1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: ncdecomp.lib,v 1.6 2005-02-23 18:10:46 levandov Exp $"; |
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3 | category="Noncommutative"; |
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4 | info=" |
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5 | LIBRARY: ncdecomp.lib Central character decomposition of a module |
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6 | AUTHORS: Viktor Levandovskyy, levandov@mathematik.uni-kl.de. |
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7 | |
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8 | OVERVIEW: |
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9 | This library presents algorithms for the central character |
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10 | decomposition of a module (in other words, a |
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11 | decomposition into generalized weight modules with respect to the center). |
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12 | Based on ideas of O. Khomenko and V. Levandovskyy. |
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13 | |
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14 | PROCEDURES: |
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15 | CentralQuot(I,G); central quotient I:G, |
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16 | CentralSaturation(M,T); central saturation ((M:T):...):T) ( = M:T^{\infty}), |
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17 | CenCharDec(I,C); central character decomposition of I w.r.t. C |
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18 | IntersectWithSub(M,Z); intersection of M with the subalgebra, generated |
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19 | by pairwise commutative elements. |
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20 | "; |
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21 | |
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22 | LIB "ncalg.lib"; |
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23 | LIB "primdec.lib"; |
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24 | /////////////////////////////////////////////////////////////////////////////// |
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25 | static proc CharKernel(list L, int i) |
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26 | { |
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27 | // compute \cup L[j], j!=i |
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28 | int sL = size(L); |
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29 | if ( (i<=0) || (i>sL)) { return(0); } |
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30 | int j; |
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31 | list Li; |
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32 | if (i ==1 ) |
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33 | { |
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34 | Li = L[2..sL]; |
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35 | } |
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36 | if (i ==sL ) |
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37 | { |
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38 | Li = L[1..sL-1]; |
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39 | } |
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40 | if ( (i>1) && (i < sL)) |
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41 | { |
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42 | Li = L[1..i-1]; |
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43 | for (j=i+1; j<=sL; j++) |
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44 | { |
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45 | Li[j-1] = L[j]; |
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46 | } |
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47 | } |
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48 | // print("intersecting kernels..."); |
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49 | module Cres = intersect(Li[1..size(Li)]); |
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50 | return(Cres); |
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51 | } |
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52 | /////////////////////////////////////////////////////////////////////////////// |
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53 | static proc CentralQuotPoly(module M, poly g) |
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54 | { |
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55 | // here an elimination of components should be used ! |
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56 | int N=nrows(M); // M = A^N /I_M |
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57 | module @M; |
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58 | int i,j; |
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59 | for(i=1; i<=N; i++) |
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60 | { |
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61 | @M=@M,g*gen(i); |
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62 | } |
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63 | @M = simplify(@M,2); |
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64 | @M = @M,M; |
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65 | module S = syz(@M); |
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66 | matrix s = S; |
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67 | module T; |
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68 | vector t; |
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69 | for(i=1; i<=ncols(s); i++) |
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70 | { |
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71 | t = 0*gen(N); |
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72 | for(j=1; j<=N; j++) |
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73 | { |
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74 | t = t + s[j,i]*gen(j); |
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75 | } |
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76 | T[i] = t; |
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77 | } |
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78 | T = simplify(T,2); |
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79 | return(T); |
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80 | } |
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81 | /////////////////////////////////////////////////////////////////////////////// |
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82 | static proc MyIsEqual(module A, module B) |
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83 | { |
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84 | // both A and B are submodules of free module |
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85 | option(redSB); |
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86 | option(redTail); |
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87 | if (attrib(A,"isSB")!=1) |
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88 | { |
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89 | A = std(A); |
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90 | } |
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91 | if (attrib(B,"isSB")!=1) |
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92 | { |
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93 | B = std(B); |
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94 | } |
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95 | int ANSWER = 1; |
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96 | if ( ( ncols(A) == ncols(B) ) && ( nrows(A) == nrows(B) ) ) |
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97 | { |
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98 | module @AB = A-B; |
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99 | @AB = simplify(@AB,2); |
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100 | if (@AB[1]!=0) { ANSWER = 0; } |
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101 | } |
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102 | else { ANSWER = 0; } |
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103 | return(ANSWER); |
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104 | } |
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105 | /////////////////////////////////////////////////////////////////////////////// |
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106 | proc CentralQuot(module I, ideal G) |
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107 | "USAGE: CentralQuot(M, T), for a module M and an ideal T, |
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108 | RETURN: module of the central quotient I:G, |
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109 | NOTE: the output module is not necessarily a Groebner basis, |
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110 | SEE ALSO: CentralSaturation, CenCharDec |
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111 | EXAMPLE: example CentralQuot; shows examples |
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112 | "{ |
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113 | int i; |
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114 | list @L; |
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115 | for(i=1; i<=size(G); i++) |
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116 | { |
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117 | @L[i] = CentralQuotPoly(I,G[i]); |
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118 | } |
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119 | module @I = intersect(@L[1..size(G)]); |
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120 | if (nrows(@I)==1) |
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121 | { |
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122 | @I = ideal(@I); |
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123 | } |
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124 | return(@I); |
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125 | } |
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126 | example |
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127 | { "EXAMPLE:"; echo = 2; |
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128 | option(returnSB); |
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129 | def a = sl2(); |
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130 | setring a; |
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131 | ideal I = e3,f3,h3-4*h; |
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132 | I = std(I); |
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133 | poly C=4*e*f+h^2-2*h; |
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134 | ideal G = (C-8)*(C-24); |
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135 | ideal R = CentralQuot(I,G); |
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136 | R; |
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137 | } |
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138 | /////////////////////////////////////////////////////////////////////////////// |
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139 | proc CentralSaturation(module M, ideal T) |
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140 | "USAGE: CentralSaturation(M, T), for a module M and an ideal T, |
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141 | RETURN: module of the central saturation of M by T (also denoted by M:T^{\infty}), |
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142 | NOTE: the output module is not necessarily a Groebner basis, |
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143 | SEE ALSO: CentralQuot, CenCharDec |
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144 | EXAMPLE: example CentralSaturation; shows examples |
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145 | "{ |
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146 | option(redSB); |
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147 | option(redTail); |
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148 | option(returnSB); |
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149 | module Q=0; |
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150 | module S=M; |
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151 | while ( !MyIsEqual(Q,S) ) |
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152 | { |
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153 | Q = CentralQuot(S, T); |
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154 | S = CentralQuot(Q, T); |
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155 | } |
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156 | if (nrows(Q)==1) |
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157 | { |
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158 | Q = ideal(Q); |
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159 | } |
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160 | // Q = std(Q); |
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161 | return(Q); |
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162 | } |
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163 | example |
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164 | { "EXAMPLE:"; echo = 2; |
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165 | option(returnSB); |
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166 | def a = sl2(); |
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167 | setring a; |
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168 | ideal I = e3,f3,h3-4*h; |
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169 | I = std(I); |
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170 | poly C=4*e*f+h^2-2*h; |
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171 | ideal G = C*(C-8); |
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172 | ideal R = CentralSaturation(I,G); |
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173 | R=std(R); |
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174 | vdim(R); |
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175 | R; |
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176 | } |
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177 | /////////////////////////////////////////////////////////////////////////////// |
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178 | proc CenCharDec(module I, list Center) |
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179 | "USAGE: CenCharDec(I, C); I a module, C a list of generators of the center; |
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180 | RETURN: a list L, where each entry consists of three records: |
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181 | @* L[*][1] is the central character in terms of central elements, |
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182 | @* L[*][2] is the Groebner basis of the corresponding weight module, |
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183 | @* L[*][3] is the K-dimension of the weight module; |
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184 | NOTE: some modules have no finite decomposition (in such case one |
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185 | gets warning message) |
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186 | SEE ALSO: CentralQuot, CentralSaturation |
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187 | EXAMPLE: example CenCharDec; shows examples |
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188 | " |
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189 | { |
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190 | // M = A/I |
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191 | //1. Find the Zariski closure of Supp_Z M |
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192 | // J = Ann_M 1 == I |
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193 | // J \cap Z: |
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194 | option(redSB); |
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195 | option(redTail); |
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196 | option(returnSB); |
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197 | def @A = basering; |
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198 | setring @A; |
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199 | int sZ=size(Center); |
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200 | int i,j; |
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201 | poly t=1; |
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202 | for(i=1; i<=nvars(@A); i++) |
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203 | { |
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204 | t=t*var(i); |
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205 | } |
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206 | ring @Z=0,(@z(1..sZ)),dp; |
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207 | // @Z; |
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208 | def @ZplusA = @A+@Z; |
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209 | setring @ZplusA; |
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210 | // @ZplusA; |
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211 | ideal I = imap(@A,I); |
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212 | list Center = imap(@A,Center); |
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213 | poly t = imap(@A,t); |
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214 | ideal @Ker; |
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215 | for(i=1; i<=sZ; i++) |
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216 | { |
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217 | @Ker[i]=@z(i) - Center[i]; |
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218 | } |
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219 | @Ker = @Ker,I; |
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220 | ideal @JcapZ = eliminate(@Ker,t); |
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221 | // do not forget parameters of a basering! |
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222 | string strZ="ring @@Z=("+charstr(@A)+"),(@z(1.."+string(sZ)+")),dp;"; |
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223 | // print(strZ); |
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224 | execute(strZ); |
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225 | setring @@Z; |
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226 | ideal @JcapZ = imap(@ZplusA,@JcapZ); |
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227 | @JcapZ = std(@JcapZ); |
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228 | // @JcapZ; |
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229 | int sJ = vdim(@JcapZ); |
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230 | if (sJ==-1) |
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231 | { |
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232 | "There is no finite decomposition"; |
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233 | return(0); |
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234 | } |
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235 | // print(@JcapZ); |
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236 | // 2. compute the min.ass.primes of the ideal in the center |
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237 | list @L = minAssGTZ(@JcapZ); |
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238 | int sL = size(@L); |
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239 | // print("etL:"); |
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240 | // @L; |
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241 | // exception: is sL==1, the whole ideal has unique cen.char |
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242 | if (sL ==1) |
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243 | { |
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244 | setring @A; |
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245 | map @M = @@Z,Center[1..size(Center)]; |
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246 | list L = @M(@L); |
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247 | list @R; |
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248 | @R[1] = L[1]; |
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249 | if (nrows(@R[1])==1) |
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250 | { |
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251 | @R[1] = ideal(@R[1]); |
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252 | } |
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253 | @R[2] = I; |
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254 | if (nrows(@R[2])==1) |
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255 | { |
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256 | @R[2] = ideal(@R[2]); |
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257 | } |
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258 | @R[2] = std(@R[2]); |
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259 | @R[3] = vdim(@R[2]); |
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260 | return(@R); |
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261 | } |
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262 | list @CharKer; |
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263 | for(i=1; i<=sL; i++) |
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264 | { |
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265 | @L[i] = std(@L[i]); |
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266 | } |
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267 | // 3. compute the intersections of characters |
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268 | for(i=1; i<=sL; i++) |
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269 | { |
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270 | @CharKer[i] = CharKernel(@L,i); |
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271 | } |
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272 | // print("Charker:"); |
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273 | // @CharKer; |
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274 | // 4. Go back to the algebra and compute central saturations |
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275 | setring @A; |
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276 | map @M = @@Z,Center[1..size(Center)]; |
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277 | list L = @M(@CharKer); |
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278 | list R,@R; |
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279 | for(i=1; i<=sL; i++) |
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280 | { |
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281 | @R[1] = L[i]; |
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282 | if (nrows(@R[1])==1) |
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283 | { |
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284 | @R[1] = ideal(@R[1]); |
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285 | } |
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286 | @R[2] = CentralSaturation(I,L[i]); |
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287 | if (nrows(@R[2])==1) |
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288 | { |
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289 | @R[2] = ideal(@R[2]); |
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290 | } |
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291 | @R[2] = std(@R[2]); |
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292 | @R[3] = vdim(@R[2]); |
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293 | R[i] = @R; |
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294 | } |
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295 | return(R); |
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296 | } |
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297 | example |
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298 | { "EXAMPLE:"; echo = 2; |
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299 | option(returnSB); |
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300 | def a = sl2(); // U(sl_2) in characteristic 0 |
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301 | setring a; |
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302 | ideal I = e3,f3,h3-4*h; |
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303 | I = twostd(I); // two-sided ideal generated by I |
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304 | vdim(I); // it is finite-dimensional |
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305 | list Cn = 4*e*f+h^2-2*h; // the only central element |
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306 | list T = CenCharDec(I,Cn); |
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307 | T; |
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308 | } |
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309 | /////////////////////////////////////////////////////////////////////////////// |
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310 | proc IntersectWithSub (ideal M, ideal Z) |
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311 | "USAGE: IntersectWithSub(M,Z), M an ideal, Z an ideal of pairwise commutative elements |
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312 | PURPOSE: computes an intersection of M with the subalgebra, generated by Z |
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313 | RETURN: ideal (of two--sided generators, not a Groebner basis!) |
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314 | NOTE: usually one puts generators of the center into Z |
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315 | EXAMPLE: example IntersectWithSub; shows an example |
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316 | " |
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317 | { |
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318 | // returns a submodule of M, equal to M \cap Z |
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319 | // correctness: Z should consists of pairwise |
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320 | // commutative elements |
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321 | int nz = size(Z); |
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322 | int i,j; |
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323 | poly p; |
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324 | for (i=1; i<nz; i++) |
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325 | { |
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326 | for (j=i+1; j<=nz; j++) |
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327 | { |
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328 | p = bracket(Z[i],Z[j]); |
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329 | if (p!=0) |
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330 | { |
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331 | "Error: generators of the subalgebra do not commute."; |
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332 | return(ideal(0)); |
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333 | } |
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334 | } |
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335 | } |
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336 | // main action |
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337 | def B = basering; |
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338 | setring B; |
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339 | string s1,s2; |
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340 | s1 = "ring @Z = ("; |
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341 | s2 = s1 + charstr(basering) + "),(z(1.." + string(nz)+")),Dp"; |
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342 | // s2; |
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343 | execute(s2); |
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344 | setring B; |
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345 | map F = @Z,Z; |
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346 | setring @Z; |
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347 | ideal PreM = preimage(B,F,M); |
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348 | PreM = std(PreM); |
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349 | setring B; |
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350 | ideal T = F(PreM); |
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351 | return(T); |
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352 | } |
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353 | example |
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354 | { |
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355 | "EXAMPLE:"; echo = 2; |
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356 | ring r=(0,a),(e,f,h),Dp; |
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357 | matrix @d[3][3]; |
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358 | @d[1,2]=-h; |
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359 | @d[1,3]=2e; |
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360 | @d[2,3]=-2f; |
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361 | ncalgebra(1,@d); // parametric U(sl_2) |
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362 | ideal I = e,h-a; |
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363 | ideal C; |
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364 | C[1] = h^2-2*h+4*e*f; // the center of U(sl_2) |
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365 | ideal X = IntersectWithSub(I,C); |
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366 | X; |
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367 | ideal G = e*f, h; // the biggest comm. subalgebra of U(sl_2) |
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368 | ideal Y = IntersectWithSub(I,G); |
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369 | Y; |
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370 | } |
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