ApCoCoA:Heisenberg groups

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Description

The Heisenberg group is the group of 3x3 upper triangular matrices of the form HeisenbergMatrix.png


Heisenberg groups are often used in quantum mechanics and also occurs in fourier analysis. A representation is given by:

 H(2k+1) = <a_{1},...,a_{k},b_{1},...,b_{k},c | [a_{i},b_{i}] = c, [a_{i},c] = [b_{i},c], [a_{i},b_{j}] = 1 for all i != j

Reference

Ernst Binz and Sonja Pods, Geometry of Heisenberg Groups, American Mathematical Society, 2008.

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // Number of Heisenberg group
 MEMORY.N:=1;
 
 // a invers to d and b invers to e and c invers to f
 Use ZZ/(2)[a[1..MEMORY.N],b[1..MEMORY.N],c,d[1..MEMORY.N],e[1..MEMORY.N],f];
 NC.SetOrdering("LLEX");
 
 Define CreateRelationsHeisenberg()
   Relations:=[];
 
   // add the relations of the inverse elements ad = da = be = eb = cf = fc = 1
   Append(Relations,[[c,f],[1]]);
   Append(Relations,[[f,c],[1]]);
   For Index1 := 1 To MEMORY.N Do
     Append(Relations,[[a[Index1],d[Index1]],[1]]);
     Append(Relations,[[d[Index1],a[Index1]],[1]]);
     Append(Relations,[[b[Index1],e[Index1]],[1]]);
     Append(Relations,[[e[Index1],b[Index1]],[1]]);
   EndFor;
   
   // add the relation [a_{i}, b_{i}] = c
   For Index2 := 1 To MEMORY.N Do
     Append(Relations,[[a[Index2],b[Index2],d[Index2],e[Index2]],[c]]);
   EndFor;
   
   // add the relation [a_{i}, c] = [b_i, c]
   For Index3 := 1 To MEMORY.N Do
     Append(Relations,[[a[Index3],c,d[Index3],f],[b[Index3],c,e[Index3],f]]);
   EndFor;
   
   // add the relation [a_{i}, b_{j}] = 1 for all i != j
   For Index4 := 1 To MEMORY.N Do
     For Index5 := 1 To MEMORY.N Do
       If Index4 <> Index5 Then
         Append(Relations,[[a[Index4],b[Index5],d[Index4],e[Index5]],[1]]);
       EndIf;
     Endfor;
   EndFor;
   
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsHeisenberg();
 Relations;
 Size(Relations);
 
 Gb:=NC.GB(Relations,31,1,100,1000);
 Size(Gb);

Examples in Symbolic Data Format

Heisenberg group 1
 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>a1,b1,c,d1,e1,f</vars>
 	<uptoDeg>7</uptoDeg>
 	<basis>
 	<ncpoly>c*f-1</ncpoly>
 	<ncpoly>f*c-1</ncpoly>
 	<ncpoly>a1*d1-1</ncpoly>
 	<ncpoly>d1*a1-1</ncpoly>
 	<ncpoly>b1*e1-1</ncpoly>
 	<ncpoly>e1*b1-1</ncpoly>
 	<ncpoly>a1*b1*d1*e1-c</ncpoly>
 	<ncpoly>a1*c*d1*f-b1*c*e1*f</ncpoly>
 	</basis>
 	<Comment>Heisenberg_group_1</Comment>
 </FREEALGEBRA> 
Heisenberg group 2
 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>a1,a2,b1,b2,c,d1,d2,e1,e2,f</vars>
 	<uptoDeg>5</uptoDeg>
 	<basis>
 	<ncpoly>c*f-1</ncpoly>
 	<ncpoly>f*c-1</ncpoly>
 	<ncpoly>a1*d1-1</ncpoly>
 	<ncpoly>d1*a1-1</ncpoly>
 	<ncpoly>b1*e1-1</ncpoly>
 	<ncpoly>e1*b1-1</ncpoly>
 	<ncpoly>a2*d2-1</ncpoly>
 	<ncpoly>d2*a2-1</ncpoly>
 	<ncpoly>b2*e2-1</ncpoly>
 	<ncpoly>e2*b2-1</ncpoly>
 	<ncpoly>a1*b1*d1*e1-c</ncpoly>
 	<ncpoly>a2*b2*d2*e2-c</ncpoly>
 	<ncpoly>a1*c*d1*f-b1*c*e1*f</ncpoly>
 	<ncpoly>a2*c*d2*f-b2*c*e2*f</ncpoly>
 	<ncpoly>a1*b2*d1*e2-1</ncpoly>
 	<ncpoly>a2*b1*d2*e1-1</ncpoly>
 	</basis>
 	<Comment>Heisenberg_group_2</Comment>
 </FREEALGEBRA> 
Heisenberg group 3
 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>a1,a2,a3,b1,b2,b3,c,d1,d2,d3,e1,e2,e3,f</vars>
 	<uptoDeg>4</uptoDeg>
 	<basis>
 	<ncpoly>c*f-1</ncpoly>
 	<ncpoly>f*c-1</ncpoly>
 	<ncpoly>a1*d1-1</ncpoly>
 	<ncpoly>d1*a1-1</ncpoly>
 	<ncpoly>b1*e1-1</ncpoly>
 	<ncpoly>e1*b1-1</ncpoly>
 	<ncpoly>a2*d2-1</ncpoly>
 	<ncpoly>d2*a2-1</ncpoly>
 	<ncpoly>b2*e2-1</ncpoly>
 	<ncpoly>e2*b2-1</ncpoly>
 	<ncpoly>a3*d3-1</ncpoly>
 	<ncpoly>d3*a3-1</ncpoly>
 	<ncpoly>b3*e3-1</ncpoly>
 	<ncpoly>e3*b3-1</ncpoly>
 	<ncpoly>a1*b1*d1*e1-c</ncpoly>
 	<ncpoly>a2*b2*d2*e2-c</ncpoly>
 	<ncpoly>a3*b3*d3*e3-c</ncpoly>
 	<ncpoly>a1*c*d1*f-b1*c*e1*f</ncpoly>
 	<ncpoly>a2*c*d2*f-b2*c*e2*f</ncpoly>
 	<ncpoly>a3*c*d3*f-b3*c*e3*f</ncpoly>
 	<ncpoly>a1*b2*d1*e2-1</ncpoly>
 	<ncpoly>a1*b3*d1*e3-1</ncpoly>
 	<ncpoly>a2*b1*d2*e1-1</ncpoly>
 	<ncpoly>a2*b3*d2*e3-1</ncpoly>
 	<ncpoly>a3*b1*d3*e1-1</ncpoly>
 	<ncpoly>a3*b2*d3*e2-1</ncpoly>
 	</basis>
 	<Comment>Heisenberg_group_3</Comment> 
 </FREEALGEBRA>