Difference between revisions of "ApCoCoA-1:NC.KernelOfHomomorphism"
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<command> | <command> | ||
<title>NC.KernelOfHomomorphism</title> | <title>NC.KernelOfHomomorphism</title> | ||
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<example> | <example> | ||
Use QQ[x[1..2],y[1..2],a,b]; | Use QQ[x[1..2],y[1..2],a,b]; | ||
| − | NC.SetOrdering( | + | NC.SetOrdering("ELIM"); -- we want to eliminate x[1..2],y[1..2] |
-- Group ring Q<a,b>/<F11,F12,F13> with the triangle group of order 576 | -- Group ring Q<a,b>/<F11,F12,F13> with the triangle group of order 576 | ||
F11:=[[a^2], [-1]]; | F11:=[[a^2], [-1]]; | ||
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F26:=[[y[1],y[2]],[-y[2],y[1]]]; | F26:=[[y[1],y[2]],[-y[2],y[1]]]; | ||
-- Q-group algebra homomorphism phi: Q<a,b>/<F11,F12,F13> -->Q<x[1],y[1],x[2],y[2]>/<F21,F22,F23,F24,F25,F26> | -- Q-group algebra homomorphism phi: Q<a,b>/<F11,F12,F13> -->Q<x[1],y[1],x[2],y[2]>/<F21,F22,F23,F24,F25,F26> | ||
| − | -- defined by | + | -- defined by mapping a to x[1] and b to 0 |
D1:=[[a],[-x[1]]]; -- map a to x[1] | D1:=[[a],[-x[1]]]; -- map a to x[1] | ||
D2:=[[b]]; -- map b to 0 | D2:=[[b]]; -- map b to 0 | ||
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-- hence the group <a,b: b^2=(ab)^2=1> is infinite. | -- hence the group <a,b: b^2=(ab)^2=1> is infinite. | ||
Use QQ[a,b,y]; | Use QQ[a,b,y]; | ||
| − | NC.SetOrdering( | + | NC.SetOrdering("ELIM"); -- we want to eliminate a and b |
F1:=[[b^2],[-1]]; -- b^2-1 | F1:=[[b^2],[-1]]; -- b^2-1 | ||
F2:=[[a,b,a,b],[-1]]; -- (ab)^2-1 | F2:=[[a,b,a,b],[-1]]; -- (ab)^2-1 | ||
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------------------------------- | ------------------------------- | ||
Use QQ[a,b,y]; | Use QQ[a,b,y]; | ||
| − | NC.SetOrdering( | + | NC.SetOrdering("ELIM"); -- we want to eliminate a and b |
G1:=[[a^3],[-1]]; -- a^3-1 | G1:=[[a^3],[-1]]; -- a^3-1 | ||
G2:=[[b^2],[-1]]; -- b^2-1 | G2:=[[b^2],[-1]]; -- b^2-1 | ||
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</description> | </description> | ||
<seealso> | <seealso> | ||
| − | <see>Use</see> | + | <see>ApCoCoA-1:Use|Use</see> |
| − | <see>NC.FindPolys</see> | + | <see>ApCoCoA-1:NC.FindPolys|NC.FindPolys</see> |
| − | <see>NC.GB</see> | + | <see>ApCoCoA-1:NC.GB|NC.GB</see> |
| − | <see>NC.SetOrdering</see> | + | <see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see> |
| − | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |
</seealso> | </seealso> | ||
<types> | <types> | ||
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<key>NC.KernelOfHomomorphism</key> | <key>NC.KernelOfHomomorphism</key> | ||
<key>KernelOfHomomorphism</key> | <key>KernelOfHomomorphism</key> | ||
| − | <wiki-category>Package_ncpoly</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category> |
</command> | </command> | ||
Latest revision as of 13:34, 29 October 2020
| This article is about a function from ApCoCoA-1. |
NC.KernelOfHomomorphism
The kernel of an algebra homomorphism.
Syntax
Description
Proposition (Kernel of an Algebra Homomorphism): Let I be a two-sided ideal in the free monoid ring K<x[1],...,x[n]>, and let J be a two-sided ideal in the free monoid ring K<y[1],...,y[m]>. Moreover, let g_1,...,g_m be polynomials in K<x[1],...,x[n]>, and let phi: K<y[1],...,y[m]>/J-->K<x[1],...,x[n]>/I be a homomorphism of K-algebras defined by phi(y[i]+J)=g_i+I for i=1,...,m. We form the free monoid ring K<x[1],...,x[n],y[1],...,y[m]>, and let D be the diagonal ideal generated by the set {y[1]-g_1,...,y[m]-g_m}. Then we have ker(phi)=((D+J) intersets K<y[1],...,y[m]>)+I.
Example
Use QQ[x[1..2],y[1..2],a,b];
NC.SetOrdering("ELIM"); -- we want to eliminate x[1..2],y[1..2]
-- Group ring Q<a,b>/<F11,F12,F13> with the triangle group of order 576
F11:=[[a^2], [-1]];
F12:=[[b^3], [-1]];
F13:=[[a,b,a,b,a,b^2,a,b^2,a,b,a,b,a,b^2,a,b^2], [-1]];
-- Group ring Q<x[1],y[1],x[2],y[2]>/<F21,F22,F23,F24,F25,F26> with the fundamental group of 3-manifold of order 72
F21:=[[x[1]^3],[y[1]^3]]; --x[1]^3=y[1]^3
F22:=[[x[1]^3],[-y[1],x[2],y[1],x[2]]]; --x[1]^3p=(y[1]x[1]^-1)^2
F23:=[[x[2],x[1]],[-1]];
F24:=[[x[1],x[2]],[-x[2],x[1]]];
F25:=[[y[2],y[1]],[-1]];
F26:=[[y[1],y[2]],[-y[2],y[1]]];
-- Q-group algebra homomorphism phi: Q<a,b>/<F11,F12,F13> -->Q<x[1],y[1],x[2],y[2]>/<F21,F22,F23,F24,F25,F26>
-- defined by mapping a to x[1] and b to 0
D1:=[[a],[-x[1]]]; -- map a to x[1]
D2:=[[b]]; -- map b to 0
G:=[D1,D2,F21,F22,F23,F24,F25,F26];
Gb:=NC.GB(G);
KGb:=NC.FindPolys(Gb,[a,b]);
Concat(KGb,[F11,F12,F13]); -- a generating sy[1]stem of the ker(phi)
[[[b]], [[a^18], [-1]], [[a^2], [-1]], [[b^3], [-1]],
[[a, b, a, b, a, b^2, a, b^2, a, b, a, b, a, b^2, a, b^2], [-1]]]
-------------------------------
Corollary (Minimal Polynomial): Let phi: K[y]-->K<x[1],...,x[n]>/I be a K-algebra homomorphism given by phi(y)=g+I. Then g+I is algebraic over K if and only if ker(phi) is not zero. Moreover, if g+I is algebraic over K, then the unique monic generating polynomial of the ideal ker(phi) is the minimal polynomial of g+I over K.
Example
-- The following example shows that a is transcendental over Q in the algebra Q<a,b>/<b^2-1,(ab)^2-1>,
-- hence the group <a,b: b^2=(ab)^2=1> is infinite.
Use QQ[a,b,y];
NC.SetOrdering("ELIM"); -- we want to eliminate a and b
F1:=[[b^2],[-1]]; -- b^2-1
F2:=[[a,b,a,b],[-1]]; -- (ab)^2-1
-- Construct a Q-algebra homomorphism phi: Q[y]--> Q<a,b>/<b^2-1,(ab)^2-1>
-- defined by mapping y to a
D:=[[y],[-a]]; -- y-a
G:=[D,F1,F2];
Gb:=NC.GB(G);
Gb;
NC.FindPolys(Gb,[y]);
[[[a], [-y]], [[b^2], [-1]], [[y, b, y], [-b]]]
-------------------------------
[ ]
-------------------------------
Use QQ[a,b,y];
NC.SetOrdering("ELIM"); -- we want to eliminate a and b
G1:=[[a^3],[-1]]; -- a^3-1
G2:=[[b^2],[-1]]; -- b^2-1
G3:=[[a,b,a,b],[-1]]; -- (ab)^2-1
-- Construct a Q-algebra homomorphism phi: Q[y]--> Q<a,b>/<a^3-1,b^2-1,(ab)^2-1>
-- defined by mapping y to bab
D:=[[y],[-b,a,b]]; -- y-bab
G:=[D,G1,G2,G3];
Gb:=NC.GB(G);
NC.FindPolys(Gb,[y]);
[[[y^3], [-1]]]
-------------------------------
See also
