1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: nctools.lib,v 1.25 2007-05-13 18:29:43 levandov Exp $"; |
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3 | category="Noncommutative"; |
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4 | info=" |
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5 | LIBRARY: nctools.lib General tools for noncommutative algebras |
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6 | AUTHORS: Levandovskyy V., levandov@mathematik.uni-kl.de, |
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7 | @* Lobillo, F.J., jlobillo@ugr.es, |
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8 | @* Rabelo, C., crabelo@ugr.es. |
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9 | |
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10 | SUPPORT: DFG (Deutsche Forschungsgesellschaft) and Metodos algebraicos y efectivos en grupos cuanticos, BFM2001-3141, MCYT, Jose Gomez-Torrecillas (Main researcher). |
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11 | |
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12 | MAIN PROCEDURES: |
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13 | Gweights(r); compute weights for a compatible ordering in a G-algebra, |
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14 | weightedRing(r); change the ordering of a ring to a weighted one, |
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15 | ndcond(); the ideal of non-degeneracy conditions in G-algebra, |
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16 | Weyl([p]); create Weyl algebra structure in a basering (two different realizations), |
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17 | makeWeyl(n, [p]); return n-th Weyl algebra in (x(i),D(i)) presentation, |
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18 | makeHeisenberg(N, [p,d]); return n-th Heisenberg algebra in (x(i),y(i),h) realization, |
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19 | Exterior(); return qring, the exterior algebra of a basering, |
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20 | findimAlgebra(M,[r]); create finite dimensional algebra structure from the basering and the multiplication matrix M, |
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21 | SuperCommutative([b,e, Q]); return qring, the super-commutative algebra over a basering, |
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22 | rightStd(I); compute a right Groebner basis of an ideal, |
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23 | |
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24 | AUXILIARY PROCEDURES: |
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25 | ncRelations(r); recover the non-commutative relations of a G-algebra, |
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26 | isCentral(p); check for the commutativity of a polynomial in the G-algebra, |
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27 | isNC(); check whether basering is noncommutative, |
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28 | UpOneMatrix(N); return NxN matrix with 1's in the whole upper triagle, |
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29 | AltVarStart() return first alternating variable of a super-commutative algebra, |
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30 | AltVarEnd() return last alternating variable of a super-commutative algebra, |
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31 | IsSCA() checks whether current ring is a super-commutative algebra |
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32 | "; |
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33 | |
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34 | |
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35 | LIB "ring.lib"; // for rootofUnity |
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36 | LIB "poly.lib"; // for newtonDiag |
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37 | LIB "general.lib"; // for sort |
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38 | // LIB "ncalg.lib"; |
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39 | |
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40 | /////////////////////////////////////////////////////////////////////////////// |
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41 | |
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42 | // This procedure computes a weights vector for a G-algebra r |
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43 | |
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44 | proc Gweights(def r) |
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45 | "USAGE: Gweights(r); r a ring or a square matrix |
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46 | RETURN: intvec |
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47 | PURPOSE: compute the weight vector for the following G-algebra: |
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48 | @* for r itself, if it is of the type ring, |
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49 | @* or for a G-algebra, defined by the square polynomial matrix r |
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50 | THEORY: @code{Gweights} returns a vector, which must be used to redefine the G-algebra. If the input is a matrix and the output is the zero vector then there is not a G-algebra structure associated to these relations with respect to the given variables. Another possibility is to use @code{weightedRing} to obtain directly the G-algebra with the new weighted ordering. |
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51 | EXAMPLE: example Gweights; shows examples |
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52 | SEE ALSO: weightedRing |
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53 | "{ |
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54 | matrix tails; |
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55 | int novalid=0; |
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56 | if (typeof(r)=="ring") //a ring is admissible as input |
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57 | { |
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58 | setring r; |
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59 | def l = ncRelations(r); |
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60 | tails = l[2]; // l=C,D we need D, the tails of the relations |
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61 | } |
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62 | else |
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63 | { |
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64 | if ( (typeof(r)=="matrix") || (typeof(r)=="intmat") ) |
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65 | { |
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66 | if ( nrows(r)==ncols(r) ) //the input is a square matrix |
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67 | { |
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68 | tails = matrix(r); |
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69 | } |
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70 | else |
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71 | { |
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72 | novalid = 1; |
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73 | } |
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74 | } |
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75 | else |
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76 | { |
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77 | novalid=1; |
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78 | } |
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79 | } |
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80 | if (novalid==0) |
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81 | { |
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82 | intmat IM = SimplMat(tails); |
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83 | if ( size(IM)>1 ) |
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84 | { |
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85 | int n = ncols(tails); |
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86 | int m = nrows(IM)-1; |
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87 | int m1 = 0; |
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88 | int m2 = m; |
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89 | int m3 = 0; |
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90 | ring simplexring=(real,10),(x),lp;// The simplex procedure requires a basering of this type |
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91 | matrix M = IM; |
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92 | list sol = simplex (M,m,n,m1,m2,m3); |
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93 | return(weightvector(sol)); |
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94 | } |
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95 | else |
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96 | { |
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97 | "Invalid input"; //usually because the input is a one variable ring |
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98 | return(); |
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99 | } |
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100 | } |
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101 | else |
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102 | { |
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103 | "The input must be a ring or a square matrix"; |
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104 | return(); |
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105 | } |
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106 | } |
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107 | example |
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108 | { |
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109 | "EXAMPLE:";echo=2; |
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110 | ring r = (0,q),(a,b,c,d),lp; |
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111 | matrix C[4][4]; |
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112 | C[1,2]=q; C[1,3]=q; C[1,4]=1; C[2,3]=1; C[2,4]=q; C[3,4]=q; |
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113 | matrix D[4][4]; |
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114 | D[1,4]=(q-1/q)*b*c; |
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115 | ncalgebra(C,D); |
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116 | r; |
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117 | Gweights(r); |
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118 | Gweights(D); |
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119 | } |
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120 | |
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121 | /////////////////////////////////////////////////////////////////////////////// |
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122 | |
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123 | // This procedure take a ring r, call to Gweights(r) and use the output |
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124 | // of Gweights(r) to make a change of order in r |
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125 | // The output is a new ring, equal to r but the order |
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126 | // r must be a G-algebra |
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127 | |
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128 | proc weightedRing(def r) |
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129 | "USAGE: weightedRing(r); r a ring |
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130 | RETURN: ring |
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131 | PURPOSE: equip the variables of a ring with such weights,that the relations of new ring (with weighted variables) satisfies the ordering condition for G-algebras |
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132 | NOTE: activate this ring with the \"setring\" command |
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133 | EXAMPLE: example weightedRing; shows examples |
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134 | SEE ALSO: Gweights |
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135 | "{ |
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136 | def wv=Gweights(r); |
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137 | if (typeof(wv)=="intvec") |
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138 | { |
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139 | setring r; |
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140 | int n=nvars(r); |
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141 | // Generating an nxn-intmat order |
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142 | intmat m[n][n]; |
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143 | m[1,1]=wv[1]; |
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144 | int i; |
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145 | for (i=2; i<=n; i++) |
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146 | { |
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147 | m[1,i]=wv[i]; |
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148 | m[i,n+2-i]=1; |
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149 | } |
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150 | // End of generation. |
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151 | def lr=ncRelations(r); |
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152 | string newringstring="ring newring=("+charstr(r)+"),("+varstr(r)+"),M("+string(m)+")"; |
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153 | execute (newringstring); |
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154 | def lnewring=imap(r,lr); |
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155 | ncalgebra(lnewring[1],lnewring[2]); |
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156 | return(newring); |
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157 | } |
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158 | else |
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159 | { |
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160 | "Invalid input.";//usually because the input is a one variable ring |
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161 | return(); |
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162 | } |
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163 | } |
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164 | example |
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165 | { |
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166 | "EXAMPLE:";echo=2; |
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167 | ring r = (0,q),(a,b,c,d),lp; |
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168 | matrix C[4][4]; |
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169 | C[1,2]=q; C[1,3]=q; C[1,4]=1; C[2,3]=1; C[2,4]=q; C[3,4]=q; |
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170 | matrix D[4][4]; |
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171 | D[1,4]=(q-1/q)*b*c; |
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172 | ncalgebra(C,D); |
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173 | r; |
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174 | def t=weightedRing(r); |
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175 | setring t; t; |
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176 | } |
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177 | |
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178 | /////////////////////////////////////////////////////////////////////////////// |
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179 | |
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180 | // This procedure computes ei+ej-f with f running in Newton(pij) and deletes the zero rows |
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181 | |
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182 | static proc Cij(intmat M, int i,j) |
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183 | { |
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184 | M=(-1)*M; |
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185 | int nc=ncols(M); |
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186 | intvec N; |
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187 | int k; |
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188 | for (k=1; k<=nrows(M); k++) |
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189 | { |
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190 | M[k,i]=M[k,i]+1; |
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191 | M[k,j]=M[k,j]+1; |
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192 | if (intvec(M[k,1..nc])!=0) |
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193 | { |
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194 | N=N,intvec(M[k,1..nc]); |
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195 | } // we only want non-zero rows |
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196 | } |
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197 | if (size(N)>1) |
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198 | { |
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199 | N=N[2..size(N)]; // Deleting the zero added in the definition of N |
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200 | M=intmat(N,size(N)/nc,nc); // Conversion from vector to matrix |
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201 | } |
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202 | else |
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203 | { |
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204 | intmat M[1][1]=0; |
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205 | } |
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206 | return (M); |
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207 | } |
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208 | |
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209 | /////////////////////////////////////////////////////////////////////////////// |
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210 | |
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211 | // This procedure run over the matrix of pij calculating Cij |
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212 | |
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213 | static proc Ct(matrix P) |
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214 | { |
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215 | int k = ncols(P); |
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216 | intvec T = 0; |
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217 | int i,j; |
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218 | // int notails=1; |
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219 | def S; |
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220 | for (j=2; j<=k; j++) |
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221 | { |
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222 | for (i=1; i<j; i++) |
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223 | { |
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224 | if ( P[i,j] != 0 ) |
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225 | { |
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226 | // notails=0; |
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227 | S = newtonDiag(P[i,j]); |
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228 | S = Cij(S,i,j); |
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229 | if ( size(S)>1 ) |
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230 | { |
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231 | T = T,S; |
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232 | } |
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233 | } |
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234 | } |
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235 | } |
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236 | if ( size(T)==1 ) |
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237 | { |
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238 | intmat C[1][1] = 0; |
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239 | } |
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240 | else |
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241 | { |
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242 | T=T[2..size(T)]; // Deleting the zero added in the definition of T |
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243 | intmat C = intmat(T,size(T)/k,k); // Conversion from vector to matrix |
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244 | } |
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245 | return (C); |
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246 | } |
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247 | |
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248 | /////////////////////////////////////////////////////////////////////////////// |
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249 | |
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250 | // The purpose of this procedure is to produce the input matrix required by simplex procedure |
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251 | |
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252 | static proc SimplMat(matrix P) |
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253 | { |
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254 | intmat C=Ct(P); |
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255 | if (size(C)>1) |
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256 | { |
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257 | int r = nrows(C); |
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258 | int n = ncols(C); |
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259 | int f = 1+n+r; |
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260 | intmat M[f][n+1]=0; |
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261 | int i; |
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262 | for (i=2; i<=(n+1); i++) |
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263 | { |
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264 | M[1,i]=-1; // (0,-1,-1,-1,...) objective function in the first row |
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265 | } |
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266 | for (i=2; i<=f; i++) {M[i,1]=1;} // All the independent terms are 1 |
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267 | for (i=2; i<=(n+1); i++) {M[i,i]=-1;} // wi>=1 is an identity matrix |
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268 | M[(n+2)..f,2..(n+1)]=(-1)*intvec(C); // <wi,a> >= 1, a in C ... |
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269 | } |
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270 | else |
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271 | { |
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272 | int n = ncols(P); |
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273 | int f = 1+n; |
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274 | intmat M[f][n+1]=0; |
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275 | int i; |
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276 | for (i=2; i<=(n+1); i++) {M[1,i]=-1;} // (0,-1,-1,-1,...) objective function in the first row |
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277 | for (i=2; i<=f; i++) {M[i,1]=1;} // All the independent terms are 1 |
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278 | for (i=2; i<=(n+1); i++) {M[i,i]=-1;} // wi>=1 is an identity matrix |
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279 | } |
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280 | return (M); |
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281 | } |
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282 | |
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283 | /////////////////////////////////////////////////////////////////////////////// |
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284 | |
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285 | // This procedure generates a nice output of the simplex method consisting of a vector |
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286 | // with the solutions. The vector is ordered. |
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287 | |
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288 | static proc weightvector(list l) |
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289 | "ASSUME: l is the output of simplex. |
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290 | RETURN: if there is a solution, an intvec with it will be returned" |
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291 | { |
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292 | matrix m=l[1]; |
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293 | intvec nv=l[3]; |
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294 | int sol=l[2]; |
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295 | int rows=nrows(m); |
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296 | int N=l[6]; |
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297 | intmat wv[1][N]=0; |
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298 | int i; |
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299 | if (sol) |
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300 | { |
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301 | "no solution satisfies the given constraints"; |
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302 | } |
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303 | else |
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304 | { |
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305 | for ( i = 2; i <= rows; i++ ) |
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306 | { |
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307 | if ( nv[i-1] <= N ) |
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308 | { |
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309 | wv[1,nv[i-1]]=int(m[i,1]); |
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310 | } |
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311 | } |
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312 | } |
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313 | return (intvec(wv)); |
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314 | } |
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315 | |
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316 | |
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317 | |
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318 | /////////////////////////////////////////////////////////////////////////////// |
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319 | |
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320 | // This procedure recover the non-conmutative relations (matrices C and D) |
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321 | |
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322 | proc ncRelations(def r) |
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323 | "USAGE: ncRelations(r); r a ring |
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324 | RETURN: list L with two elements, both elements are of type matrix: |
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325 | @* L[1] = matrix of coefficients C, |
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326 | @* L[2] = matrix of polynomials D |
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327 | PURPOSE: recover the noncommutative relations via matrices C and D from |
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328 | a noncommutative ring |
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329 | SEE ALSO: ringlist, G-algebras |
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330 | EXAMPLE: example ncRelations; shows examples |
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331 | "{ |
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332 | list l; |
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333 | if (typeof(r)=="ring") |
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334 | { |
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335 | int n=nvars(r); |
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336 | matrix C[n][n]=0; |
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337 | matrix D[n][n]=0; |
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338 | poly f; poly g; |
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339 | if (n>1) |
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340 | { |
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341 | int i,j; |
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342 | for (i=2; i<=n; i++) |
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343 | { |
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344 | for (j=1; j<i; j++) |
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345 | { |
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346 | f=var(i)*var(j); // yx=c*xy+... |
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347 | g=var(j)*var(i); // xy |
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348 | while (C[j,i]==0) |
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349 | { |
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350 | if (leadmonom(f)==leadmonom(g)) |
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351 | { |
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352 | C[j,i]=leadcoef(f); |
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353 | D[j,i]=D[j,i]+f-lead(f); |
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354 | } |
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355 | else |
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356 | { |
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357 | D[j,i]=D[j,i]+lead(f); |
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358 | f=f-lead(f); |
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359 | } |
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360 | } |
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361 | } |
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362 | } |
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363 | l=C,D; |
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364 | } |
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365 | else { "The ring must have two or more variables"; } |
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366 | } |
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367 | else { "The input must be of a type ring";} |
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368 | return (l); |
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369 | } |
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370 | example |
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371 | { |
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372 | "EXAMPLE:";echo=2; |
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373 | ring r = 0,(x,y,z),dp; |
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374 | matrix C[3][3]=0,1,2,0,0,-1,0,0,0; |
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375 | print(C); |
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376 | matrix D[3][3]=0,1,2y,0,0,-2x+y+1; |
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377 | print(D); |
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378 | ncalgebra(C,D); |
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379 | r; |
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380 | def l=ncRelations(r); |
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381 | print (l[1]); |
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382 | print (l[2]); |
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383 | } |
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384 | |
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385 | /////////////////////////////////////////////////////////////////////////////// |
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386 | |
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387 | proc findimAlgebra(matrix M, list #) |
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388 | "USAGE: findimAlgebra(M,[r]); M a matrix, r an optional ring |
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389 | RETURN: nothing |
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390 | PURPOSE: define a finite dimensional algebra structure on a ring |
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391 | NOTE: the matrix M is used to define the relations x(j)*x(i) = M[i,j] in the |
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392 | basering (by default) or in the optional ring r. |
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393 | @* The procedure equips the ring with the noncommutative structure. |
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394 | @* The procedure exports the ideal (not a two-sided Groebner basis!), called @code{fdQuot}, for further qring definition. |
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395 | THEORY: finite dimensional algebra can be represented as a factor algebra |
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396 | of a G-algebra modulo certain two-sided ideal. The relations of a f.d. algebra are thus naturally divided into two groups: firstly, the relations |
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397 | on the variables of the ring, making it into G-algebra and the rest of them, which constitute the ideal which will be factored out. |
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398 | EXAMPLE: example findimAlgebra; shows examples |
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399 | " |
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400 | { |
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401 | if (size(#) >0) |
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402 | { |
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403 | if ( typeof(#[1])!="ring" ) { return();} |
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404 | else |
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405 | { |
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406 | def @R1 = #[1]; |
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407 | setring @R1; |
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408 | } |
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409 | } |
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410 | int i,j; |
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411 | int n=nvars(basering); |
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412 | poly p; |
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413 | ideal I; |
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414 | number c; |
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415 | matrix C[n][n]; |
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416 | matrix D[n][n]; |
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417 | for (i=1; i<=n; i++) |
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418 | { |
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419 | for (j=i; j<=n; j++) |
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420 | { |
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421 | p=var(i)*var(j)-M[i,j]; |
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422 | if ( (size(I)==1) && (I[1]==0) ) { I=p; } |
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423 | else { I=I,p; } |
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424 | if (j>i) |
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425 | { |
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426 | if ((M[i,j]!=0) && (M[j,i]!=0)) |
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427 | { |
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428 | c = leadcoef(M[j,i])/leadcoef(M[i,j]); |
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429 | } |
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430 | else |
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431 | { |
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432 | c = 1; |
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433 | } |
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434 | C[i,j]=c; |
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435 | D[i,j]= - M[j,i] +c*M[i,j]; |
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436 | } |
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437 | } |
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438 | } |
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439 | ncalgebra(C,D); |
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440 | ideal fdQuot = I; |
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441 | export fdQuot; |
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442 | } |
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443 | example |
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444 | { |
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445 | "EXAMPLE:";echo=2; |
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446 | ring r=(0,a,b),(x(1..3)),dp; |
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447 | matrix S[3][3]; |
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448 | S[2,3]=a*x(1); S[3,2]=-b*x(1); |
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449 | findimAlgebra(S); |
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450 | fdQuot = twostd(fdQuot); |
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451 | qring Qr = fdQuot; |
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452 | Qr; |
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453 | } |
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454 | |
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455 | /////////////////////////////////////////////////////////////////////////////// |
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456 | |
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457 | proc isCentral(poly p, list #) |
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458 | "USAGE: isCentral(p); p poly |
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459 | RETURN: int, 1 if p commutes with all variables and 0 otherwise |
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460 | PURPOSE: check whether p is central in a basering (that is, commutes with every generator of a ring) |
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461 | NOTE: if @code{printlevel} > 0, the procedure displays intermediate information (by default, @code{printlevel}=0 ) |
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462 | EXAMPLE: example isCentral; shows examples |
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463 | "{ |
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464 | //v an integer (with v!=0, procedure will be verbose) |
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465 | int N = nvars(basering); |
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466 | int in; |
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467 | int flag = 1; |
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468 | poly q = 0; |
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469 | for (in=1; in<=N; in++) |
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470 | { |
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471 | q = p*var(in)-var(in)*p; |
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472 | if (q!=0) |
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473 | { |
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474 | if ( (size(#) >0 ) || (printlevel>0) ) |
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475 | { |
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476 | "Noncentral at:", var(in); |
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477 | } |
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478 | flag = 0; |
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479 | } |
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480 | } |
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481 | return(flag); |
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482 | } |
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483 | example |
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484 | { |
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485 | "EXAMPLE:";echo=2; |
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486 | ring r=0,(x,y,z),dp; |
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487 | matrix D[3][3]=0; |
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488 | D[1,2]=-z; |
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489 | D[1,3]=2*x; |
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490 | D[2,3]=-2*y; |
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491 | ncalgebra(1,D); // this is U(sl_2) |
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492 | poly c = 4*x*y+z^2-2*z; |
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493 | printlevel = 0; |
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494 | isCentral(c); |
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495 | poly h = x*c; |
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496 | printlevel = 1; |
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497 | isCentral(h); |
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498 | } |
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499 | |
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500 | /////////////////////////////////////////////////////////////////////////////// |
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501 | |
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502 | proc UpOneMatrix(int N) |
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503 | "USAGE: UpOneMatrix(n); n an integer |
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504 | RETURN: intmat |
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505 | PURPOSE: compute an n x n matrix with 1's in the whole upper triangle |
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506 | NOTE: helpful for setting noncommutative algebras with complicated |
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507 | coefficient matrices |
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508 | EXAMPLE: example UpOneMatrix; shows examples |
---|
509 | "{ |
---|
510 | int ii,jj; |
---|
511 | intmat U[N][N]=0; |
---|
512 | for (ii=1;ii<N;ii++) |
---|
513 | { |
---|
514 | for (jj=ii+1;jj<=N;jj++) |
---|
515 | { |
---|
516 | U[ii,jj]=1; |
---|
517 | } |
---|
518 | } |
---|
519 | return(U); |
---|
520 | } |
---|
521 | example |
---|
522 | { |
---|
523 | "EXAMPLE:";echo=2; |
---|
524 | ring r = (0,q),(x,y,z),dp; |
---|
525 | matrix C = UpOneMatrix(3); |
---|
526 | C[1,3] = q; |
---|
527 | print(C); |
---|
528 | ncalgebra(C,0); |
---|
529 | r; |
---|
530 | } |
---|
531 | |
---|
532 | /////////////////////////////////////////////////////////////////////////////// |
---|
533 | proc ndcond(list #) |
---|
534 | "USAGE: ndcond(); |
---|
535 | RETURN: ideal |
---|
536 | PURPOSE: compute the non-degeneracy conditions of the basering |
---|
537 | NOTE: if @code{printlevel} > 0, the procedure displays intermediate information (by default, @code{printlevel}=0 ) |
---|
538 | EXAMPLE: example ndcond; shows examples |
---|
539 | " |
---|
540 | { |
---|
541 | // internal documentation, for tests etc |
---|
542 | // 1st arg: v an optional integer (if v!=0, will be verbose) |
---|
543 | // if the second argument is given, produces ndc wrt powers x^N |
---|
544 | int N = 1; |
---|
545 | int Verbose = 0; |
---|
546 | if ( size(#)>=1 ) { Verbose = int(#[1]); } |
---|
547 | if ( size(#)>=2 ) { N = int(#[2]); } |
---|
548 | Verbose = ((Verbose) || (printlevel>0)); |
---|
549 | int cnt = 1; |
---|
550 | int numvars = nvars(basering); |
---|
551 | int a,b,c; |
---|
552 | poly p = 1; |
---|
553 | ideal res = 0; |
---|
554 | for (cnt=1; cnt<=N; cnt++) |
---|
555 | { |
---|
556 | if (Verbose) { "Processing degree :",cnt;} |
---|
557 | for (a=1; a<=numvars-2; a++) |
---|
558 | { |
---|
559 | for (b=a+1; b<=numvars-1; b++) |
---|
560 | { |
---|
561 | for(c=b+1; c<=numvars; c++) |
---|
562 | { |
---|
563 | p = (var(c)^cnt)*(var(b)^cnt); |
---|
564 | p = p*(var(a)^cnt); |
---|
565 | p = p-(var(c)^cnt)*((var(b)^cnt)*(var(a)^cnt)); |
---|
566 | if (Verbose) {a,".",b,".",c,".";} |
---|
567 | if (p!=0) |
---|
568 | { |
---|
569 | if ( res==0 ) |
---|
570 | { |
---|
571 | res[1] = p; |
---|
572 | } |
---|
573 | else |
---|
574 | { |
---|
575 | res = res,p; |
---|
576 | } |
---|
577 | if (Verbose) { "failed:",p; } |
---|
578 | } |
---|
579 | } |
---|
580 | } |
---|
581 | } |
---|
582 | if (Verbose) { "done"; } |
---|
583 | } |
---|
584 | return(res); |
---|
585 | } |
---|
586 | example |
---|
587 | { |
---|
588 | "EXAMPLE:";echo=2; |
---|
589 | ring r = (0,q1,q2),(x,y,z),dp; |
---|
590 | matrix C[3][3]; |
---|
591 | C[1,2]=q2; C[1,3]=q1; C[2,3]=1; |
---|
592 | matrix D[3][3]; |
---|
593 | D[1,2]=x; D[1,3]=z; |
---|
594 | ncalgebra(C,D); |
---|
595 | r; |
---|
596 | ideal j=ndcond(); // the silent version |
---|
597 | j; |
---|
598 | printlevel=1; |
---|
599 | ideal i=ndcond(); // the verbose version |
---|
600 | i; |
---|
601 | } |
---|
602 | |
---|
603 | |
---|
604 | /////////////////////////////////////////////////////////////////////////////// |
---|
605 | proc Weyl(list #) |
---|
606 | "USAGE: Weyl([p]); p an optional integer |
---|
607 | RETURN: nothing |
---|
608 | PURPOSE: create a Weyl algebra structure on a basering |
---|
609 | NOTE: suppose the number of variables of a basering is 2k. |
---|
610 | (if this number is odd, an error message will be returned) |
---|
611 | @* by default, the procedure treats first k variables as coordinates x_i and the last k as differentials d_i |
---|
612 | @* if nonzero p is given, the procedure treats 2k variables of a basering as k pairs (x_i,d_i), i.e. variables with odd numbers are treated as coordinates and with even numbers as differentials |
---|
613 | SEE ALSO: makeWeyl |
---|
614 | EXAMPLE: example Weyl; shows examples |
---|
615 | " |
---|
616 | { |
---|
617 | //there are two possibilities for choosing the PBW basis. |
---|
618 | //The variables have names x(i) for coordinates and d(i) for partial |
---|
619 | // differentiations. By default, the procedure |
---|
620 | //creates a ring, where the variables are ordered as x(1..n),d(1..n). the |
---|
621 | // tensor product-like realization x(1),d(1),x(2),d(2),... is used. |
---|
622 | string rname=nameof(basering); |
---|
623 | if ( rname == "basering") // i.e. no ring has been set yet |
---|
624 | { |
---|
625 | "You have to call the procedure from the ring"; |
---|
626 | return(); |
---|
627 | } |
---|
628 | int @chr = 0; |
---|
629 | if ( size(#) > 0 ) |
---|
630 | { |
---|
631 | if ( typeof( #[1] ) == "int" ) |
---|
632 | { |
---|
633 | @chr = #[1]; |
---|
634 | } |
---|
635 | } |
---|
636 | int nv = nvars(basering); |
---|
637 | int N = nv div 2; |
---|
638 | if ((nv % 2) != 0) |
---|
639 | { |
---|
640 | "Cannot create Weyl structure for an odd number of generators"; |
---|
641 | return(); |
---|
642 | } |
---|
643 | matrix @D[nv][nv]; |
---|
644 | int i; |
---|
645 | for ( i=1; i<=N; i++ ) |
---|
646 | { |
---|
647 | if ( @chr==0 ) // default |
---|
648 | { |
---|
649 | @D[i,N+i]=1; |
---|
650 | } |
---|
651 | else |
---|
652 | { |
---|
653 | @D[2*i-1,2*i]=1; |
---|
654 | } |
---|
655 | } |
---|
656 | ncalgebra(1,@D); |
---|
657 | return(); |
---|
658 | } |
---|
659 | example |
---|
660 | { |
---|
661 | "EXAMPLE:";echo=2; |
---|
662 | ring A1=0,(x(1..2),d(1..2)),dp; |
---|
663 | Weyl(); |
---|
664 | A1; |
---|
665 | kill A1; |
---|
666 | ring B1=0,(x1,d1,x2,d2),dp; |
---|
667 | Weyl(1); |
---|
668 | B1; |
---|
669 | } |
---|
670 | |
---|
671 | /////////////////////////////////////////////////////////////////////////////// |
---|
672 | proc makeHeisenberg(int N, list #) |
---|
673 | "USAGE: makeHeisenberg(n, [p,d]); int n (setting 2n+1 variables), optional int p (field characteristic), optional int d (power of h in the commutator) |
---|
674 | RETURN: nothing |
---|
675 | PURPOSE: create an n-th Heisenberg algebra in the variables x(1),y(1),...,x(n),y(n),h |
---|
676 | SEE ALSO: makeWeyl |
---|
677 | NOTE: activate this ring with the \"setring\" command |
---|
678 | EXAMPLE: example makeHeisenberg; shows examples |
---|
679 | " |
---|
680 | { |
---|
681 | int @chr = 0; |
---|
682 | int @deg = 1; |
---|
683 | if ( size(#) > 0 ) |
---|
684 | { |
---|
685 | if ( typeof( #[1] ) == "int" ) |
---|
686 | { |
---|
687 | @chr = #[1]; |
---|
688 | } |
---|
689 | } |
---|
690 | if ( size(#) > 1 ) |
---|
691 | { |
---|
692 | if ( typeof( #[2] ) == "int" ) |
---|
693 | { |
---|
694 | @deg = #[2]; |
---|
695 | if (@deg <1) { @deg = 1; } |
---|
696 | } |
---|
697 | } |
---|
698 | ring @@r=@chr,(x(1..N),y(1..N),h),lp; |
---|
699 | matrix D[2*N+1][2*N+1]; |
---|
700 | int i; |
---|
701 | for (i=1;i<=N;i++) |
---|
702 | { |
---|
703 | D[i,N+i]=h^@deg; |
---|
704 | } |
---|
705 | ncalgebra(1,D); |
---|
706 | return(@@r); |
---|
707 | } |
---|
708 | example |
---|
709 | { |
---|
710 | "EXAMPLE:";echo=2; |
---|
711 | def a = makeHeisenberg(2); |
---|
712 | setring a; a; |
---|
713 | def H3 = makeHeisenberg(3, 7, 2); |
---|
714 | setring H3; H3; |
---|
715 | } |
---|
716 | |
---|
717 | /////////////////////////////////////////////////////////////////////////////// |
---|
718 | proc SuperCommutative(list #) |
---|
719 | "USAGE: SuperCommutative([b,[e, [Q]]]); |
---|
720 | RETURN: qring |
---|
721 | PURPOSE: create the super-commutative algebra over a basering, |
---|
722 | NOTE: activate this qring with the \"setring\" command |
---|
723 | THEORY: given a basering, this procedure introduces the anticommutative relations x(j)x(i)=-x(i)x(j) for all e>=j>i>=b, |
---|
724 | @* moreover, creates a factor algebra modulo the two-sided ideal, generated by x(b)^2, ..., x(e)^2[ + Q] |
---|
725 | EXAMPLE: example SuperCommutative; shows examples |
---|
726 | " |
---|
727 | { |
---|
728 | string rname=nameof(basering); |
---|
729 | if ( rname == "basering") // i.e. no ring has been set yet |
---|
730 | { |
---|
731 | ERROR("You have to call the procedure from the ring"); |
---|
732 | return(); |
---|
733 | } |
---|
734 | int N = nvars(basering); |
---|
735 | |
---|
736 | int b = 1; |
---|
737 | int e = N; |
---|
738 | |
---|
739 | def saveRing = basering; |
---|
740 | ideal Q = 0; |
---|
741 | |
---|
742 | if(size(#)>0) |
---|
743 | { |
---|
744 | if(typeof(#[1]) != "int") |
---|
745 | { |
---|
746 | ERROR("First argument 'b' must be an integer!"); |
---|
747 | return(); |
---|
748 | } |
---|
749 | b = #[1]; |
---|
750 | } |
---|
751 | |
---|
752 | if(size(#)>1) |
---|
753 | { |
---|
754 | if(typeof(#[2]) != "int") |
---|
755 | { |
---|
756 | ERROR("Last argument 'e' must be an integer!"); |
---|
757 | return(); |
---|
758 | } |
---|
759 | e = #[2]; |
---|
760 | } |
---|
761 | |
---|
762 | if(size(#)>2) |
---|
763 | { |
---|
764 | if(typeof(#[3]) != "ideal") |
---|
765 | { |
---|
766 | ERROR("Last argument 'Q' must be an ideal!"); |
---|
767 | return(); |
---|
768 | } |
---|
769 | Q = #[3]; |
---|
770 | } |
---|
771 | |
---|
772 | list CurrRing = ringlist(basering); |
---|
773 | def @R = ring(CurrRing); |
---|
774 | setring @R; // @R; |
---|
775 | |
---|
776 | matrix @E = UpOneMatrix(N); |
---|
777 | |
---|
778 | int i, j; |
---|
779 | |
---|
780 | for ( i = b; i < e; i++ ) |
---|
781 | { |
---|
782 | for ( j = i+1; j <= e; j++ ) |
---|
783 | { |
---|
784 | @E[i, j] = -1; |
---|
785 | } |
---|
786 | } |
---|
787 | |
---|
788 | ncalgebra(@E, 0); |
---|
789 | |
---|
790 | ideal Q = fetch(saveRing, Q); |
---|
791 | j = ncols(Q) + 1; |
---|
792 | |
---|
793 | for ( i=e; i>=b; i--, j++ ) |
---|
794 | { |
---|
795 | Q[j] = var(i)^2; |
---|
796 | } |
---|
797 | Q = twostd(Q); |
---|
798 | qring @EA = Q; |
---|
799 | |
---|
800 | // "Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "]."; |
---|
801 | return(@EA); |
---|
802 | } |
---|
803 | example |
---|
804 | { |
---|
805 | "EXAMPLE:";echo=2; |
---|
806 | ring R = 0,(x(1..4)),dp; // global! |
---|
807 | def ER = SuperCommutative(); // the same as Exterior (b = 1, e = N) |
---|
808 | setring ER; ER; |
---|
809 | "Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "]."; |
---|
810 | kill R; kill ER; |
---|
811 | ring R = 0,(x(1..4)),(lp(1), dp(3)); // global! |
---|
812 | def ER = SuperCommutative(2); // b = 2, e = N |
---|
813 | setring ER; ER; |
---|
814 | "Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "]."; |
---|
815 | kill R; kill ER; |
---|
816 | ring R = 0,(x(1..6)),(ls(2), dp(2), lp(2)); // local! |
---|
817 | def ER = SuperCommutative(3,4); // b = 3, e = 4 |
---|
818 | setring ER; ER; |
---|
819 | "Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "]."; |
---|
820 | kill R; kill ER; |
---|
821 | } |
---|
822 | |
---|
823 | |
---|
824 | static proc ParseSCA() |
---|
825 | " |
---|
826 | RETURN: list {AltVarStart, AltVarEnd} is currRing is SCA, returns undef otherwise. |
---|
827 | " |
---|
828 | { |
---|
829 | def saveRing = basering; |
---|
830 | list L = ringlist(saveRing); |
---|
831 | |
---|
832 | if( size(L)!=6 ) |
---|
833 | { |
---|
834 | return("The current ring is commutative!"); |
---|
835 | } |
---|
836 | |
---|
837 | module D = simplify(L[6], 2 + 4); |
---|
838 | |
---|
839 | if( size(D)>0 ) |
---|
840 | { |
---|
841 | return("The current ring is not SCA! (D!=0)"); |
---|
842 | } |
---|
843 | |
---|
844 | int i, j; |
---|
845 | int N = nvars(saveRing); |
---|
846 | |
---|
847 | int b = N+1; |
---|
848 | int e = -1; |
---|
849 | |
---|
850 | matrix C = L[5]; |
---|
851 | poly c; |
---|
852 | |
---|
853 | for( i = 1; i < N; i++ ) |
---|
854 | { |
---|
855 | for( j = i+1; j <= N; j++ ) |
---|
856 | { |
---|
857 | c = C[i, j]; |
---|
858 | |
---|
859 | if( c == -1 ) |
---|
860 | { |
---|
861 | if(i < b) |
---|
862 | { |
---|
863 | b = i; |
---|
864 | } |
---|
865 | |
---|
866 | if(j > e) |
---|
867 | { |
---|
868 | e = j; |
---|
869 | } |
---|
870 | } else |
---|
871 | { // should commute |
---|
872 | if( c!=1 ) |
---|
873 | { |
---|
874 | return("The current ring is not SCA! (C["+ string(i)+"," + string(j)+"]!=1)"); |
---|
875 | } |
---|
876 | } |
---|
877 | } |
---|
878 | } |
---|
879 | |
---|
880 | if( (b > N) || (e < 1)) |
---|
881 | { |
---|
882 | return("The current ring is commutative!"); |
---|
883 | } |
---|
884 | |
---|
885 | for( i = 1; i < N; i++ ) |
---|
886 | { |
---|
887 | for( j = i+1; j <= N; j++ ) |
---|
888 | { |
---|
889 | c = C[i, j]; |
---|
890 | |
---|
891 | if( (b <= i) && (j <= e) ) // S <= i < j <= E |
---|
892 | { // anticommutative part |
---|
893 | if( c!= -1 ) |
---|
894 | { |
---|
895 | return("The current ring is not SCA! (C["+ string(i)+"," + string(j)+"]!=-1)"); |
---|
896 | } |
---|
897 | } else |
---|
898 | { // should commute |
---|
899 | if( c!=1 ) |
---|
900 | { |
---|
901 | return("The current ring is not SCA! (C["+ string(i)+"," + string(j)+"]!=1)"); |
---|
902 | } |
---|
903 | } |
---|
904 | } |
---|
905 | } |
---|
906 | |
---|
907 | list LL = list(L[1], L[2], L[3], ideal(0), L[5], L[6]); |
---|
908 | ideal Q = L[4]; |
---|
909 | // "Q = ", string(Q); |
---|
910 | |
---|
911 | def E = ring(LL); |
---|
912 | setring E; // not a qring! |
---|
913 | // E; |
---|
914 | |
---|
915 | ideal Q = fetch(saveRing, Q); // should belong to E! |
---|
916 | Q = twostd(Q); |
---|
917 | |
---|
918 | // "Q = ", string(Q); |
---|
919 | |
---|
920 | for( i = b; i <= e; i++ ) |
---|
921 | { |
---|
922 | if( NF(var(i)^2, Q) != 0 ) |
---|
923 | { |
---|
924 | setring saveRing; |
---|
925 | return("The current ring is not SCA! (Wrong quotient ideal)"); |
---|
926 | } |
---|
927 | } |
---|
928 | |
---|
929 | //////////////////////////////////////////////////////////////////////// |
---|
930 | // ok. it is a SCA!!! |
---|
931 | |
---|
932 | ideal QQ; |
---|
933 | |
---|
934 | for( i = e; i >= b; i-- ) |
---|
935 | { |
---|
936 | QQ[i - b + 1] = var(i)^2; |
---|
937 | } |
---|
938 | |
---|
939 | QQ = twostd(QQ); |
---|
940 | Q = simplify(NF(Q, QQ), 1 + 2 + 4); |
---|
941 | |
---|
942 | setring saveRing; |
---|
943 | |
---|
944 | ideal QQ = fetch(E, Q); |
---|
945 | |
---|
946 | return(list(b, e, QQ)); |
---|
947 | } |
---|
948 | |
---|
949 | |
---|
950 | |
---|
951 | |
---|
952 | /////////////////////////////////////////////////////////////////////////////// |
---|
953 | proc AltVarStart() |
---|
954 | "USAGE: AltVarStart(); |
---|
955 | RETURN: int |
---|
956 | PURPOSE: returns the number of the first alternating variable of basering |
---|
957 | NOTE: basering should be a super-commutative algebra! |
---|
958 | EXAMPLE: example AltVarStart; shows examples |
---|
959 | " |
---|
960 | { |
---|
961 | def l = ParseSCA(); |
---|
962 | |
---|
963 | if( typeof(l) != "string" ) |
---|
964 | { |
---|
965 | return(l[1]); |
---|
966 | } |
---|
967 | |
---|
968 | ERROR(l); |
---|
969 | return(); |
---|
970 | } |
---|
971 | example |
---|
972 | { |
---|
973 | "EXAMPLE:";echo=2; |
---|
974 | ring R = 0,(x(1..4)),dp; // global! |
---|
975 | def ER = SuperCommutative(2); // (b = 2, e = N) |
---|
976 | setring ER; ER; |
---|
977 | "Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "]."; |
---|
978 | } |
---|
979 | |
---|
980 | /////////////////////////////////////////////////////////////////////////////// |
---|
981 | proc AltVarEnd() |
---|
982 | "USAGE: AltVarStart(); |
---|
983 | RETURN: int |
---|
984 | PURPOSE: returns the number of the last alternating variable of basering |
---|
985 | NOTE: basering should be a super-commutative algebra! |
---|
986 | EXAMPLE: example AltVarEnd; shows examples |
---|
987 | " |
---|
988 | { |
---|
989 | def l = ParseSCA(); |
---|
990 | |
---|
991 | if( typeof(l) != "string" ) |
---|
992 | { |
---|
993 | return(l[2]); |
---|
994 | } |
---|
995 | |
---|
996 | ERROR(l); |
---|
997 | return(); |
---|
998 | } |
---|
999 | example |
---|
1000 | { |
---|
1001 | "EXAMPLE:";echo=2; |
---|
1002 | ring R = 0,(x(1..4)),dp; // global! |
---|
1003 | def ER = SuperCommutative(2); // (b = 2, e = N) |
---|
1004 | setring ER; ER; |
---|
1005 | "Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "]."; |
---|
1006 | } |
---|
1007 | |
---|
1008 | /////////////////////////////////////////////////////////////////////////////// |
---|
1009 | proc IsSCA() |
---|
1010 | "USAGE: IsSCA(); |
---|
1011 | RETURN: int |
---|
1012 | PURPOSE: returns 1 if basering is a supercommutative algebra and 0 otherwise. |
---|
1013 | NOTE: no error message! |
---|
1014 | EXAMPLE: example IsSCA; shows examples |
---|
1015 | " |
---|
1016 | { |
---|
1017 | def l = ParseSCA(); |
---|
1018 | |
---|
1019 | if( typeof(l) != "string" ) |
---|
1020 | { |
---|
1021 | return(1); |
---|
1022 | } |
---|
1023 | |
---|
1024 | return(0); |
---|
1025 | } |
---|
1026 | example |
---|
1027 | { |
---|
1028 | "EXAMPLE:";echo=2; |
---|
1029 | ///////////////////////////////////////////////////////////////////// |
---|
1030 | ring R = 0,(x(1..4)),dp; // commutative |
---|
1031 | if(IsSCA()) |
---|
1032 | { "Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "]."; } |
---|
1033 | else |
---|
1034 | { "Not a supercommutative algebra!!!"; } |
---|
1035 | kill R; |
---|
1036 | ///////////////////////////////////////////////////////////////////// |
---|
1037 | ring R = 0,(x(1..4)),dp; |
---|
1038 | ncalgebra(1, 0); // still commutative! |
---|
1039 | if(IsSCA()) |
---|
1040 | { "Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "]."; } |
---|
1041 | else |
---|
1042 | { "Not a supercommutative algebra!!!"; } |
---|
1043 | kill R; |
---|
1044 | ///////////////////////////////////////////////////////////////////// |
---|
1045 | ring R = 0,(x(1..4)),dp; |
---|
1046 | list CurrRing = ringlist(R); |
---|
1047 | def ER = ring(CurrRing); |
---|
1048 | setring ER; // R; |
---|
1049 | |
---|
1050 | matrix E = UpOneMatrix(nvars(R)); |
---|
1051 | |
---|
1052 | int i, j; int b = 2; int e = 3; |
---|
1053 | |
---|
1054 | for ( i = b; i < e; i++ ) |
---|
1055 | { |
---|
1056 | for ( j = i+1; j <= e; j++ ) |
---|
1057 | { |
---|
1058 | E[i, j] = -1; |
---|
1059 | } |
---|
1060 | } |
---|
1061 | |
---|
1062 | ncalgebra(E,0); |
---|
1063 | if(IsSCA()) |
---|
1064 | { "Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "]."; } |
---|
1065 | else |
---|
1066 | { "Not a supercommutative algebra!!!"; } |
---|
1067 | kill R; |
---|
1068 | kill ER; |
---|
1069 | ///////////////////////////////////////////////////////////////////// |
---|
1070 | ring R = 0,(x(1..4)),dp; |
---|
1071 | def ER = SuperCommutative(2); // (b = 2, e = N) |
---|
1072 | setring ER; ER; |
---|
1073 | if(IsSCA()) |
---|
1074 | { "This is a SCA! Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "]."; } |
---|
1075 | else |
---|
1076 | { "Not a supercommutative algebra!!!"; } |
---|
1077 | kill R; |
---|
1078 | kill ER; |
---|
1079 | |
---|
1080 | } |
---|
1081 | |
---|
1082 | |
---|
1083 | |
---|
1084 | /////////////////////////////////////////////////////////////////////////////// |
---|
1085 | proc Exterior(list #) |
---|
1086 | "USAGE: Exterior(); |
---|
1087 | RETURN: qring |
---|
1088 | PURPOSE: create the exterior algebra of a basering |
---|
1089 | NOTE: activate this qring with the \"setring\" command |
---|
1090 | THEORY: given a basering, this procedure introduces the anticommutative relations x(j)x(i)=-x(i)x(j) for all j>i, |
---|
1091 | @* moreover, creates a factor algebra modulo the two-sided ideal, generated by x(i)^2 for all i |
---|
1092 | EXAMPLE: example Exterior; shows examples |
---|
1093 | " |
---|
1094 | { |
---|
1095 | string rname=nameof(basering); |
---|
1096 | if ( rname == "basering") // i.e. no ring has been set yet |
---|
1097 | { |
---|
1098 | "You have to call the procedure from the ring"; |
---|
1099 | return(); |
---|
1100 | } |
---|
1101 | int N = nvars(basering); |
---|
1102 | string NewRing = "ring @R=("+charstr(basering)+"),("+varstr(basering)+"),("+ordstr(basering)+");"; |
---|
1103 | execute(NewRing); |
---|
1104 | matrix @E = UpOneMatrix(N); |
---|
1105 | @E = -1*(@E); |
---|
1106 | ncalgebra(@E,0); |
---|
1107 | int i; |
---|
1108 | ideal Q; |
---|
1109 | for ( i=1; i<=N; i++ ) |
---|
1110 | { |
---|
1111 | Q[i] = var(i)^2; |
---|
1112 | } |
---|
1113 | Q = twostd(Q); |
---|
1114 | qring @EA = Q; |
---|
1115 | return(@EA); |
---|
1116 | } |
---|
1117 | example |
---|
1118 | { |
---|
1119 | "EXAMPLE:";echo=2; |
---|
1120 | ring R = 0,(x(1..3)),dp; |
---|
1121 | def ER = Exterior(); |
---|
1122 | setring ER; |
---|
1123 | ER; |
---|
1124 | } |
---|
1125 | |
---|
1126 | /////////////////////////////////////////////////////////////////////////////// |
---|
1127 | proc makeWeyl(int n, list #) |
---|
1128 | "USAGE: makeWeyl(n,[p]); n an integer, n>0; p an optional integer (field characteristic) |
---|
1129 | RETURN: ring |
---|
1130 | PURPOSE: create an n-th Weyl algebra |
---|
1131 | NOTE: activate this ring with the \"setring\" command. |
---|
1132 | @* The presentation of an n-th Weyl algebra is classical: D(i)x(i)=x(i)D(i)+1, |
---|
1133 | @* where x(i) correspond to coordinates and D(i) to partial differentiations, i=1,...,n. |
---|
1134 | SEE ALSO: Weyl |
---|
1135 | EXAMPLE: example makeWeyl; shows examples |
---|
1136 | "{ |
---|
1137 | if (n<1) |
---|
1138 | { |
---|
1139 | print("Incorrect input"); |
---|
1140 | return(); |
---|
1141 | } |
---|
1142 | int @p = 0; |
---|
1143 | if ( size(#) > 0 ) |
---|
1144 | { |
---|
1145 | if ( typeof( #[1] ) == "int" ) |
---|
1146 | { |
---|
1147 | @p = #[1]; |
---|
1148 | } |
---|
1149 | } |
---|
1150 | if (n ==1) |
---|
1151 | { |
---|
1152 | ring @rr = @p,(x,D),dp; |
---|
1153 | } |
---|
1154 | else |
---|
1155 | { |
---|
1156 | ring @rr = @p,(x(1..n),D(1..n)),dp; |
---|
1157 | } |
---|
1158 | setring @rr; |
---|
1159 | Weyl(); |
---|
1160 | return(@rr); |
---|
1161 | } |
---|
1162 | example |
---|
1163 | { "EXAMPLE:"; echo = 2; |
---|
1164 | def a = makeWeyl(3); |
---|
1165 | setring a; |
---|
1166 | a; |
---|
1167 | } |
---|
1168 | |
---|
1169 | ////////////////////////////////////////////////////////////////////// |
---|
1170 | proc isNC() |
---|
1171 | "USAGE: isNC(); |
---|
1172 | PURPOSE: check whether a basering is commutative or not |
---|
1173 | RETURN: int, 1 if basering is noncommutative and 0 otherwise |
---|
1174 | EXAMPLE: example isNC; shows examples |
---|
1175 | "{ |
---|
1176 | string rname=nameof(basering); |
---|
1177 | if ( rname == "basering") // i.e. no ring has been set yet |
---|
1178 | { |
---|
1179 | "You have to call the procedure from the ring"; |
---|
1180 | return(); |
---|
1181 | } |
---|
1182 | int n = nvars(basering); |
---|
1183 | int i,j; |
---|
1184 | poly p; |
---|
1185 | for (i=1; i<n; i++) |
---|
1186 | { |
---|
1187 | for (j=i+1; j<=n; j++) |
---|
1188 | { |
---|
1189 | p = var(j)*var(i) - var(i)*var(j); |
---|
1190 | if (p!=0) { return(1);} |
---|
1191 | } |
---|
1192 | } |
---|
1193 | return(0); |
---|
1194 | } |
---|
1195 | example |
---|
1196 | { "EXAMPLE:"; echo = 2; |
---|
1197 | def a = makeWeyl(2); |
---|
1198 | setring a; |
---|
1199 | isNC(); |
---|
1200 | kill a; |
---|
1201 | ring r = 17,(x(1..7)),dp; |
---|
1202 | isNC(); |
---|
1203 | kill r; |
---|
1204 | } |
---|
1205 | |
---|
1206 | |
---|
1207 | proc rightStd(def I) |
---|
1208 | "USAGE: rightStd(I); I an ideal/ module |
---|
1209 | PURPOSE: compute a right Groebner basis of I |
---|
1210 | RETURN: the same type as input |
---|
1211 | EXAMPLE: example rightStd; shows examples |
---|
1212 | " |
---|
1213 | { |
---|
1214 | def A = basering; |
---|
1215 | def Aopp = opposite(A); |
---|
1216 | setring Aopp; |
---|
1217 | def Iopp = oppose(A,I); |
---|
1218 | def Jopp = groebner(Iopp); |
---|
1219 | setring A; |
---|
1220 | def J = oppose(Aopp,Jopp); |
---|
1221 | return(J); |
---|
1222 | } |
---|
1223 | example |
---|
1224 | { "EXAMPLE:"; echo = 2; |
---|
1225 | LIB "ncalg.lib"; |
---|
1226 | def A = makeUsl(2); |
---|
1227 | setring A; |
---|
1228 | ideal I = e2,f; |
---|
1229 | option(redSB); |
---|
1230 | option(redTail); |
---|
1231 | ideal LI = std(I); |
---|
1232 | LI; |
---|
1233 | ideal RI = rightStd(I); |
---|
1234 | RI; |
---|
1235 | } |
---|
1236 | |
---|
1237 | |
---|
1238 | proc rightSyz(def I) |
---|
1239 | "USAGE: rightSyz(I); I an ideal/ module |
---|
1240 | PURPOSE: compute a right syzygy module of I |
---|
1241 | RETURN: the same type as input |
---|
1242 | EXAMPLE: example rightSyz; shows examples |
---|
1243 | " |
---|
1244 | { |
---|
1245 | def A = basering; |
---|
1246 | def Aopp = opposite(A); |
---|
1247 | setring Aopp; |
---|
1248 | def Iopp = oppose(A,I); |
---|
1249 | def Jopp = syz(Iopp); |
---|
1250 | setring A; |
---|
1251 | def J = oppose(Aopp,Jopp); |
---|
1252 | return(J); |
---|
1253 | } |
---|
1254 | example |
---|
1255 | { "EXAMPLE:"; echo = 2; |
---|
1256 | ring r = 0,(x,d),dp; |
---|
1257 | ncalgebra(1,1); // the first Weyl algebra |
---|
1258 | ideal I = x,d; |
---|
1259 | module LS = syz(I); |
---|
1260 | print(LS); |
---|
1261 | module RS = rightSyz(I); |
---|
1262 | print(RS); |
---|
1263 | } |
---|
1264 | |
---|
1265 | |
---|
1266 | proc rightNF(def v, def M) |
---|
1267 | "USAGE: rightNF(I); v a poly/vector, M an ideal/module |
---|
1268 | PURPOSE: compute a right normal form of v w.r.t. M |
---|
1269 | RETURN: poly/vector (as of the 1st argument) |
---|
1270 | EXAMPLE: example rightNF; shows examples |
---|
1271 | " |
---|
1272 | { |
---|
1273 | def A = basering; |
---|
1274 | def Aopp = opposite(A); |
---|
1275 | setring Aopp; |
---|
1276 | def vopp = oppose(A,v); |
---|
1277 | def Mopp = oppose(A,M); |
---|
1278 | Mopp = std(Mopp); |
---|
1279 | def wopp = NF(vopp,Mopp); |
---|
1280 | setring A; |
---|
1281 | def w = oppose(Aopp,wopp); |
---|
1282 | w = simplify(w,2); // skip zeros in ideal/module |
---|
1283 | return(w); |
---|
1284 | } |
---|
1285 | example |
---|
1286 | { "EXAMPLE:"; echo = 2; |
---|
1287 | LIB "ncalg.lib"; |
---|
1288 | ring r = 0,(x,d),dp; |
---|
1289 | ncalgebra(1,1); // Weyl algebra |
---|
1290 | ideal I = x; I = std(I); |
---|
1291 | poly p = x*d+1; |
---|
1292 | NF(p,I); // left normal form |
---|
1293 | rightNF(p,I); // right normal form |
---|
1294 | } |
---|
1295 | |
---|
1296 | // ********************************** |
---|
1297 | // * NF: Example for vector/module: * |
---|
1298 | // ********************************** |
---|
1299 | // module M = [x,0],[0,d]; M = std(M); |
---|
1300 | // vector v = (x*d+1)*[1,1]; |
---|
1301 | // print(NF(v,M)); |
---|
1302 | // print(rightNF(v,M)); |
---|
1303 | |
---|
1304 | proc rightModulo(def M, def N) |
---|
1305 | "USAGE: rightModulo(M,N); M,N are ideals/modules |
---|
1306 | PURPOSE: compute a right representation of the module (M+N)/N |
---|
1307 | RETURN: module |
---|
1308 | ASSUME: M,N are presentation matrices for right modules |
---|
1309 | EXAMPLE: example rightModulo; shows examples |
---|
1310 | " |
---|
1311 | { |
---|
1312 | def A = basering; |
---|
1313 | def Aopp = opposite(A); |
---|
1314 | setring Aopp; |
---|
1315 | def Mopp = oppose(A,M); |
---|
1316 | def Nopp = oppose(A,N); |
---|
1317 | def Kopp = modulo(Mopp,Nopp); |
---|
1318 | setring A; |
---|
1319 | def K = oppose(Aopp,Kopp); |
---|
1320 | return(K); |
---|
1321 | } |
---|
1322 | example |
---|
1323 | { "EXAMPLE:"; echo = 2; |
---|
1324 | LIB "ncalg.lib"; |
---|
1325 | def A = makeUsl(2); |
---|
1326 | setring A; |
---|
1327 | option(redSB); |
---|
1328 | option(redTail); |
---|
1329 | ideal I = e2,f2,h2-1; |
---|
1330 | I = twostd(I); |
---|
1331 | print(matrix(I)); |
---|
1332 | ideal E = std(e); |
---|
1333 | ideal TL = e,h-1; // the result of left modulo |
---|
1334 | TL; |
---|
1335 | ideal T = rightModulo(E,I); |
---|
1336 | T = rightStd(T+I); |
---|
1337 | T = rightStd(rightNF(T,I)); // make the output canonic |
---|
1338 | T; |
---|
1339 | } |
---|
1340 | |
---|
1341 | ////////////////////////////////////////////////////////////////////// |
---|