1 | version="$Id: involut.lib,v 1.16 2009-04-09 12:04:41 seelisch Exp $"; |
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2 | category="Noncommutative"; |
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3 | info=" |
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4 | LIBRARY: involut.lib Computations and operations with involutions |
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5 | AUTHORS: Oleksandr Iena, yena@mathematik.uni-kl.de, |
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6 | @* Markus Becker, mbecker@mathematik.uni-kl.de, |
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7 | @* Viktor Levandovskyy, levandov@mathematik.uni-kl.de |
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8 | |
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9 | THEORY: Involution is an anti-isomorphism of a noncommutative algebra with the |
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10 | property that applied an involution twice, one gets an identity. Involution is linear with respect to the ground field. In this library we compute linear involutions, distinguishing the case of a diagonal matrix (such involutions are called homothetic) and a general one. |
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11 | |
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12 | SUPPORT: Forschungsschwerpunkt 'Mathematik und Praxis' (Project of Dr. E. Zerz |
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13 | and V. Levandovskyy), Uni Kaiserslautern |
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14 | |
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15 | NOTE: This library provides algebraic tools for computations and operations |
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16 | with algebraic involutions and linear automorphisms of non-commutative algebras |
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17 | |
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18 | PROCEDURES: |
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19 | findInvo(); computes linear involutions on a basering; |
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20 | findInvoDiag(); computes homothetic (diagonal) involutions on a basering; |
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21 | findAuto(); computes linear automorphisms of a basering; |
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22 | ncdetection(); computes an ideal, presenting an involution map on some particular noncommutative algebras; |
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23 | involution(m,theta); applies the involution to an object. |
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24 | "; |
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25 | |
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26 | LIB "ncalg.lib"; |
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27 | LIB "poly.lib"; |
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28 | LIB "primdec.lib"; |
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29 | /////////////////////////////////////////////////////////////////////////////// |
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30 | proc ncdetection() |
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31 | "USAGE: ncdetection(); |
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32 | RETURN: ideal, representing an involution map |
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33 | PURPOSE: compute classical involutions (i.e. acting rather on operators than on variables) for some particular noncommutative algebras |
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34 | ASSUME: the procedure is aimed at non-commutative algebras with differential, shift or advance operators arising in Control Theory. |
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35 | It has to be executed in a ring. |
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36 | EXAMPLE: example ncdetection; shows an example |
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37 | "{ |
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38 | // in this procedure an involution map is generated from the NCRelations |
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39 | // that will be used in the function involution |
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40 | // in dieser proc. wird eine matrix erzeugt, die in der i-ten zeile die indices |
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41 | // der differential-, shift- oder advance-operatoren enthaelt mit denen die i-te |
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42 | // variable nicht kommutiert. |
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43 | if ( nameof(basering)=="basering" ) |
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44 | { |
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45 | "No current ring defined."; |
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46 | return(ideal(0)); |
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47 | } |
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48 | def r = basering; |
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49 | setring r; |
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50 | int i,j,k,LExp; |
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51 | int NVars = nvars(r); |
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52 | matrix rel = ncRelations(r)[2]; |
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53 | intmat M[NVars][3]; |
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54 | int NRows = nrows(rel); |
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55 | intvec v,w; |
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56 | poly d,d_lead; |
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57 | ideal I; |
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58 | map theta; |
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59 | for( j=NRows; j>=2; j-- ) |
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60 | { |
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61 | if( rel[j] == w ) //the whole column is zero |
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62 | { |
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63 | j--; |
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64 | continue; |
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65 | } |
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66 | for( i=1; i<j; i++ ) |
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67 | { |
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68 | if( rel[i,j]==1 ) //relation of type var(j)*var(i) = var(i)*var(j) +1 |
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69 | { |
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70 | M[i,1]=j; |
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71 | } |
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72 | if( rel[i,j] == -1 ) //relation of type var(i)*var(j) = var(j)*var(i) -1 |
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73 | { |
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74 | M[j,1]=i; |
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75 | } |
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76 | d = rel[i,j]; |
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77 | d_lead = lead(d); |
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78 | v = leadexp(d_lead); //in the next lines we check wether we have a relation of differential or shift type |
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79 | LExp=0; |
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80 | for(k=1; k<=NVars; k++) |
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81 | { |
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82 | LExp = LExp + v[k]; |
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83 | } |
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84 | // if( (d-d_lead != 0) || (LExp > 1) ) |
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85 | if ( ( (d-d_lead) != 0) || (LExp > 1) || ( (LExp==0) && ((d_lead>1) || (d_lead<-1)) ) ) |
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86 | { |
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87 | return(theta); |
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88 | } |
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89 | |
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90 | if( v[j] == 1) //relation of type var(j)*var(i) = var(i)*var(j) -lambda*var(j) |
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91 | { |
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92 | if (leadcoef(d) < 0) |
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93 | { |
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94 | M[i,2] = j; |
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95 | } |
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96 | else |
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97 | { |
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98 | M[i,3] = j; |
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99 | } |
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100 | } |
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101 | if( v[i]==1 ) //relation of type var(j)*var(i) = var(i)*var(j) -lambda*var(i) |
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102 | { |
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103 | if (leadcoef(d) > 0) |
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104 | { |
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105 | M[j,2] = i; |
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106 | } |
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107 | else |
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108 | { |
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109 | M[j,3] = i; |
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110 | } |
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111 | } |
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112 | } |
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113 | } |
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114 | // from here on, the map is computed |
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115 | for(i=1;i<=NVars;i++) |
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116 | { |
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117 | I=I+var(i); |
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118 | } |
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119 | |
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120 | for(i=1;i<=NVars;i++) |
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121 | { |
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122 | if( M[i,1..3]==(0,0,0) ) |
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123 | { |
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124 | i++; |
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125 | continue; |
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126 | } |
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127 | if( M[i,1]!=0 ) |
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128 | { |
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129 | if( (M[i,2]!=0) && (M[i,3]!=0) ) |
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130 | { |
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131 | I[M[i,1]] = -var(M[i,1]); |
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132 | I[M[i,2]] = var(M[i,3]); |
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133 | I[M[i,3]] = var(M[i,2]); |
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134 | } |
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135 | if( (M[i,2]==0) && (M[i,3]==0) ) |
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136 | { |
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137 | I[M[i,1]] = -var(M[i,1]); |
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138 | } |
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139 | if( ( (M[i,2]!=0) && (M[i,3]==0) )|| ( (M[i,2]!=0) && (M[i,3]==0) ) |
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140 | ) |
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141 | { |
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142 | I[i] = -var(i); |
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143 | } |
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144 | } |
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145 | else |
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146 | { |
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147 | if( (M[i,2]!=0) && (M[i,3]!=0) ) |
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148 | { |
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149 | I[i] = -var(i); |
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150 | I[M[i,2]] = var(M[i,3]); |
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151 | I[M[i,3]] = var(M[i,2]); |
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152 | } |
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153 | else |
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154 | { |
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155 | I[i] = -var(i); |
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156 | } |
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157 | } |
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158 | } |
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159 | return(I); |
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160 | } |
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161 | example |
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162 | { |
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163 | "EXAMPLE:"; echo = 2; |
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164 | ring R = 0,(x,y,z,D(1..3)),dp; |
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165 | matrix D[6][6]; |
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166 | D[1,4]=1; D[2,5]=1; D[3,6]=1; |
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167 | def r = nc_algebra(1,D); setring r; |
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168 | map F = r; |
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169 | ncdetection(); |
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170 | F; // F(F) should be maxideal(1) |
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171 | kill r, R; |
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172 | //---------------------------------------- |
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173 | ring R=0,(x,S),dp; |
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174 | def r = nc_algebra(1,-S); setring r; |
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175 | map F = r; |
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176 | ncdetection(); |
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177 | F; // F(F) should be maxideal(1) |
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178 | kill r, R; |
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179 | //---------------------------------------- |
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180 | ring R=0,(x,D(1),S),dp; |
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181 | matrix D[3][3]; |
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182 | D[1,2]=1; D[1,3]=-S; |
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183 | def r = nc_algebra(1,D); setring r; |
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184 | map F = r; |
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185 | ncdetection(); |
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186 | F; // F(F) should be maxideal(1) |
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187 | } |
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188 | |
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189 | static proc In_Poly(poly mm, list l, int NVars) |
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190 | // applies the involution to the poly mm |
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191 | // entries of a list l are images of variables under invo |
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192 | // more general than invo_poly; used in many rings setting |
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193 | { |
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194 | int i,j; |
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195 | intvec v; |
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196 | poly pp, zz; |
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197 | poly nn = 0; |
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198 | i = 1; |
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199 | while(mm[i]!=0) |
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200 | { |
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201 | v = leadexp(mm[i]); |
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202 | zz = 1; |
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203 | for( j=NVars; j>=1; j--) |
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204 | { |
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205 | if (v[j]!=0) |
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206 | { |
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207 | pp = l[j]; |
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208 | zz = zz*(pp^v[j]); |
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209 | } |
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210 | } |
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211 | nn = nn + (leadcoef(mm[i])*zz); |
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212 | i++; |
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213 | } |
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214 | return(nn); |
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215 | } |
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216 | |
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217 | static proc Hom_Poly(poly mm, list l, int NVars) |
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218 | // applies the endomorphism to the poly mm |
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219 | // entries of a list l are images of variables under endo |
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220 | // should not be replaced by map-based stuff! used in |
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221 | // many rings setting |
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222 | { |
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223 | int i,j; |
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224 | intvec v; |
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225 | poly pp, zz; |
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226 | poly nn = 0; |
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227 | i = 1; |
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228 | while(mm[i]!=0) |
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229 | { |
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230 | v = leadexp(mm[i]); |
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231 | zz = 1; |
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232 | for( j=NVars; j>=1; j--) |
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233 | { |
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234 | if (v[j]!=0) |
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235 | { |
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236 | pp = l[j]; |
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237 | zz = (pp^v[j])*zz; |
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238 | } |
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239 | } |
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240 | nn = nn + (leadcoef(mm[i])*zz); |
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241 | i++; |
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242 | } |
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243 | return(nn); |
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244 | } |
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245 | |
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246 | static proc invo_poly(poly m, map theta) |
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247 | // applies the involution map theta to m, where m=polynomial |
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248 | { |
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249 | // compatibility: |
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250 | ideal l = ideal(theta); |
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251 | int i; |
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252 | list L; |
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253 | for (i=1; i<=size(l); i++) |
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254 | { |
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255 | L[i] = l[i]; |
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256 | } |
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257 | int nv = nvars(basering); |
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258 | return (In_Poly(m,L,nv)); |
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259 | // if (m==0) { return(m); } |
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260 | // int i,j; |
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261 | // intvec v; |
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262 | // poly p,z; |
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263 | // poly n = 0; |
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264 | // i = 1; |
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265 | // while(m[i]!=0) |
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266 | // { |
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267 | // v = leadexp(m[i]); |
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268 | // z =1; |
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269 | // for(j=nvars(basering); j>=1; j--) |
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270 | // { |
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271 | // if (v[j]!=0) |
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272 | // { |
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273 | // p = var(j); |
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274 | // p = theta(p); |
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275 | // z = z*(p^v[j]); |
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276 | // } |
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277 | // } |
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278 | // n = n + (leadcoef(m[i])*z); |
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279 | // i++; |
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280 | // } |
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281 | // return(n); |
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282 | } |
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283 | /////////////////////////////////////////////////////////////////////////////////// |
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284 | proc involution(m, map theta) |
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285 | "USAGE: involution(m, theta); m is a poly/vector/ideal/matrix/module, theta is a map |
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286 | RETURN: object of the same type as m |
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287 | PURPOSE: applies the involution, presented by theta to the object m |
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288 | THEORY: for an involution theta and two polynomials a,b from the algebra, theta(ab) = theta(b) theta(a); theta is linear with respect to the ground field |
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289 | NOTE: This is generalized ''theta(m)'' for data types unsupported by ''map''. |
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290 | EXAMPLE: example involution; shows an example |
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291 | "{ |
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292 | // applies the involution map theta to m, |
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293 | // where m= vector, polynomial, module, matrix, ideal |
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294 | int i,j; |
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295 | intvec v; |
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296 | poly p,z; |
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297 | if (typeof(m)=="poly") |
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298 | { |
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299 | return (invo_poly(m,theta)); |
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300 | } |
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301 | if ( typeof(m)=="ideal" ) |
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302 | { |
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303 | ideal n; |
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304 | for (i=1; i<=size(m); i++) |
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305 | { |
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306 | n[i] = invo_poly(m[i], theta); |
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307 | } |
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308 | return(n); |
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309 | } |
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310 | if (typeof(m)=="vector") |
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311 | { |
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312 | for(i=1; i<=size(m); i++) |
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313 | { |
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314 | m[i] = invo_poly(m[i], theta); |
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315 | } |
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316 | return (m); |
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317 | } |
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318 | if ( (typeof(m)=="matrix") || (typeof(m)=="module")) |
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319 | { |
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320 | matrix n = matrix(m); |
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321 | int @R=nrows(n); |
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322 | int @C=ncols(n); |
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323 | for(i=1; i<=@R; i++) |
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324 | { |
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325 | for(j=1; j<=@C; j++) |
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326 | { |
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327 | if (m[i,j]!=0) |
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328 | { |
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329 | n[i,j] = invo_poly( m[i,j], theta); |
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330 | } |
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331 | } |
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332 | } |
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333 | if (typeof(m)=="module") |
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334 | { |
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335 | return (module(n)); |
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336 | } |
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337 | else // matrix |
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338 | { |
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339 | return(n); |
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340 | } |
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341 | } |
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342 | // if m is not of the supported type: |
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343 | "Error: unsupported argument type!"; |
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344 | return(); |
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345 | } |
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346 | example |
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347 | { |
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348 | "EXAMPLE:";echo = 2; |
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349 | ring R = 0,(x,d),dp; |
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350 | def r = nc_algebra(1,1); setring r; // Weyl-Algebra |
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351 | map F = r,x,-d; |
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352 | F(F); // should be maxideal(1) for an involution |
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353 | poly f = x*d^2+d; |
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354 | poly If = involution(f,F); |
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355 | f-If; |
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356 | poly g = x^2*d+2*x*d+3*x+7*d; |
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357 | poly tg = -d*x^2-2*d*x+3*x-7*d; |
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358 | poly Ig = involution(g,F); |
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359 | tg-Ig; |
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360 | ideal I = f,g; |
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361 | ideal II = involution(I,F); |
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362 | II; |
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363 | I - involution(II,F); |
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364 | module M = [f,g,0],[g,0,x^2*d]; |
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365 | module IM = involution(M,F); |
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366 | print(IM); |
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367 | print(M - involution(IM,F)); |
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368 | } |
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369 | /////////////////////////////////////////////////////////////////////////////////// |
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370 | static proc new_var() |
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371 | //generates a string of new variables |
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372 | { |
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373 | |
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374 | int NVars=nvars(basering); |
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375 | int i,j; |
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376 | // string s="@_1_1"; |
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377 | string s="a11"; |
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378 | for(i=1; i<=NVars; i++) |
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379 | { |
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380 | for(j=1; j<=NVars; j++) |
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381 | { |
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382 | if(i*j!=1) |
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383 | { |
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384 | s = s+ ","+NVAR(i,j); |
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385 | }; |
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386 | }; |
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387 | }; |
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388 | return(s); |
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389 | }; |
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390 | |
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391 | static proc NVAR(int i, int j) |
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392 | { |
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393 | // return("@_"+string(i)+"_"+string(j)); |
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394 | return("a"+string(i)+string(j)); |
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395 | }; |
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396 | /////////////////////////////////////////////////////////////////////////////////// |
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397 | static proc new_var_special() |
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398 | //generates a string of new variables |
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399 | { |
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400 | int NVars=nvars(basering); |
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401 | int i; |
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402 | // string s="@_1_1"; |
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403 | string s="a11"; |
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404 | for(i=2; i<=NVars; i++) |
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405 | { |
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406 | s = s+ ","+NVAR(i,i); |
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407 | }; |
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408 | return(s); |
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409 | }; |
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410 | /////////////////////////////////////////////////////////////////////////////////// |
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411 | static proc RelMatr() |
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412 | // returns the matrix of relations |
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413 | // only Lie-type relations x_j x_i= x_i x_j + .. are taken into account |
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414 | { |
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415 | int i,j; |
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416 | int NVars = nvars(basering); |
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417 | matrix Rel[NVars][NVars]; |
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418 | for(i=1; i<NVars; i++) |
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419 | { |
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420 | for(j=i+1; j<=NVars; j++) |
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421 | { |
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422 | Rel[i,j]=var(j)*var(i)-var(i)*var(j); |
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423 | }; |
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424 | }; |
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425 | return(Rel); |
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426 | }; |
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427 | ///////////////////////////////////////////////////////////////// |
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428 | proc findInvo() |
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429 | "USAGE: findInvo(); |
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430 | RETURN: a ring containing a list L of pairs, where |
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431 | @* L[i][1] = ideal; a Groebner Basis of an i-th associated prime, |
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432 | @* L[i][2] = matrix, defining a linear map, with entries, reduced with respect to L[i][1] |
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433 | PURPOSE: computed the ideal of linear involutions of the basering |
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434 | NOTE: for convenience, the full ideal of relations @code{idJ} |
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435 | and the initial matrix with indeterminates @code{matD} are exported in the output ring |
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436 | SEE ALSO: findInvoDiag, involution |
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437 | EXAMPLE: example findInvo; shows examples |
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438 | |
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439 | "{ |
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440 | def @B = basering; //save the name of basering |
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441 | int NVars = nvars(@B); //number of variables in basering |
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442 | int i, j; |
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443 | |
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444 | matrix Rel = RelMatr(); //the matrix of relations |
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445 | |
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446 | string @ss = new_var(); //string of new variables |
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447 | string Par = parstr(@B); //string of parameters in old ring |
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448 | |
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449 | if (Par=="") // if there are no parameters |
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450 | { |
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451 | execute("ring @@@KK=0,("+varstr(@B)+","+@ss+"), dp;"); //new ring with new variables |
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452 | } |
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453 | else //if there exist parameters |
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454 | { |
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455 | execute("ring @@@KK=(0,"+Par+") ,("+varstr(@B)+","+@ss+"), dp;");//new ring with new variables |
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456 | }; |
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457 | |
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458 | matrix Rel = imap(@B, Rel); //consider the matrix of relations in new ring |
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459 | |
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460 | int Sz = NVars*NVars+NVars; // number of variables in new ring |
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461 | |
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462 | matrix M[Sz][Sz]; //to be the matrix of relations in new ring |
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463 | |
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464 | for(i=1; i<NVars; i++) //initialize that matrix of relations |
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465 | { |
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466 | for(j=i+1; j<=NVars; j++) |
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467 | { |
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468 | M[i,j] = Rel[i,j]; |
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469 | }; |
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470 | }; |
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471 | |
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472 | def @@K = nc_algebra(1, M); setring @@K; //now new ring @@K become a noncommutative ring |
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473 | |
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474 | list l; //list to define an involution |
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475 | poly @@F; |
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476 | for(i=1; i<=NVars; i++) //initializing list for involution |
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477 | { |
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478 | @@F=0; |
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479 | for(j=1; j<=NVars; j++) |
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480 | { |
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481 | execute( "@@F = @@F+"+NVAR(i,j)+"*"+string( var(j) )+";" ); |
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482 | }; |
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483 | l=l+list(@@F); |
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484 | }; |
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485 | |
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486 | matrix N = imap(@@@KK,Rel); |
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487 | |
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488 | for(i=1; i<NVars; i++)//get matrix by applying the involution to relations |
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489 | { |
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490 | for(j=i+1; j<=NVars; j++) |
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491 | { |
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492 | N[i,j]= l[j]*l[i] - l[i]*l[j] + In_Poly( N[i,j], l, NVars); |
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493 | }; |
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494 | }; |
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495 | kill l; |
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496 | //--------------------------------------------- |
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497 | //get the ideal of coefficients of N |
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498 | ideal J; |
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499 | ideal idN = simplify(ideal(N),2); |
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500 | J = ideal(coeffs( idN, var(1) ) ); |
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501 | for(i=2; i<=NVars; i++) |
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502 | { |
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503 | J = ideal( coeffs( J, var(i) ) ); |
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504 | }; |
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505 | J = simplify(J,2); |
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506 | //------------------------------------------------- |
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507 | if ( Par=="" ) //initializes the ring of relations |
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508 | { |
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509 | execute("ring @@KK=0,("+@ss+"), dp;"); |
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510 | } |
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511 | else |
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512 | { |
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513 | execute("ring @@KK=(0,"+Par+"),("+@ss+"), dp;"); |
---|
514 | }; |
---|
515 | ideal J = imap(@@K,J); // ideal, considered in @@KK now |
---|
516 | string snv = "["+string(NVars)+"]"; |
---|
517 | execute("matrix @@D"+snv+snv+"="+@ss+";"); // matrix with entries=new variables |
---|
518 | |
---|
519 | J = J, ideal( @@D*@@D-matrix( freemodule(NVars) ) ); // add the condition that involution to square is just identity |
---|
520 | J = simplify(J,2); // without extra zeros |
---|
521 | list mL = minAssGTZ(J); // components not in GB |
---|
522 | int sL = size(mL); |
---|
523 | option(redSB); // important for reduced GBs |
---|
524 | option(redTail); |
---|
525 | matrix IM = @@D; // involution map |
---|
526 | list L = list(); // the answer |
---|
527 | list TL; |
---|
528 | ideal tmp = 0; |
---|
529 | for (i=1; i<=sL; i++) // compute GBs of components |
---|
530 | { |
---|
531 | TL = list(); |
---|
532 | TL[1] = std(mL[i]); |
---|
533 | tmp = NF( ideal(IM), TL[1] ); |
---|
534 | TL[2] = matrix(tmp, NVars,NVars); |
---|
535 | L[i] = TL; |
---|
536 | } |
---|
537 | export(L); // main export |
---|
538 | ideal idJ = J; // debug-comfortable exports |
---|
539 | matrix matD = @@D; |
---|
540 | export(idJ); |
---|
541 | export(matD); |
---|
542 | return(@@KK); |
---|
543 | } |
---|
544 | example |
---|
545 | { "EXAMPLE:"; echo = 2; |
---|
546 | def a = makeWeyl(1); |
---|
547 | setring a; // this algebra is a first Weyl algebra |
---|
548 | a; |
---|
549 | def X = findInvo(); |
---|
550 | setring X; // ring with new variables, corr. to unknown coefficients |
---|
551 | X; |
---|
552 | L; |
---|
553 | // look at the matrix in the new variables, defining the linear involution |
---|
554 | print(L[1][2]); |
---|
555 | L[1][1]; // where new variables obey these relations |
---|
556 | idJ; |
---|
557 | } |
---|
558 | /////////////////////////////////////////////////////////////////////////// |
---|
559 | proc findInvoDiag() |
---|
560 | "USAGE: findInvoDiag(); |
---|
561 | RETURN: a ring together with a list of pairs L, where |
---|
562 | @* L[i][1] = ideal; a Groebner Basis of an i-th associated prime, |
---|
563 | @* L[i][2] = matrix, defining a linear map, with entries, reduced with respect to L[i][1] |
---|
564 | PURPOSE: compute homothetic (diagonal) involutions of the basering |
---|
565 | NOTE: for convenience, the full ideal of relations @code{idJ} |
---|
566 | and the initial matrix with indeterminates @code{matD} are exported in the output ring |
---|
567 | SEE ALSO: findInvo, involution |
---|
568 | EXAMPLE: example findInvoDiag; shows examples |
---|
569 | "{ |
---|
570 | def @B = basering; //save the name of basering |
---|
571 | int NVars = nvars(@B); //number of variables in basering |
---|
572 | int i, j; |
---|
573 | |
---|
574 | matrix Rel = RelMatr(); //the matrix of relations |
---|
575 | |
---|
576 | string @ss = new_var_special(); //string of new variables |
---|
577 | string Par = parstr(@B); //string of parameters in old ring |
---|
578 | |
---|
579 | if (Par=="") // if there are no parameters |
---|
580 | { |
---|
581 | execute("ring @@@KK=0,("+varstr(@B)+","+@ss+"), dp;"); //new ring with new variables |
---|
582 | } |
---|
583 | else //if there exist parameters |
---|
584 | { |
---|
585 | execute("ring @@@KK=(0,"+Par+") ,("+varstr(@B)+","+@ss+"), dp;");//new ring with new variables |
---|
586 | }; |
---|
587 | |
---|
588 | matrix Rel = imap(@B, Rel); //consider the matrix of relations in new ring |
---|
589 | |
---|
590 | int Sz = 2*NVars; // number of variables in new ring |
---|
591 | |
---|
592 | matrix M[Sz][Sz]; //to be the matrix of relations in new ring |
---|
593 | for(i=1; i<NVars; i++) //initialize that matrix of relations |
---|
594 | { |
---|
595 | for(j=i+1; j<=NVars; j++) |
---|
596 | { |
---|
597 | M[i,j] = Rel[i,j]; |
---|
598 | }; |
---|
599 | }; |
---|
600 | |
---|
601 | def @@K = nc_algebra(1, M); setring @@K; //now new ring @@K become a noncommutative ring |
---|
602 | |
---|
603 | list l; //list to define an involution |
---|
604 | |
---|
605 | for(i=1; i<=NVars; i++) //initializing list for involution |
---|
606 | { |
---|
607 | execute( "l["+string(i)+"]="+NVAR(i,i)+"*"+string( var(i) )+";" ); |
---|
608 | |
---|
609 | }; |
---|
610 | matrix N = imap(@@@KK,Rel); |
---|
611 | |
---|
612 | for(i=1; i<NVars; i++)//get matrix by applying the involution to relations |
---|
613 | { |
---|
614 | for(j=i+1; j<=NVars; j++) |
---|
615 | { |
---|
616 | N[i,j]= l[j]*l[i] - l[i]*l[j] + In_Poly( N[i,j], l, NVars); |
---|
617 | }; |
---|
618 | }; |
---|
619 | kill l; |
---|
620 | //--------------------------------------------- |
---|
621 | //get the ideal of coefficients of N |
---|
622 | |
---|
623 | ideal J; |
---|
624 | ideal idN = simplify(ideal(N),2); |
---|
625 | J = ideal(coeffs( idN, var(1) ) ); |
---|
626 | for(i=2; i<=NVars; i++) |
---|
627 | { |
---|
628 | J = ideal( coeffs( J, var(i) ) ); |
---|
629 | }; |
---|
630 | J = simplify(J,2); |
---|
631 | //------------------------------------------------- |
---|
632 | |
---|
633 | if ( Par=="" ) //initializes the ring of relations |
---|
634 | { |
---|
635 | execute("ring @@KK=0,("+@ss+"), dp;"); |
---|
636 | } |
---|
637 | else |
---|
638 | { |
---|
639 | execute("ring @@KK=(0,"+Par+"),("+@ss+"), dp;"); |
---|
640 | }; |
---|
641 | |
---|
642 | ideal J = imap(@@K,J); // ideal, considered in @@KK now |
---|
643 | |
---|
644 | matrix @@D[NVars][NVars]; // matrix with entries=new variables to square i.e. @@D=@@D^2 |
---|
645 | for(i=1;i<=NVars;i++) |
---|
646 | { |
---|
647 | execute("@@D["+string(i)+","+string(i)+"]="+NVAR(i,i)+";"); |
---|
648 | }; |
---|
649 | J = J, ideal( @@D*@@D - matrix( freemodule(NVars) ) ); // add the condition that involution to square is just identity |
---|
650 | J = simplify(J,2); // without extra zeros |
---|
651 | |
---|
652 | list mL = minAssGTZ(J); // components not in GB |
---|
653 | int sL = size(mL); |
---|
654 | option(redSB); // important for reduced GBs |
---|
655 | option(redTail); |
---|
656 | matrix IM = @@D; // involution map |
---|
657 | list L = list(); // the answer |
---|
658 | list TL; |
---|
659 | ideal tmp = 0; |
---|
660 | for (i=1; i<=sL; i++) // compute GBs of components |
---|
661 | { |
---|
662 | TL = list(); |
---|
663 | TL[1] = std(mL[i]); |
---|
664 | tmp = NF( ideal(IM), TL[1] ); |
---|
665 | TL[2] = matrix(tmp, NVars,NVars); |
---|
666 | L[i] = TL; |
---|
667 | } |
---|
668 | export(L); |
---|
669 | ideal idJ = J; // debug-comfortable exports |
---|
670 | matrix matD = @@D; |
---|
671 | export(idJ); |
---|
672 | export(matD); |
---|
673 | return(@@KK); |
---|
674 | } |
---|
675 | example |
---|
676 | { "EXAMPLE:"; echo = 2; |
---|
677 | def a = makeWeyl(1); |
---|
678 | setring a; // this algebra is a first Weyl algebra |
---|
679 | a; |
---|
680 | def X = findInvoDiag(); |
---|
681 | setring X; // ring with new variables, corresponding to unknown coefficients |
---|
682 | X; |
---|
683 | // print matrices, defining linear involutions |
---|
684 | print(L[1][2]); // a first matrix: we see it is constant |
---|
685 | print(L[2][2]); // and a second possible matrix; it is constant too |
---|
686 | L; // let us take a look on the whole list |
---|
687 | idJ; |
---|
688 | } |
---|
689 | ///////////////////////////////////////////////////////////////////// |
---|
690 | proc findAuto(int n) |
---|
691 | "USAGE: findAuto(n); n an integer |
---|
692 | RETURN: a ring together with a list of pairs L, where |
---|
693 | @* L[i][1] = ideal; a Groebner Basis of an i-th associated prime, |
---|
694 | @* L[i][2] = matrix, defining a linear map, with entries, reduced with respect to L[i][1] |
---|
695 | PURPOSE: compute the ideal of linear automorphisms of the basering, given by a matrix, n-th power of which gives identity (i.e. unipotent matrix) |
---|
696 | NOTE: if n=0, a matrix, defining an automorphism is not assumed to be unipotent but just non-degenerate. A nonzero parameter @code{@@p} is introduced as the value of the determinant of the matrix above. |
---|
697 | @* For convenience, the full ideal of relations @code{idJ} and the initial matrix with indeterminates @code{matD} are mutually exported in the output ring |
---|
698 | SEE ALSO: findInvo |
---|
699 | EXAMPLE: example findAuto; shows examples |
---|
700 | "{ |
---|
701 | if ((n<0 ) || (n==1)) |
---|
702 | { |
---|
703 | "The index of unipotency is too small."; |
---|
704 | return(0); |
---|
705 | } |
---|
706 | def @B = basering; //save the name of basering |
---|
707 | int NVars = nvars(@B); //number of variables in basering |
---|
708 | int i, j; |
---|
709 | |
---|
710 | matrix Rel = RelMatr(); //the matrix of relations |
---|
711 | |
---|
712 | string @ss = new_var(); //string of new variables |
---|
713 | string Par = parstr(@B); //string of parameters in old ring |
---|
714 | |
---|
715 | if (Par=="") // if there are no parameters |
---|
716 | { |
---|
717 | execute("ring @@@K=0,("+varstr(@B)+","+@ss+"), dp;"); //new ring with new variables |
---|
718 | } |
---|
719 | else //if there exist parameters |
---|
720 | { |
---|
721 | execute("ring @@@K=(0,"+Par+") ,("+varstr(@B)+","+@ss+"), dp;");//new ring with new variables |
---|
722 | }; |
---|
723 | |
---|
724 | matrix Rel = imap(@B, Rel); //consider the matrix of relations in new ring |
---|
725 | |
---|
726 | int Sz = NVars*NVars+NVars; // number of variables in new ring |
---|
727 | |
---|
728 | matrix M[Sz][Sz]; //to be the matrix of relations in new ring |
---|
729 | |
---|
730 | for(i=1; i<NVars; i++) //initialize that matrix of relations |
---|
731 | { |
---|
732 | for(j=i+1; j<=NVars; j++) |
---|
733 | { |
---|
734 | M[i,j] = Rel[i,j]; |
---|
735 | }; |
---|
736 | }; |
---|
737 | |
---|
738 | def @@K = nc_algebra(1, M); setring @@K; //now new ring @@K become a noncommutative ring |
---|
739 | |
---|
740 | list l; //list to define a homomorphism(isomorphism) |
---|
741 | poly @@F; |
---|
742 | for(i=1; i<=NVars; i++) //initializing list for involution |
---|
743 | { |
---|
744 | @@F=0; |
---|
745 | for(j=1; j<=NVars; j++) |
---|
746 | { |
---|
747 | execute( "@@F = @@F+"+NVAR(i,j)+"*"+string( var(j) )+";" ); |
---|
748 | }; |
---|
749 | l=l+list(@@F); |
---|
750 | }; |
---|
751 | |
---|
752 | matrix N = imap(@@@K,Rel); |
---|
753 | |
---|
754 | for(i=1; i<NVars; i++)//get matrix by applying the homomorphism to relations |
---|
755 | { |
---|
756 | for(j=i+1; j<=NVars; j++) |
---|
757 | { |
---|
758 | N[i,j]= l[j]*l[i] - l[i]*l[j] - Hom_Poly( N[i,j], l, NVars); |
---|
759 | }; |
---|
760 | }; |
---|
761 | kill l; |
---|
762 | //--------------------------------------------- |
---|
763 | //get the ideal of coefficients of N |
---|
764 | ideal J; |
---|
765 | ideal idN = simplify(ideal(N),2); |
---|
766 | J = ideal(coeffs( idN, var(1) ) ); |
---|
767 | for(i=2; i<=NVars; i++) |
---|
768 | { |
---|
769 | J = ideal( coeffs( J, var(i) ) ); |
---|
770 | }; |
---|
771 | J = simplify(J,2); |
---|
772 | //------------------------------------------------- |
---|
773 | if (( Par=="" ) && (n!=0)) //initializes the ring of relations |
---|
774 | { |
---|
775 | execute("ring @@KK=0,("+@ss+"), dp;"); |
---|
776 | } |
---|
777 | if (( Par=="" ) && (n==0)) //initializes the ring of relations |
---|
778 | { |
---|
779 | execute("ring @@KK=(0,@p),("+@ss+"), dp;"); |
---|
780 | } |
---|
781 | if ( Par!="" ) |
---|
782 | { |
---|
783 | execute("ring @@KK=(0,"+Par+"),("+@ss+"), dp;"); |
---|
784 | }; |
---|
785 | // execute("setring @@KK;"); |
---|
786 | // basering; |
---|
787 | ideal J = imap(@@K,J); // ideal, considered in @@KK now |
---|
788 | string snv = "["+string(NVars)+"]"; |
---|
789 | execute("matrix @@D"+snv+snv+"="+@ss+";"); // matrix with entries=new variables |
---|
790 | |
---|
791 | if (n>=2) |
---|
792 | { |
---|
793 | J = J, ideal( @@D*@@D-matrix( freemodule(NVars) ) ); // add the condition that homomorphism to square is just identity |
---|
794 | } |
---|
795 | if (n==0) |
---|
796 | { |
---|
797 | J = J, det(@@D)-@p; // det of non-unipotent matrix is nonzero |
---|
798 | } |
---|
799 | J = simplify(J,2); // without extra zeros |
---|
800 | list mL = minAssGTZ(J); // components not in GB |
---|
801 | int sL = size(mL); |
---|
802 | option(redSB); // important for reduced GBs |
---|
803 | option(redTail); |
---|
804 | matrix IM = @@D; // map |
---|
805 | list L = list(); // the answer |
---|
806 | list TL; |
---|
807 | ideal tmp = 0; |
---|
808 | for (i=1; i<=sL; i++)// compute GBs of components |
---|
809 | { |
---|
810 | TL = list(); |
---|
811 | TL[1] = std(mL[i]); |
---|
812 | tmp = NF( ideal(IM), TL[1] ); |
---|
813 | TL[2] = matrix(tmp,NVars, NVars); |
---|
814 | L[i] = TL; |
---|
815 | } |
---|
816 | export(L); |
---|
817 | ideal idJ = J; // debug-comfortable exports |
---|
818 | matrix matD = @@D; |
---|
819 | export(idJ); |
---|
820 | export(matD); |
---|
821 | return(@@KK); |
---|
822 | } |
---|
823 | example |
---|
824 | { "EXAMPLE:"; echo = 2; |
---|
825 | def a = makeWeyl(1); |
---|
826 | setring a; // this algebra is a first Weyl algebra |
---|
827 | a; |
---|
828 | def X = findAuto(2); // in contrast to findInvo look for automorphisms |
---|
829 | setring X; // ring with new variables - unknown coefficients |
---|
830 | X; |
---|
831 | size(L); // we have (size(L)) families in the answer |
---|
832 | // look at matrices, defining linear automorphisms: |
---|
833 | print(L[1][2]); // a first one: we see it is the identity |
---|
834 | print(L[2][2]); // and a second possible matrix; it is diagonal |
---|
835 | // L; // we can take a look on the whole list, too |
---|
836 | idJ; |
---|
837 | kill X; kill a; |
---|
838 | //----------- find all the linear automorphisms -------------------- |
---|
839 | //----------- use the call findAuto(0) -------------------- |
---|
840 | ring R = 0,(x,s),dp; |
---|
841 | def r = nc_algebra(1,s); setring r; // the shift algebra |
---|
842 | s*x; // the only relation in the algebra is: |
---|
843 | def Y = findAuto(0); |
---|
844 | setring Y; |
---|
845 | size(L); // here, we have 1 parametrized family |
---|
846 | print(L[1][2]); // here, @p is a nonzero parameter |
---|
847 | det(L[1][2]-@p); // check whether determinante is zero |
---|
848 | } |
---|