Ricardo

PRINTPOL_strg

Sun, 20 Jul 2014 19:06:48 -0300

This function prints a non-commutative polynomial as an input to a tex file.

PRINTPOL_strg:=function(pol) # rel é do tipo rel = [mon, pol].
local polstrg, ind;

polstrg:=String(pol);

if polstrg = "0" then
   Print(polstrg);
   #return polstrg;
else
   polstrg:=SplitString(polstrg, "+");
   Sort(polstrg, ORDER);
   polstrg:=Reversed(polstrg);
   for ind in [1..Size(polstrg)] do
       if EvalString(COEF_STRG(polstrg[ind])) = 1   then
           polstrg[ind]:=ReplacedString(polstrg[ind], "(1)", "" ) ;
       elif EvalString(COEF_STRG(polstrg[ind])) = -1   then
           polstrg[ind]:=ReplacedString(polstrg[ind], "(-1)", "-" ) ;
       else
           polstrg[ind]:= ReplacedString(polstrg[ind], COEF_STRG(polstrg[ind]), String(EvalString(COEF_STRG(polstrg[ind]))));
       fi;
   od;

   polstrg:=JoinStringsWithSeparator(polstrg, "\n+");
   polstrg:=ReplacedString(polstrg, "*", "" ) ;
   polstrg:=ReplacedString(polstrg, "\n+-", "\n-" ) ;

   Print(polstrg);
   #return polstrg;
fi;
end;

ORDERING

Wed, 19 Feb 2014 15:56:33 -0300

This file provides a non-standard ordering for the monomials in the cubic algebra. The order is

U < V < U^2 < V^2;
syllable length of monomials;
total length;
reverse lexicographical ordering.

ORDERING:=function(strg1, strg2)
local s1, s2, syl1, syl2, deg1, deg2, degV1, degV2, s1r, s2r, ind ;

s1:=ReplacedString( MON_STRG( STRETCH(strg1)), "*", "");
s2:=ReplacedString( MON_STRG( STRETCH(strg2)), "*", "");

if s1="UU" and s2="V" then return true;fi;

syl1:=1;
for ind in [1..Size(s1)-1] do
   if s1[ind] <> s1[ind+1] then
       syl1:=syl1+1;
   fi;
od;

syl2:=1;
for ind in [1..Size(s2)-1] do
   if s2[ind] <> s2[ind+1] then
       syl2:=syl2+1;
   fi;
od;

deg1:=Size(s1);
deg2:=Size(s2);
degV1:=Size(ReplacedString(s1, "U", ""));
degV2:=Size(ReplacedString(s2, "U", ""));

s1r:=Reversed(s1);
s2r:=Reversed(s2);
s1r:=ReplacedString(s1r, "UU", "B");
s1r:=ReplacedString(s1r, "VV", "D");
s1r:=ReplacedString(s1r, "U", "A");
s1r:=ReplacedString(s1r, "V", "C");
s2r:=ReplacedString(s2r, "UU", "B");
s2r:=ReplacedString(s2r, "VV", "D");
s2r:=ReplacedString(s2r, "U", "A");
s2r:=ReplacedString(s2r, "V", "C");

if syl1 < syl2 then
   return true;
elif syl1 = syl2 and deg1 < deg2 then
   return true;
elif syl1 = syl2 and deg1=deg2 and degV1 < degV2 then
   return true;
elif syl1 = syl2 and deg1=deg2 and degV1 = degV2 and s1r < s2r then
   return true;
else
   return false;
fi;
end;

readall

Tue, 18 Feb 2014 18:13:42 -0300

This file is provided to call the others functions and set some parameters.

FIELD:=Rationals;
LoadPackage("GBNP");
A:=FreeAssociativeAlgebraWithOne(FIELD, "U", "V", "T");
U:=A.1;V:=A.2;T:=A.3;
GBNP.ConfigPrint("U", "V", "T");

lvar:=["U","V","T"];var:=['U', 'V', 'T'];
Read("MON_STRG");
Read("COEF_STRG");
Read("STRETCH");
Read("SUBS");
Read("SUBSC");
Read("POL2REL");
Read("FREESUB");
Read("SEARCHC");
Read("SEARCH");
Read("CHECKBYMAT");
Read("PRINTPOL");
Read("PRINTPOL_strg");
Read("CHECK_GEN");
Read("matrep");
Read("MULTREL");

rel

Tue, 18 Feb 2014 17:07:07 -0300

This routine produces a PDF file of relations from readrel.
This function calls another function: MULTREL, PRINTPOL;
This function calls external function: pdflatex, rm.

u3:=[U^3, 0*U];
v3:=[V^3, 0*V];

make_rel:=[
"$(UV + U + V)^3=0$",

"\\textcolor{red}{$ (1) \\times V^2 \\leftleftarrows (1) \\leftarrow (1) $}",

"\\textcolor{red}{$ U^2 \\times (1) \\leftleftarrows (1) \\leftarrow (1) $}",

"\\textcolor{red}{$ (2) \\times U^2 \\leftleftarrows (2) \\leftarrow (2)$}",

"\\textcolor{red}{$ U^2 \\times (2) \\leftleftarrows (2) \\leftarrow (2)$}",

"\\textcolor{red}{$ U^2 \\times (4) \\leftleftarrows (4) \\leftarrow (4)$}",

"\\textcolor{red}{$ (4) \\times V^2 \\leftleftarrows (4) \\leftarrow (4)$}",

"\\textcolor{red}{$ (6) \\leftarrow (U \\mapsto U^2 -U) $}",

"\\textcolor{red}{$ (6) \\leftarrow (V \\mapsto V^2 -V) $}",

"\\textcolor{red}{$ \\left\\{ U^2 \\times (8) - (8) \\times U^2\\right\\} \\leftleftarrows (5)                \n \\leftleftarrows (4) \\leftarrow (5) \\leftarrow (4) \\Longleftrightarrow U^2VU^2 \\in Center(B) $}",

"\\textcolor{red}{$ \\left\\{ (6) - U^2 \\times (8) \\right\\} \\leftleftarrows (4) \\leftarrow (4)             \\Longleftrightarrow U^2V^2U^2 \\in Center(B) $}",

"\\textcolor{red}{$ U \\times (4)   \\leftarrow (4)$}",

"\\textcolor{red}{$ (4) \\times V   \\leftarrow (4) \\leftarrow (8)$}",

"\\textcolor{red}{$ U \\times (8)   \\leftarrow (4)$}",

"\\textcolor{red}{$ (8) \\times V   \\leftarrow (4) \\leftarrow (8)$}",

"\\textcolor{red}{$ (15) \\leftarrow (U \\leftrightarrow V)    \\leftarrow (8)$}",

"\\textcolor{red}{$ (14) \\leftarrow (U \\leftrightarrow V)    \\leftarrow (8)$}",

"\\textcolor{red}{$ (13) \\leftarrow (U \\leftrightarrow V)    \\leftarrow (8)$}",

"\\textcolor{red}{$ (12) \\leftarrow (U \\leftrightarrow V)    \\leftarrow (8)$}",

"\\textcolor{red}{$ V \\times (12) \\leftarrow (5) \\leftarrow (8) \\leftarrow (16) \\leftarrow (17) \\leftarrow (18) $}",

"\\textcolor{red}{$ (12) \\times U \\leftarrow (5) $}",

"\\textcolor{red}{$ (12) \\times V \\leftarrow (4) \\leftarrow (8) \\leftarrow (12) \\leftarrow (13) \\leftarrow (15) $}",

"\\textcolor{red}{$ U \\times (13) \\leftarrow (4) $}",

"\\textcolor{red}{$ V \\times (13) \\leftarrow (5) \\leftarrow (8) \\leftarrow (16) \\leftarrow (17) \\leftarrow (18) $}",

"\\textcolor{red}{$ (13) \\times U \\leftarrow (5) \\leftarrow (6) $}",

"\\textcolor{red}{$ (13) \\times V \\leftarrow (4) \\leftarrow (8) \\leftarrow (12) \\leftarrow (13) \\leftarrow (15) $}",

"\\textcolor{red}{$ V \\times (14) \\leftarrow (5) \\leftarrow (8) \\leftarrow (9) \\leftarrow (18) $}",

"\\textcolor{red}{$ (14) \\times U \\leftarrow (18) \\leftarrow (19) $}",

"\\textcolor{red}{$ V \\times (15) \\leftarrow (5) \\leftarrow (7) \\leftarrow (8) \\leftarrow (9) \\leftarrow (18) $}",

"\\textcolor{red}{$ (15) \\times U \\leftarrow (18) \\leftarrow (19) $}",

"\\textcolor{red}{$ (15) \\times V \\leftarrow (4) \\leftarrow (8) \\leftarrow (9) \\leftarrow (15) $}",

"\\textcolor{red}{$ V \\times (16) \\leftarrow (5) \\leftarrow (8) \\leftarrow (16) \\leftarrow (17) \\leftarrow (18) $}",

"\\textcolor{red}{$ (16) \\times U \\leftarrow (5) $}",

"\\textcolor{red}{$ (16) \\times V \\leftarrow (4) \\leftarrow (8) \\leftarrow (12) \\leftarrow (14) \\leftarrow (15) \\leftarrow (20) $}",

"\\textcolor{red}{$ U \\times (17) \\leftarrow (4) $}",

"\\textcolor{red}{$ V \\times (17) \\leftarrow (5) \\leftarrow (8) \\leftarrow (16) \\leftarrow (17) \\leftarrow (18) $}",

"\\textcolor{red}{$ (17) \\times V \\leftarrow (4) \\leftarrow (8) \\leftarrow (12) \\leftarrow (14) \\leftarrow (15) \\leftarrow (24) $}",
#38
"\\textcolor{red}{$ U \\times (18) \\leftarrow (13) \\leftarrow (15) \\leftarrow (28) \\leftarrow (32) \\leftarrow (34)$}",

"\\textcolor{red}{$ V \\times (18) \\leftarrow (5) \\leftarrow (8) \\leftarrow (9) \\leftarrow (18) $}",

"\\textcolor{red}{$ (19) \\times V \\leftarrow (4) \\leftarrow (8) \\leftarrow (9) \\leftarrow (15)$}",

"\\textcolor{red}{$ (38) \\leftarrow (U \\mapsto U^2 -U) \\leftarrow (6) \\leftarrow (38) $}",

"\\textcolor{red}{$ U \\times (38) \\leftarrow (6) \\leftarrow (13) \\leftarrow (15) \\leftarrow (26) \\leftarrow (34) \\leftarrow (37) \\leftarrow (38) $}",

"\\textcolor{red}{$ \\dfrac{1}{3} \\times (38) $}",

"\\textcolor{red}{$ \\dfrac{1}{6} \\times (41) $}",

"\\textcolor{red}{$ \\dfrac{1}{3} \\times (42) \\leftarrow (44) $}",
#46
"\\textcolor{red}{$ (1) \\leftarrow (U \\leftrightarrow V) \\leftarrow (2) \\leftarrow (3) \\leftarrow (4) \n \\leftarrow (16) \\leftarrow (17) \\leftarrow (18) \\leftarrow (43) \n           \\leftarrow (43) \\leftarrow (44)$}",

"\\textcolor{red}{$ (7) \\leftarrow (U \\mapsto U^2 + 2U) \\leftarrow (7) \\leftarrow (9) $}",

"\\textcolor{red}{$ U \\times (47)$}",

#49
"\\textcolor{red}{$ (47) \\leftarrow (U \\leftrightarrow V) $}",

"\\textcolor{red}{$ (48) \\leftarrow (U \\leftrightarrow V) $}",

"\\textcolor{red}{$ (9) \\leftarrow (V \\mapsto V^2 + 2V) \\leftarrow (6) \\leftarrow (9) \\leftarrow (47) \\leftarrow (48) \\leftarrow(49) $}",

"\\textcolor{red}{$ (51) \\times V \\leftarrow (4) \\leftarrow (8) \\leftarrow (49) $}",

"\\textcolor{red}{$ (8) \\leftarrow \\dfrac{1}{6} \\times (51) $}",

"\\textcolor{black}{$ u([X,Y])^3=0 \\leftarrow \\{B^7 = 0\\} \\leftarrow (1) \\leftarrow (2) \\leftarrow (3) \n \\leftarrow (5) \\leftarrow (12)\\leftarrow (13) \\leftarrow (14) \\leftarrow (15) \\leftarrow (20) \\leftarrow (24) \\leftarrow (28) \\leftarrow (30) \\leftarrow (32) \\leftarrow (34) \n \\leftarrow (42) \\leftarrow (43) \\leftarrow (44) \\leftarrow (46) $}",

"\\textcolor{black}{$ u(X^2Y)^3=u(XY^2)^3=0 $}",

"\\textcolor{red}{$ (9) \\leftarrow (U \\mapsto U^2 + 2U) \\leftarrow (7) \\leftarrow (9) $}",

"\\textcolor{red}{$ U \\times (48) $}",

"\\textcolor{red}{$ (49) \\leftarrow ( U \\leftrightarrow V) $}",

"\\textcolor{green}{$ (48) \\leftarrow (49) $}",

"\\textcolor{red}{$ (51) \\leftarrow ( U \\leftrightarrow V) $}",

"\\textcolor{red}{$ (9) \\leftarrow (V \\mapsto V^2 + 2V) \\leftarrow (6) \\leftarrow (9) \\leftarrow (49) \n   \\leftarrow (51) \\leftarrow (52) $}",

"\\textcolor{red}{$ (53) \\times V \\leftarrow (4) \\leftarrow (8) \\leftarrow (52) $}",

"\\textcolor{red}{$ \\dfrac{1}{3} \\times (49) $}",

"\\textcolor{red}{$ \\dfrac{1}{3} \\times (50) $}",

"\\textcolor{red}{$ \\dfrac{1}{3} \\times (51) $}",

"\\textcolor{red}{$ \\dfrac{1}{3} \\times (52) $}",

"\\textcolor{red}{$ \\dfrac{1}{6} \\times (53) $}",

"\\textcolor{red}{$ \\dfrac{1}{6} \\times (54) $}",

"\\textcolor{black}{$ u([X,Y])^3=0 \\leftarrow \\{B^7 = 0\\} \\leftarrow (20) \\leftarrow (22) \\leftarrow (24) \\leftarrow (28) \\leftarrow (30) \\leftarrow (32) \\leftarrow (34) \n \\leftarrow (37) \\leftarrow (38) \\leftarrow (41) \\leftarrow (42) \\leftarrow (44) \\leftarrow (48) \\leftarrow (51) \n \\leftarrow (53) \\leftarrow (54) $}",

"\\textcolor{red}{$ \\dfrac{1}{6} \\times (61) $}",

];

rel:= [

[ (1)*(U*V)^3, (-1)*U^2*V+(-1)*U*V*U+(-1)*U*V^2+(-1)*V*U^2+(-1)*V*U*V+(-1)*V^2*U+(-1)*U^2*V*U+(-1)*U^2*V^2+(-1)*U*V*U^2+(-2)*(U*V)^2+(-1)*U*V^2*U+(-1)*V*U^2*V+(
    -1)*(V*U)^2+(-1)*V*U*V^2+(-1)*V^2*U*V+(-1)*U*(U*V)^2+(-1)*U*V*U^2*V+(-1)*(U*V)^2*U+(-1)*(U*V)^2*V+(-1)*U*V^2*U*V+(-1)*(V*U)^2*V ],

[ (1)*(U*V)^2*U, (-1)*U^2*V+(-1)*U*V*U+(-1)*U*V^2+(-1)*V*U^2+(-1)*V*U*V+(-1)*V^2*U+(-1)*U^2*V*U+(-1)*U*V*U^2+(-1)*(U*V)^2+(-1)*U*V^2*U+(-1)*(V*U)^2 ],

[ (1)*(V*U)^2*V, (-1)*U^2*V+(-1)*U*V*U+(-1)*U*V^2+(-1)*V*U^2+(-1)*V*U*V+(-1)*V^2*U+(-1)*(U*V)^2+(-1)*V*U^2*V+(-1)*(V*U)^2+(-1)*V*U*V^2+(-1)*V^2*U*V ],

[ (1)*(U*V)^2, (-1)*U^2*V+(-1)*U*V*U+(-1)*U*V^2+(-1)*V*U^2+(-1)*V*U*V+(-1)*V^2*U+(1)*V^2*U^2 ],

[ (1)*(V*U)^2, (-1)*U^2*V+(-1)*U*V*U+(-1)*U*V^2+(-1)*V*U^2+(-1)*V*U*V+(-1)*V^2*U+(1)*U^2*V^2 ],

[ (1)*U^2*V^2*U^2, (1)*U^2*V+(1)*U*V*U+(1)*U*V^2+(1)*V*U^2+(1)*V*U*V+(1)*V^2*U+(-1)*U^2*V*U+(-1)*U^2*V^2+(-1)*U*V*U^2+(-1)*U*V^2*U+(-1)*V^2*U^2+(1)*U^2*V*U^2+(1)*U^2*V^2*U+(1)*U*V^2*U^2 ],

[ (1)*V^2*U^2*V^2, (1)*U^2*V+(1)*U*V*U+(1)*U*V^2+(1)*V*U^2+(1)*V*U*V+(1)*V^2*U+(-1)*U^2*V^2+(-1)*V*U^2*V+(
    -1)*V*U*V^2+(-1)*V^2*U^2+(-1)*V^2*U*V+(1)*V*U^2*V^2+(1)*V^2*U^2*V+(1)*V^2*U*V^2 ],

[ (1)*V*U^2*V, (-1)*U^2*V+(-1)*U*V*U+(1)*U*V^2+(-1)*V*U^2+(1)*V*U*V+(1)*V^2*U+(1)*U*V^2*U ],

[ (1)*V^2*U*V^2, (1)*U^2*V+(1)*U*V*U+(-1)*U*V^2+(1)*V*U^2+(-1)*V*U*V+(-1)*V^2*U+(-1)*U^2*V*U+(-1)*U*V*U^2+(1)*V*U*V^2+(1)*V^2*U*V+(1)*U^2*V*U^2 ],

[ (1)*(U^2*V)^2, (1)*(V*U^2)^2 ],

[ (1)*(U^2*V)^2, (-2)*U^2*V*U^2+(1)*U^2*V^2*U^2 ],

[ (1)*U*(U*V)^2, (1)*U^2*V+(1)*U*V*U+(1)*U*V^2+(1)*V*U^2+(1)*V*U*V+(1)*V^2*U+(-1)*U^2*V*U+(-1)*U^2*V^2+(
    -1)*U*V*U^2+(-1)*U*V^2*U+(-1)*V^2*U^2+(1)*U*V^2*U^2 ],

[ (1)*(U*V)^2*V, (2)*U^2*V+(2)*U*V*U+(2)*V*U^2+(-1)*U^2*V^2+(-1)*U*V^2*U+(-1)*V*U*V^2+(-1)*V^2*U^2+(-1)*V^2*U*V+(1)*V^2*U^2*V ],

[ (1)*U*V*U^2*V, (-1)*U^2*V+(-1)*U*V*U+(-1)*U*V^2+(-1)*V*U^2+(-1)*V*U*V+(-1)*V^2*U+(-1)*U^2*V*U+(1)*U^2*V^2+(-1)*U*V*U^2+(1)*U*V^2*U+(1)*V^2*U^2+(1)*U^2*V^2*U ],

[ (1)*U*V^2*U*V, (-2)*U^2*V+(-2)*U*V*U+(-2)*V*U^2+(1)*U^2*V^2+(1)*U*V^2*U+(-1)*V*U*V^2+(1)*V^2*U^2+(-1)*V^2*U*V+(
    1)*V*U^2*V^2 ],

[ (1)*V*U^2*V*U, (-1)*U^2*V+(-1)*U*V*U+(-1)*U*V^2+(-1)*V*U^2+(-1)*V*U*V+(-1)*V^2*U+(-1)*U^2*V*U+(1)*U^2*V^2+(
    -1)*U*V*U^2+(1)*U*V^2*U+(1)*V^2*U^2+(1)*U*V^2*U^2 ],

[ (1)*(V*U)^2*U, (1)*U^2*V+(1)*U*V*U+(1)*U*V^2+(1)*V*U^2+(1)*V*U*V+(1)*V^2*U+(-1)*U^2*V*U+(-1)*U^2*V^2+(
    -1)*U*V*U^2+(-1)*U*V^2*U+(-1)*V^2*U^2+(1)*U^2*V^2*U ],

[ (1)*V*U*V^2*U, (-2)*U^2*V+(-2)*U*V*U+(-2)*V*U^2+(1)*U^2*V^2+(1)*U*V^2*U+(-1)*V*U*V^2+(1)*V^2*U^2+(-1)*V^2*U*V+(1)*V^2*U^2*V ],

[ (1)*V*(V*U)^2, (2)*U^2*V+(2)*U*V*U+(2)*V*U^2+(-1)*U^2*V^2+(-1)*U*V^2*U+(-1)*V*U*V^2+(-1)*V^2*U^2+(-1)*V^2*U*V+(1)*V*U^2*V^2 ],

[ (1)*V*U*(U*V)^2, (2)*U^2*V*U+(2)*U*V*U^2+(2)*V*U*V^2+(2)*V^2*U*V+(-1)*U^2*V^2*U+(-1)*U*V^2*U^2+(-1)*V*U^2*V^2+(-1)*V^2*U^2*V+(1)*V*U*V^2*U^2 ],

[ (1)*U*(U*V)^2*U, (-1)*U^2*V+(-1)*U*V*U+(-1)*U*V^2+(-1)*V*U^2+(-1)*V*U*V+(-1)*V^2*U+(1)*U^2*V*U+(1)*U^2*V^2+(1)*U*V*U^2+(1)*U*V^2*U+(1)*V^2*U^2+(-1)*U^2*V*U^2+(-1)*U^2*V^2*U+(
    -1)*U*V^2*U^2 ],

[ (1)*U*(U*V)^2*V, (2)*U^2*V*U+(2)*U*V*U^2+(2)*V*U*V^2+(2)*V^2*U*V+(-1)*U^2*V^2*U+(-1)*U*V^2*U^2+(-1)*V*U^2*V^2+(-1)*V^2*U^2*V+(1)*U*V^2*U^2*V ],

[ (1)*(U^2*V)^2, (1)*U^2*V+(1)*U*V*U+(1)*U*V^2+(1)*V*U^2+(1)*V*U*V+(1)*V^2*U+(-1)*U^2*V*U+(-1)*U^2*V^2+(-1)*U*V*U^2+(-1)*U*V^2*U+(-1)*V^2*U^2+(-1)*U^2*V*U^2+(1)*U^2*V^2*U+(
    1)*U*V^2*U^2 ],

[ (1)*(V*U)^2*U*V, (2)*U^2*V*U+(2)*U*V*U^2+(-2)*V*U*V^2+(-2)*V^2*U*V+(-1)*U^2*V^2*U+(-1)*U*V^2*U^2+(1)*V*U^2*V^2+(1)*V^2*U^2*V+(1)*V*U^2*V^2*U ],

[ (1)*(U*V*U)^2, (2)*U^2*V+(2)*U*V*U+(2)*U*V^2+(2)*V*U^2+(2)*V*U*V+(2)*V^2*U+(-2)*U^2*V*U+(-2)*U^2*V^2+(-2)*U*V*U^2+(-2)*U*V^2*U+(-2)*V^2*U^2+(2)*U^2*V^2*U+(2)*U*V^2*U^2 ],

[ (1)*U*V*U^2*V^2, (2)*U^2*V*U+(2)*U*V*U^2+(-2)*V*U*V^2+(-2)*V^2*U*V+(-1)*U^2*V^2*U+(-1)*U*V^2*U^2+(1)*V*U^2*V^2+(1)*V^2*U^2*V+(1)*U^2*V^2*U*V ],

[ (1)*(V*U)^2*V^2, (-3)*U^2*V+(-3)*U*V*U+(1)*U*V^2+(-3)*V*U^2+(1)*V*U*V+(1)*V^2*U+(1)*U^2*V*U+(1)*U^2*V^2+(1)*U*V*U^2+(1)*U*V^2*U+(1)*V^2*U^2+(-1)*U^2*V*U^2+(-1)*V*U^2*V^2+(
    -1)*V^2*U^2*V ],

[ (1)*(U*V)^2*V*U, (2)*U^2*V*U+(2)*U*V*U^2+(2)*V*U*V^2+(2)*V^2*U*V+(-1)*U^2*V^2*U+(-1)*U*V^2*U^2+(-1)*V*U^2*V^2+(-1)*V^2*U^2*V+(1)*V^2*U^2*V*U ],

[ (1)*(V*U*V)^2, (4)*U^2*V+(4)*U*V*U+(4)*V*U^2+(-2)*U^2*V^2+(-2)*U*V^2*U+(-2)*V*U*V^2+(-2)*V^2*U^2+(-2)*V^2*U*V+(2)*V*U^2*V^2+(2)*V^2*U^2*V ],

[ (1)*U*V*(V*U)^2, (-2)*U^2*V*U+(-2)*U*V*U^2+(2)*V*U*V^2+(2)*V^2*U*V+(1)*U^2*V^2*U+(1)*U*V^2*U^2+(-1)*V*U^2*V^2+(-1)*V^2*U^2*V+(1)*V*U^2*V^2*U ],

[ (1)*(U*V^2)^2, (1)*U^2*V+(1)*U*V*U+(1)*U*V^2+(1)*V*U^2+(1)*V*U*V+(1)*V^2*U+(1)*U^2*V*U+(-1)*U^2*V^2+(1)*U*V*U^2+(-1)*U*V^2*U+(-2)*V*U*V^2+(-1)*V^2*U^2+(-2)*V^2*U*V+(
    -1)*U^2*V*U^2+(1)*V*U^2*V^2+(1)*V^2*U^2*V ],

[ (1)*V^2*U^2*V*U, (2)*U^2*V*U+(2)*U*V*U^2+(-2)*V*U*V^2+(-2)*V^2*U*V+(-1)*U^2*V^2*U+(-1)*U*V^2*U^2+(1)*V*U^2*V^2+(1)*V^2*U^2*V+(1)*V*U*V^2*U^2 ],

[ (1)*(V*U^2)^2, (1)*U^2*V+(1)*U*V*U+(1)*U*V^2+(1)*V*U^2+(1)*V*U*V+(1)*V^2*U+(-1)*U^2*V*U+(-1)*U^2*V^2+(-1)*U*V*U^2+(-1)*U*V^2*U+(-1)*V^2*U^2+(-1)*U^2*V*U^2+(1)*U^2*V^2*U+(
    1)*U*V^2*U^2 ],

[ (1)*U*V^2*U^2*V, (4)*V*U*V^2+(4)*V^2*U*V+(-2)*V*U^2*V^2+(-2)*V^2*U^2*V+(1)*V*U*V^2*U^2 ],

[ (1)*(U*V)^2*U^2, (-1)*U^2*V+(-1)*U*V*U+(-1)*U*V^2+(-1)*V*U^2+(-1)*V*U*V+(-1)*V^2*U+(1)*U^2*V*U+(1)*U^2*V^2+(1)*U*V*U^2+(1)*U*V^2*U+(1)*V^2*U^2+(-1)*U^2*V*U^2+(-1)*U^2*V^2*U+(
    -1)*U*V^2*U^2 ],

[ (1)*V*(V*U)^2*U, (2)*U^2*V*U+(2)*U*V*U^2+(2)*V*U*V^2+(2)*V^2*U*V+(-1)*U^2*V^2*U+(-1)*U*V^2*U^2+(-1)*V*U^2*V^2+(-1)*V^2*U^2*V+(1)*V*U^2*V^2*U ],

[ (1)*U^2*V^2*U*V, (-4)*V*U*V^2+(-4)*V^2*U*V+(2)*V*U^2*V^2+(2)*V^2*U^2*V+(1)*V*U^2*V^2*U ],

[ (3)*V*U^2*V^2, (-6)*U^2*V*U+(-6)*U*V*U^2+(6)*V*U*V^2+(6)*V^2*U*V+(3)*U^2*V^2*U+(3)*U*V^2*U^2+(-3)*V^2*U^2*V ],

[ (1)*(V^2*U)^2, (1)*U^2*V+(1)*U*V*U+(1)*U*V^2+(1)*V*U^2+(1)*V*U*V+(1)*V^2*U+(1)*U^2*V*U+(-1)*U^2*V^2+(1)*U*V*U^2+(-1)*U*V^2*U+(-2)*V*U*V^2+(-1)*V^2*U^2+(-2)*V^2*U*V+(-1)*U^2*V*U^2+(1)*V*U^2*V^2+(
    1)*V^2*U^2*V ],

[ (1)*V*(V*U)^2*V, (-3)*U^2*V+(-3)*U*V*U+(1)*U*V^2+(-3)*V*U^2+(1)*V*U*V+(1)*V^2*U+(1)*U^2*V*U+(1)*U^2*V^2+(1)*U*V*U^2+(1)*U*V^2*U+(1)*V^2*U^2+(-1)*U^2*V*U^2+(-1)*V*U^2*V^2+(-1)*V^2*U^2*V ],

[ (6)*U*V^2*U^2, (-6)*U^2*V+(-6)*U*V*U+(-6)*U*V^2+(-6)*V*U^2+(-6)*V*U*V+(-6)*V^2*U+(6)*U^2*V*U+(6)*U^2*V^2+(6)*U*V*U^2+(6)*U*V^2*U+(6)*V^2*U^2+(6)*U^2*V*U^2+(-6)*U^2*V^2*U ],

[ (3)*V*U*V^2*U^2, (3)*U^2*V+(3)*U*V*U+(3)*U*V^2+(3)*V*U^2+(3)*V*U*V+(3)*V^2*U+(-15)*U^2*V*U+(-3)*U^2*V^2+(-15)*U*V*U^2+(-3)*U*V^2*U+(-3)*V^2*U^2+(-3)*U^2*V*U^2+(9)*U^2*V^2*U+(9)*U*V^2*U^2+(
    -3)*V*U^2*V^2*U ],

[ (1)*V*U^2*V^2, (-2)*U^2*V*U+(-2)*U*V*U^2+(2)*V*U*V^2+(2)*V^2*U*V+(1)*U^2*V^2*U+(1)*U*V^2*U^2+(-1)*V^2*U^2*V ],

[ (1)*U*V^2*U^2, (-1)*U^2*V+(-1)*U*V*U+(-1)*U*V^2+(-1)*V*U^2+(-1)*V*U*V+(-1)*V^2*U+(1)*U^2*V*U+(1)*U^2*V^2+(1)*U*V*U^2+(1)*U*V^2*U+(1)*V^2*U^2+(1)*U^2*V*U^2+(-1)*U^2*V^2*U ],

[ (1)*V*U*V^2*U^2, (-2)*U^2*V+(-2)*U*V*U+(-2)*U*V^2+(-2)*V*U^2+(-2)*V*U*V+(-2)*V^2*U+(-2)*U^2*V*U+(2)*U^2*V^2+(-2)*U*V*U^2+(2)*U*V^2*U+(2)*V^2*U^2+(2)*U^2*V*U^2+(-1)*V*U^2*V^2*U ],

[ (1)*(V*U)^3, (2)*U^2*V+(2)*U*V*U+(2)*U*V^2+(2)*V*U^2+(2)*V*U*V+(2)*V^2*U+(2)*U^2*V*U+(-2)*U^2*V^2+(2)*U*V*U^2+(-2)*U*V^2*U+(-2)*V^2*U^2+(-2)*U^2*V*U^2 ],

[ (3)*U*V*U^2, (-3)*U^2*V*U ],

[ (3)*U^2*V*U^2, 0*U ],

[ (3)*V*U*V^2, (-3)*V^2*U*V ],

[ (3)*V^2*U*V^2, 0*U],

[ (6)*V*U*V, (6)*U^2*V+(6)*U*V*U+(-6)*U*V^2+(6)*V*U^2+(-6)*V^2*U ],

[ (6)*U*V^2*U, (12)*U^2*V+(12)*U*V*U+(12)*V*U^2+(-6)*U^2*V^2+(-6)*V^2*U^2 ],

[ (1)*V*U^2*V, (1)*U*V^2*U ],

[ (6)*V*U^2*V^2*U, 0*V ],

];
rel_pos:=List([1..Size(rel)], i -> Concatenation(rel[i],[i]));
LogTo();
LogTo("label1");
Print("label1:=[];", "\n");
for i in [1..Size(rel)] do
Print("label1[",i ,"]:= \"");
PrintNP(GP2NP(rel_pos[i][1]));
Print("\"; ", "\n");
od;
LogTo();
Exec("sed ':a;N;s/\\n//g;ta' label1 > label");
Read("label");
label:=List(label1,i -> ReplacedString(i, "^", ""));
label:=List(label,i -> ReplacedString(i, " ", ""));
Exec("rm label label1");

###################################
###################################
###################################
###################################
var:=['U', 'V'];
ORDERP:=function(mon1, mon2)#função para ordenar monomios (lex e grau) strgi deve ser monomios senão erro.
return ORDERING(String(mon1), String(mon2));
end;

ORDERP_L:=function(lista1, lista2)#esta função usa ORDERP para ordenar um lista pela primeira coordenada.
return ORDERP(lista1[1], lista2[1]);
end;

rel_pos:=List([1..Size(rel)], i -> Concatenation(rel[i],[i]));
#Sort(rel_pos, ORDERP_L);
#rel_pos:=Reversed(rel_pos);
###################################
###################################
###################################
###################################

LogTo();
LogTo("rel.tex");
Print(
"\\documentclass{book}", "\n",
"\\usepackage{amssymb}", "\n",
"\\usepackage{amsmath}", "\n",
"\\usepackage{amsthm}", "\n",
"\\usepackage{graphicx,xcolor}", "\n",
"\\usepackage[normalem]{ulem}", "\n",
"\\begin{document}", "\n\n\n"
);

Print("\\begin{enumerate}", "\n");
for i in [1..Size(rel)] do
   Print("\\item (",i , ")", ":= ",make_rel[rel_pos[i][3]],"\n","\\label{eq:", label[i], "}\\\\", "\n");
   Print("$", "\n");
   PrintNP(GP2NP(rel_pos[i][1]));
   Print( "\\;=\\;", "\n");
   PRINTPOL(rel_pos[i][2]);
   Print("$;","\\\\ \\\\ \n");
#########################
#########################
#   Print("\\item[(",rel_pos[i][3],"):=]", make_rel[rel_pos[i][3]], "\\\\", "\n");
#   Print("$", "\n");
#   PRINTPOL_strg(rel_pos[i][1]);
#   Print( "\\;=\\;", "\n");
#   PRINTPOL_strg(rel_pos[i][2]);
#   Print("$;","\\\\ \\\\ \n");
od;

Print("\\end{enumerate}", "\n");

#Print("\\[ \\left< x, y\\;\\;|\\;\\; (x-1)^3, (y-1)^3, (xy-1)^3, (x^{-1}y -1)^3\\right>\\]", "\n");

l:=[];
for i in [1..Size(rel)] do
   if CHECKBYMAT(rel[i]) = false then
       Add(l, i);
   fi;
od;

if l=[] then
   Print("{\\Huge Todas equa\\c c\\~oes s\\~ao verdadeiras.}", "\\newline \\newline" ,"\n");
else
   for i in l do
       Print("(",i,")=falsa, ", "\n" );
   od;
fi;

Print("\\newpage", "\n");

# Proximos comando servem para eliminar os monomios que são escrito em função dos seus antecessores na ordem lex degree (ORDERP).
l:=[];
for i in [1..6] do
   l[i]:=List([1..i], j -> [U, V]);
od;

l:=List(l, Cartesian);
l:=List(l, i -> List(i, Product));
l:=Union(l); #lista dos monomios
for i in [1..Size(l)] do
   if SUBS(l[i],[U^3, 0*U])<> l[i] then
       l[i]:= SUBS(l[i],[U^3, 0*U]);
   elif SUBS(l[i],[V^3, 0*V])<> l[i] then
       l[i]:= SUBS(l[i],[V^3, 0*U]);
   fi;
od;
l:=Filtered(l, i -> i<>0*i);
Sort(l, ORDERP);

lmon:=[];
cont:=0;
for i in [1..Size(l)] do
   laux:=[];
   lf:=Filtered(rel, relation -> relation[1] = l[i]);
   for relf in lf do
       if POTENBASIS(relf, l) = true then
           Add(laux, true);# aqui deve ser não lmon
       break;
       fi;
   od;
   if true in laux then
       lmon[i]:=[l[i], true];
       cont:=cont+1;
   else
       lmon[i]:=[l[i], false];
   fi;
od;

for mon in lmon do
   if mon[2] = true then
       Print("$", "\n", "\\textcolor{red}{\\bf", "\n");
       PRINTPOL(mon[1]);
       Print(",}$ \\;\\; \n");
   else
       Print("$", "\n");
       PRINTPOL(mon[1]);
       Print("$,\\;\\; \n");
   fi;
od;


Print("\\newline \\newline","\n" ,"{\\Huge Dimens\\~ao Algebra $\\leq$ ", Size(l)-cont, "}\\\\", "\n\n");

##################
################### proximo loops sao para mostrar q B^7=0.

l1:=List([1..7], j -> [U, V]);
l1:=Cartesian(l1);
l1:=List(l1, Product);

for i in [1..Size(l1)] do
   if SUBS(l1[i],[U^3, 0*U])<> l1[i] then
       l1[i]:= SUBS(l1[i],[U^3, 0*U]);
   elif SUBS(l1[i],[V^3, 0*V])<> l1[i] then
       l1[i]:= SUBS(l1[i],[V^3, 0*U]);
   fi;
od;
l1:=Filtered(l1, i -> i<>0*i);
Sort(l1, ORDERP);

lf:=Filtered(rel, relation -> relation[2] = 0*relation[2]);
lf:=List(lf, i -> i[1]);

for mon in l1 do
   if mon in lf then
       Print("$", "\n", "\\textcolor{red}{\\bf", "\n");
       PRINTPOL(mon);
       Print( ",}$ \\;\\; \n");
   else
       Print("$", "\n");
       PRINTPOL(mon);
       Print("$,\\;\\; \n");
   fi;
od;

Print("\\newpage", "\\center{\\bf \\Huge Monomios Nulos}", "\n", "\\newline \\newline");
l:=[];
for i in [1..6] do
   l[i]:=List([1..i], j -> ["u", "v"]);
od;

l:=List(l, Cartesian);
l1:=[];for x in l do l1:=Concatenation(l1,x);od;
l1:=List(l1, i-> JoinStringsWithSeparator(i, "*") );

l1:=Filtered(l1, i -> PositionSublist(i, "u*u*u")=fail);
l1:=Filtered(l1, i -> PositionSublist(i, "v*v*v")=fail);
for mon in l1 do
   if EvalString(mon)=0*u then
       Print("$", "\n", "\\underline{\\textcolor{red}{\\bf", "\n");
       mon:=ReplacedString(mon, "u", "U");
       mon:=ReplacedString(mon, "v", "V");
       PrintNP(GP2NP(EvalString(mon)));
       Print( ",}}$ \\;\\; \n");
   else
       Print("$", "\n");
       mon:=ReplacedString(mon, "u", "U");
       mon:=ReplacedString(mon, "v", "V");
       PrintNP(GP2NP(EvalString(mon)));
       Print("$,\\;\\; \n");
   fi;
od;



Print("\\end{document}", "\n");
LogTo();

Exec("pdflatex rel.tex");
#Exec("/Applications/Adobe\\ Reader.app/Contents/MacOS/AdobeReader rel.pdf &" );
#Exec("/Applications/Safari.app/Contents/MacOS/Safari rel.pdf &" );
Exec("rm *.aux *.log MEDELETESEMPRE");

# os comandos abaixo sao para eliminar os produtos cujo peso e maior do 7 ja que B^7=0.
#pol:= ( (U^2-U)*(V^2-V) + (U^2-U) + (V^2-V) )*( U*V + U + V + One(A) ) + U*V + U + V;
#r:=M(POL2REL(pol^3), One(A));
#pol:=r[1]-r[2];
#spol:=String(pol);
#spol:=ReplacedString(spol, "U", "u");
#spol:=ReplacedString(spol, "V", "v");
#sspol:=SplitString(spol, "+");
#sspol:=Filtered(sspol, i -> not EvalString(i) in P7);
#spol:=JoinStringsWithSeparator( sspol, "+");
#spol:=ReplacedString(spol, "u", "U");
#spol:=ReplacedString(spol, "v", "V");
#pol:=EvalString(spol);
#PrintNP(GP2NP(POL2REL(pol)));

mrep

Tue, 18 Feb 2014 17:06:55 -0300

This file provides a matricial representations of the cubic algebra, here we use the GAP packge GBNP.

AA:=FreeAssociativeAlgebraWithOne(FIELD, "x", "a", "y", "b");
x:=AA.1;a:=AA.2; y:=AA.3; b:=AA.4;id:=One(AA);
rel_AA:=[a*x -id, x*a - id, b*y - id, y*b -id, (x-id)^3, (y-id)^3, (x*y - id)^3 , (a*y - id)^3 , (x^2*y - id)^3, (x*y^2 - id)^3];

rel_np:=List(rel_AA, GP2NP);
sb:=SGrobner(rel_np);
bqa:=BaseQA(sb,4, 0);

x:=MatrixQA(1, bqa, sb);
y:=MatrixQA(3, bqa, sb);
id:=x^0;

AA:= Algebra(FIELD, [ x,y ]) ;
I:=Ideal(AA, [(Comm(x,y) - x^0)^3]);
AA:=AA/I;
x:=AA.1;y:=AA.2;id:=x^0;
u:=x-id;v:=y-id;
BB:= Subalgebra(AA, [u,v]);

POL2REL

Tue, 18 Feb 2014 16:58:35 -0300

This function transforms a relation of type [pol, 0] in [mon, pol_1], where mon is a summand of largest order, where pol = mon - pol_1, for example,

gap>POL2REL(U*V^2*U - V*U^2*V);

[V*U^2*V, U*V^2*U]

gap>

This function calls other functions: ORDERING.

POL2REL:=function(pol)
local polaux, rel;

if pol = 0*pol then return [0*pol, 0*pol];fi;

polaux:=SplitString(String(pol), "+");
Sort(polaux, ORDERING);
polaux:=Reversed(polaux);

rel:=[];
if Size(polaux) = 1 then
   rel[2]:=Zero(EvalString(polaux[1]));
   if EvalString(COEF_STRG(polaux[1])) < 0 then
   rel[1]:=   -EvalString(polaux[1]);
else
   rel[1]:=   EvalString(polaux[1]);
   fi;
   return rel;
fi;


if EvalString(COEF_STRG(polaux[1])) < 0 then
   rel[1]:=   -EvalString(polaux[1]);
   rel[2]:=   EvalString(JoinStringsWithSeparator(polaux{[2..Size(polaux)]}, "+"));
else
   rel[1]:=   EvalString(polaux[1]);
   rel[2]:=   -EvalString(JoinStringsWithSeparator(polaux{[2..Size(polaux)]}, "+"));
fi;

return rel;

end;

TR

Tue, 18 Feb 2014 16:16:16 -0300

Give a non-commutative polynomial in the variables U and V, this function exchanges U and V by other polynomials, for example

gap>TR(U*V, [V,U]);

gap>(1)*V*U;

That is TR(pol, [X,Y]) have the same effect of the endomorphism U --> X and V --> Y applied to pol;
This functions calls other functions: SUBS.

TR:=function(pol, ltr)
local polaux;

polaux:=SUBS(pol, [U, T]);
polaux:=SUBS(polaux, [V, ltr[2]]);
polaux:=SUBS(polaux, [T, ltr[1]]);

return polaux;
end;

SUBSC

Tue, 18 Feb 2014 16:15:32 -0300

This routine substitutes one of the sub-summands of a given polynomial, produced in the list by the function SEARCHC.

This function calls other functions: COEF_STRG, MON_STRG and STRETCH.

SUBSC:=function(pol, rel)
local pol_s, lm, lc, rs, ind, ind1;

if pol = Zero(pol) then
   return pol;
else
   pol_s:=SplitString(String(pol), "+");
fi;
lm:=List(pol_s, mon -> MON_STRG(mon));
lc:=List(pol_s, mon -> COEF_STRG(mon));
#Print("pol_s= ",pol_s, "\n","lm= ",lm, "\n","lc= ", lc, "\n");

if rel[1] = Zero(pol) and rel[2] <> Zero(pol) then
   rs:=[];
   rs[1]:="(0)*U";
   rs[2]:=String(rel[2]);
elif rel[1] <> Zero(pol) and rel[2] = Zero(pol) then
   rs:=[];
   rs[1]:=String(rel[1]);
   rs[2]:="(0)*U";
else
   rs:=List(rel, String);
fi;

if MON_STRG(rs[1]) in lm then
   ind := Position(lm, MON_STRG(rs[1]));
   if EvalString(lc[ind]) / EvalString(    COEF_STRG(rs[1])     ) in Integers then
       lc[ind] := Concatenation( "(", String(EvalString(lc[ind]) / EvalString( COEF_STRG(rs[1]) )), ")");
       lm[ind] := Concatenation("(", rs[2], ")");
       for ind1 in [1..Size(lm)] do
           pol_s[ind1] := Concatenation( lc[ind1], "*", lm[ind1]);
       od;
       pol_s:= JoinStringsWithSeparator(pol_s, "+");
       return EvalString(pol_s);
   else
       return pol;
   fi;
else
   return pol;
fi;

end;

STRETCH

Tue, 18 Feb 2014 16:14:36 -0300

Given a non-commutative monomial written as a string, this function returns a non-commutative monomial, but this is represented without powers, for example

gap>STRETCH("V^3*U^2");

"V*V*V*U*U"

gap>

STRETCH:=function(strg_mon)
local mon, siz_mon, i, j;

mon:=GP2NP( EvalString(strg_mon) );
siz_mon:=Size(mon[1]);

for i in [1..siz_mon] do
   for j in [1..Size(mon[1][i])] do
       mon[1][i][j] := lvar[ mon[1][i][j] ];
   od;
od;

for i in [1..siz_mon] do
   mon[2][i] := Concatenation( "(", String(mon[2][i]), ")" );
od;

mon[1]:= List(mon[1], i-> JoinStringsWithSeparator(i, "*"));
mon:=List([1..siz_mon], i-> Concatenation(mon[2][i], "*", mon[1][i]));

return JoinStringsWithSeparator(mon, "+");

end;

SEARCHC

Tue, 18 Feb 2014 16:13:55 -0300

Given a polynomial this routine produces a list of summands which are substitutable from a list of relations.

ORD:=function(l1, l2) return l1[2] < l2[2];end;

FREL:=function(lrel)
local siz, nlrel, laux, x, ulrel, laux_par;

siz:=Size(lrel);
nlrel:=List([1..siz], i -> [lrel[i], i]);
ulrel:= List([1..siz], i -> lrel[i][1]);
ulrel:=Union(ulrel, ulrel);

laux:=[];
for x in ulrel do
   laux_par:=Filtered(nlrel, i -> i[1][1] = x);
   Add(laux, laux_par[Size(laux_par)]);
od;
Sort(laux, ORD);
return laux;
end;


SEARCHC:=function(pol, lrel)
local frel, x;

frel:=FREL(lrel);

for x in frel do
   if SUBSC( pol, x[1]) <> pol then
       Print("Coincident relation ", x[2], " :   ", x[1][1] , " = ??","\n");
   fi;
od;
end;