1 | ////////////////////////////////////////////////////////////////////////////// |
---|
2 | version="version gkdim.lib 4.0.0.0 Jun_2013 "; // $Id$ |
---|
3 | category="Noncommutative"; |
---|
4 | info=" |
---|
5 | LIBRARY: gkdim.lib Procedures for calculating the Gelfand-Kirillov dimension |
---|
6 | AUTHORS: Lobillo, F.J., jlobillo@ugr.es |
---|
7 | @* Rabelo, C., crabelo@ugr.es |
---|
8 | |
---|
9 | Support: 'Metodos algebraicos y efectivos en grupos cuanticos', BFM2001-3141, MCYT, Jose Gomez-Torrecillas (Main researcher). |
---|
10 | |
---|
11 | NOTE: The built-in command @code{dim}, executed for a module in @plural, computes the Gelfand-Kirillov dimension. |
---|
12 | |
---|
13 | PROCEDURES: |
---|
14 | GKdim(M); Gelfand-Kirillov dimension computation of the factor-module, whose presentation is given by the matrix M. |
---|
15 | "; |
---|
16 | |
---|
17 | /////////////////////////////////////////////////////////////////////////////////// |
---|
18 | static proc idGKdim(ideal I) |
---|
19 | "USAGE: idGKdim(I), I is a left ideal |
---|
20 | RETURN: int, the Gelfand-Kirillov dimension of the R/I |
---|
21 | NOTE: uses the dim procedure, if the factor-module is zero, -1 is returned |
---|
22 | " |
---|
23 | { |
---|
24 | if (attrib(I,"isSB")<>1) |
---|
25 | { |
---|
26 | I=std(I); |
---|
27 | } |
---|
28 | |
---|
29 | int d = dim(I); |
---|
30 | // if (d==-1) {d++;} // The GK-dimension of a finite dimensional module is zero |
---|
31 | // levandov: but for consistency, GKdim(std(1)) == -1, |
---|
32 | // mimicking the behaviour of dim() procedure. |
---|
33 | return (d); |
---|
34 | } |
---|
35 | |
---|
36 | /////////////////////////////////////////////////////////////////////////////// |
---|
37 | proc GKdim(list L) |
---|
38 | "USAGE: GKdim(L); L is a left ideal/module/matrix |
---|
39 | RETURN: int |
---|
40 | PURPOSE: compute the Gelfand-Kirillov dimension of the factor-module, whose presentation is given by L, e.g. R^r/L |
---|
41 | NOTE: if the factor-module is zero, -1 is returned |
---|
42 | EXAMPLE: example GKdim; shows examples |
---|
43 | " |
---|
44 | { |
---|
45 | def M = L[1]; |
---|
46 | int d = -1; |
---|
47 | if (typeof(M)=="ideal") |
---|
48 | { |
---|
49 | d=idGKdim(M); |
---|
50 | } |
---|
51 | else |
---|
52 | { |
---|
53 | if (typeof(M)=="matrix") |
---|
54 | { |
---|
55 | module N = module(M); |
---|
56 | kill M; |
---|
57 | module M = N; |
---|
58 | } |
---|
59 | if (typeof(M)=="module") |
---|
60 | { |
---|
61 | if (attrib(M,"isSB")<>1) |
---|
62 | { |
---|
63 | M=std(M); |
---|
64 | } |
---|
65 | d=dim(M); |
---|
66 | } |
---|
67 | else |
---|
68 | { |
---|
69 | ERROR("The input must be an ideal, a module or a matrix."); |
---|
70 | } |
---|
71 | } |
---|
72 | return (d); |
---|
73 | } |
---|
74 | example |
---|
75 | { |
---|
76 | "EXAMPLE:";echo=2; |
---|
77 | ring R = 0,(x,y,z),Dp; |
---|
78 | matrix C[3][3]=0,1,1,0,0,-1,0,0,0; |
---|
79 | matrix D[3][3]=0,0,0,0,0,x; |
---|
80 | def r = nc_algebra(C,D); setring r; |
---|
81 | r; |
---|
82 | ideal I=x; |
---|
83 | GKdim(I); |
---|
84 | ideal J=x2,y; |
---|
85 | GKdim(J); |
---|
86 | module M=[x2,y,1],[x,y2,0]; |
---|
87 | GKdim(M); |
---|
88 | ideal A = x,y,z; |
---|
89 | GKdim(A); |
---|
90 | ideal B = 1; |
---|
91 | GKdim(B); |
---|
92 | GKdim(ideal(0)) == nvars(basering); // should be true, i.e., evaluated to 1 |
---|
93 | } |
---|
94 | /////////////////////////////////////////////////////////////////////////////// |
---|
95 | proc gkdim(list L) |
---|
96 | { |
---|
97 | return(GKdim(L)); |
---|
98 | } |
---|
99 | /////////////////////////////////////////////////////////////////////////////// |
---|