\\[0.1cm]
  54. A more detailed output can be obtained by turning on the switch TR\_LIE:
  55. \begin{verbatim}
  56. ON TR_LIE;
  57. \end{verbatim}
  58. before the procedure LIENDIMCOM1 is called.\\[0.1cm]
  59. The returned list could be an input for a data bank in which mathematical
  60. relevant properties of the obtained Lie algebras are stored.\\[0.3cm]
  61. {\large\bf lie1234}\\[0.1cm]
  62. This part of the package classifies real low-dimensional Lie algebras $L$
  63. of the dimension
  64. $n:=$dim$\,L=1,2,3,4$. $L$ is also given by its structure constants $c_{ij}^k$
  65. in the basis $\{X_1,\ldots,X_n\}$ with $[X_i,X_j]=c_{ij}^k X_k$. An ARRAY
  66. LIESTRIN($n,n,n$) has to be defined and LIESTRIN($i,j,k$):=$c_{ij}^k$ for
  67. $i<j$ should be given. Then the procedure LIECLASS can be performed
  68. whose syntax is:
  69. \begin{verbatim}
  70. LIECLASS(<number>).
  71. \end{verbatim}
  72. {\tt <number>} should be the dimension of the Lie algebra $L$. The procedure
  73. stepwise simplifies the commutator relations of $L$ using properties of
  74. invariance like the dimension of the centre, of the derived algebra,
  75. unimodularity etc. The returned value has the form:
  76. \begin{verbatim}
  77. {LIEALG(n),COMTAB(m)},
  78. \end{verbatim}
  79. where $m$ corresponds to the number of the standard form (basis:
  80. $\{Y_1,\ldots,Y_n\}$) in an enumeration scheme. The corresponding enumeration
  81. schemes are listed below (cf. \cite{ntz-preprint27/92},\cite{mmpreprint1979}).
  82. In case that the standard form in the enumeration scheme depends on one (or two)
  83. parameter(s) $p_1$ (and $p_2$) the list is expanded to:
  84. \begin{verbatim}
  85. {LIEALG(n),COMTAB(m),p1,p2}.
  86. \end{verbatim}
  87. This returned value is also stored as LIE\_CLASS. The linear transformation from
  88. the basis $\{X_1,\ldots,X_n\}$ into the basis of the standard form
  89. $\{Y_1,\ldots,Y_n\}$ is given by the matrix LIEMAT:
  90. $Y_j=($LIEMAT$)_j^k X_k$.\newpage
  91. By turning on the switch TR\_LIE:
  92. \begin{verbatim}
  93. ON TR_LIE;
  94. \end{verbatim}
  95. before the procedure LIECLASS is called the output contains not only the
  96. list LIE\_CLASS but also the non-vanishing commutator relations in the
  97. standard form.\\[0.1cm]
  98. By the value $m$ and the parameters further examinations of the Lie algebra
  99. are possible, especially if in a data bank mathematical relevant properties
  100. of the enumerated standard forms are stored.\\[0.3cm]
  101. {\large\bf Enumeration schemes for lie1234}\\[0.2cm]
  102. \hspace*{0.3cm}\begin{tabular}{l|l}returned list LIE\_CLASS&
  103. the corresponding commutator relations\\[0.1cm]\hline
  104. {LIEALG(1),COMTAB(0)}&commutative case\\[0.1cm]\hline
  105. {LIEALG(2),COMTAB(0)}&commutative case\\[0.1cm]
  106. {LIEALG(2),COMTAB(1)}&$[Y_1,Y_2]=Y_2$\\[0.1cm]\hline
  107. {LIEALG(3),COMTAB(0)}&commutative case\\[0.1cm]
  108. {LIEALG(3),COMTAB(1)}&$[Y_1,Y_2]=Y_3$\\[0.1cm]
  109. {LIEALG(3),COMTAB(2)}&$[Y_1,Y_3]=Y_3$\\[0.1cm]
  110. {LIEALG(3),COMTAB(3)}&$[Y_1,Y_3]=Y_1,[Y_2,Y_3]=Y_2$\\[0.1cm]
  111. {LIEALG(3),COMTAB(4)}&$[Y_1,Y_3]=Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]
  112. {LIEALG(3),COMTAB(5)}&$[Y_1,Y_3]=-Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]
  113. {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]
  114. {LIEALG(3),COMTAB(7)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=-Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]
  115. {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
  116. {LIEALG(4),COMTAB(0)}&commutative case\\[0.1cm]
  117. {LIEALG(4),COMTAB(1)}&$[Y_1,Y_4]=Y_1$\\[0.1cm]
  118. {LIEALG(4),COMTAB(2)}&$[Y_2,Y_4]=Y_1$\\[0,1cm]
  119. {LIEALG(4),COMTAB(3)}&$[Y_1,Y_3]=Y_1,[Y_2,Y_4]=Y_2$\\[0.1cm]
  120. {LIEALG(4),COMTAB(4)}&$[Y_1,Y_3]=-Y_2,[Y_2,Y_4]=Y_2,$\\
  121. &$[Y_1,Y_4]=[Y_2,Y_3]=Y_1$\\[0.1cm]
  122. {LIEALG(4),COMTAB(5)}&$[Y_2,Y_4]=Y_2,[Y_1,Y_4]=[Y_2,Y_3]=Y_1$\\[0.1cm]
  123. {LIEALG(4),COMTAB(6)}&$[Y_2,Y_4]=Y_1,[Y_3,Y_4]=Y_2$\\[0.1cm]
  124. {LIEALG(4),COMTAB(7)}&$[Y_2,Y_4]=Y_2,[Y_3,Y_4]=Y_1$\\[0.1cm]
  125. {LIEALG(4),COMTAB(8)}&$[Y_1,Y_4]=-Y_2,[Y_2,Y_4]=Y_1$\\[0.1cm]
  126. {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]
  127. {LIEALG(4),COMTAB(10)}&$[Y_1,Y_4]=Y_1,[Y_2,Y_4]=Y_2$\\[0.1cm]
  128. {LIEALG(4),COMTAB(11)}&$[Y_1,Y_4]=Y_2,[Y_2,Y_4]=Y_1$
  129. \end{tabular}\\
  130. \hspace*{0.3cm}\begin{tabular}{l|l}returned list LIE\_CLASS&
  131. the corresponding commutator relations\\[0.1cm]\hline
  132. {LIEALG(4),COMTAB(12)}&$[Y_1,Y_4]=Y_1+Y_2,[Y_2,Y_4]=Y_2+Y_3,$\\
  133. &$[Y_3,Y_4]=Y_3$\\[0.1cm]
  134. {LIEALG(4),COMTAB(13)}&$[Y_1,Y_4]=Y_1,[Y_2,Y_4]=p_1 Y_2,[Y_3,Y_4]=p_2 Y_3,$\\
  135. &$p_1,p_2\neq 0$\\[0.1cm]
  136. {LIEALG(4),COMTAB(14)}&$[Y_1,Y_4]=p_1 Y_1+Y_2,[Y_2,Y_4]=-Y_1+p_1 Y_2,$\\
  137. &$[Y_3,Y_4]=p_2 Y_3,p_2\neq 0$\\[0.1cm]
  138. {LIEALG(4),COMTAB(15)}&$[Y_1,Y_4]=p_1 Y_1+Y_2,[Y_2,Y_4]=p_1 Y_2,$\\
  139. &$[Y_3,Y_4]=Y_3,p_1\neq 0$\\[0.1cm]
  140. {LIEALG(4),COMTAB(16)}&$[Y_1,Y_4]=2 Y_1,[Y_2,Y_3]=Y_1,$\\
  141. &$[Y_2,Y_4]=(1+p_1) Y_2,[Y_3,Y_4]=(1-p_1) Y_3,$\\
  142. &$p_1\geq 0$\\[0.1cm]
  143. {LIEALG(4),COMTAB(17)}&$[Y_1,Y_4]=2 Y_1,[Y_2,Y_3]=Y_1,$\\
  144. &$[Y_2,Y_4]=Y_2-p_1 Y_3,[Y_3,Y_4]=p_1 Y_2+Y_3,$\\
  145. &$p_1\neq 0$\\[0.1cm]
  146. {LIEALG(4),COMTAB(18)}&$[Y_1,Y_4]=2 Y_1,[Y_2,Y_3]=Y_1,$\\
  147. &$[Y_2,Y_4]=Y_2+Y_3,[Y_3,Y_4]=Y_3$\\[0.1cm]
  148. {LIEALG(4),COMTAB(19)}&$[Y_2,Y_3]=Y_1,[Y_2,Y_4]=Y_3,[Y_3,Y_4]=Y_2$\\[0.1cm]
  149. {LIEALG(4),COMTAB(20)}&$[Y_2,Y_3]=Y_1,[Y_2,Y_4]=-Y_3,[Y_3,Y_4]=Y_2$\\[0.1cm]
  150. {LIEALG(4),COMTAB(21)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=-Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]
  151. {LIEALG(4),COMTAB(22)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=Y_2,[Y_2,Y_3]=Y_1$
  152. \end{tabular}
  153. \bibliography{lie}
  154. \bibliographystyle{plain}
  155. \end{document}