gswitch.html

Function: Bigebra[gswitch] - graded switch of tensor slots

Calling Sequence:

p1 = gswitch(p2,i)
Parameters:

p2 : a tensorpolynom which is of not less thang rank i in each factor
i : the slot number (first slot from the left is 1) of the pair (i,i+1) on which the switch acts

Output:

p1 : a tensorpolynom

Global variables:

_CLIENV[_QDEF_PREFACTOR]

Description:

Given a tensor polynomial the graded switch swaps two adjaicent slots in a tensor product . In switching the factors, it takes account of the sign of the permutation. Denote the grade of a homogenous Graßmann element A by |A|. The graded switch of two homogenous elements is related to the (ungraded) switch as follows:

'( A &t B) = (-1)^(|A| |B|) (A &t B) = (-1)^(|A| |B|) (B &t A).

The action is extended by linearity to arbitrary inhomogenous elements.

The graded switch is the natural switch for the Graßmann Hopf gebra. If this switch is used in the crossed products, the co-product becomes an algebra homomorphism while the wedge product becomes a co-gebra homomorphism.

The switch of an antipodal convolution algebra can be derived [3,7] . It happens to be the graded switch in the case of the Graßmann Hopf gebra.

The graded switch makes the Grassmann co-gebra graded co-commutative .

Examples:

> restart:_CLIENV[_SILENT]:=`true`:with(Bigebra):

Warning, new definition for drop_t

Warning, new definition for gco_d_monom

Warning, new definition for gco_monom

Warning, new definition for init

Graded switch/swap of tensor factors:

> &t(e1,e2);
gswitch(%,1);

> &t(Id,Id);
gswitch(%,1);

> &t(e1,e1we2);
gswitch(%,1);

The graded switch and the antipode of a Graßmann Hopf algebra are related as follows:

> for i from 1 to 4 do
dim_V:=i:bas:=cbasis(dim_V):
X:=add(_X[i]*bas[i],i=1..2^dim_V):
Y:=add(_Y[i]*bas[i],i=1..2^dim_V):
p1:=&t(X,Y);
gsw:=&map(gswitch(p1,1),1,wedge);
agsw:=gantipode(&map(gswitch(gantipode(gantipode(p1,1),2),1),1,wedge),1);
printf("gsw=%a\n",gsw);
printf("agsw=%a\n",agsw);
printf("#####\nIn dimension %d the equation `gsw=agsw` is `%a`\n#####\n",dim_V,evalb(0=simplify(gsw-agsw)));
od:

gsw=_X[1]*_Y[1]*`&t`(Id)+_X[1]*_Y[2]*`&t`(e1)+_X[2]*_Y[1]*`&t`(e1)

agsw=_X[1]*_Y[1]*`&t`(Id)+_X[1]*_Y[2]*`&t`(e1)+_X[2]*_Y[1]*`&t`(e1)

#####

In dimension 1 the equation `gsw=agsw` is `true`

#####

gsw=_X[1]*_Y[1]*`&t`(Id)+_X[1]*_Y[2]*`&t`(e1)+_X[1]*_Y[3]*`&t`(e2)+_X[1]*_Y[4]*`&t`(e1we2)+_X[2]*_Y[1]*`&t`(e1)+_X[2]*_Y[3]*`&t`(e1we2)+_X[3]*_Y[1]*`&t`(e2)-_X[3]*_Y[2]*`&t`(e1we2)+_X[4]*_Y[1]*`&t`(e1we2)

agsw=_X[1]*_Y[1]*`&t`(Id)+_X[1]*_Y[2]*`&t`(e1)+_X[1]*_Y[3]*`&t`(e2)+_X[1]*_Y[4]*`&t`(e1we2)+_X[2]*_Y[1]*`&t`(e1)+_X[2]*_Y[3]*`&t`(e1we2)+_X[3]*_Y[1]*`&t`(e2)-_X[3]*_Y[2]*`&t`(e1we2)+_X[4]*_Y[1]*`&t`(e1we2)

#####

In dimension 2 the equation `gsw=agsw` is `true`

#####

gsw=_X[4]*_Y[1]*`&t`(e3)+_X[7]*_Y[2]*`&t`(e1we2we3)+_X[8]*_Y[1]*`&t`(e1we2we3)+_X[1]*_Y[2]*`&t`(e1)+_X[1]*_Y[1]*`&t`(Id)+_X[1]*_Y[3]*`&t`(e2)-_X[3]*_Y[2]*`&t`(e1we2)-_X[3]*_Y[6]*`&t`(e1we2we3)-_X[6]*_Y[3]*`&t`(e1we2we3)+_X[3]*_Y[1]*`&t`(e2)+_X[2]*_Y[1]*`&t`(e1)+_X[6]*_Y[1]*`&t`(e1we3)+_X[5]*_Y[4]*`&t`(e1we2we3)+_X[1]*_Y[8]*`&t`(e1we2we3)+_X[2]*_Y[7]*`&t`(e1we2we3)+_X[7]*_Y[1]*`&t`(e2we3)+_X[1]*_Y[5]*`&t`(e1we2)+_X[4]*_Y[5]*`&t`(e1we2we3)+_X[1]*_Y[6]*`&t`(e1we3)+_X[1]*_Y[7]*`&t`(e2we3)+_X[1]*_Y[4]*`&t`(e3)+_X[5]*_Y[1]*`&t`(e1we2)-_X[4]*_Y[2]*`&t`(e1we3)-_X[4]*_Y[3]*`&t`(e2we3)+_X[2]*_Y[3]*`&t`(e1we2)+_X[2]*_Y[4]*`&t`(e1we3)+_X[3]*_Y[4]*`&t`(e2we3)

agsw=_X[4]*_Y[1]*`&t`(e3)+_X[7]*_Y[2]*`&t`(e1we2we3)+_X[8]*_Y[1]*`&t`(e1we2we3)+_X[1]*_Y[2]*`&t`(e1)+_X[1]*_Y[1]*`&t`(Id)+_X[1]*_Y[3]*`&t`(e2)-_X[3]*_Y[2]*`&t`(e1we2)-_X[3]*_Y[6]*`&t`(e1we2we3)-_X[6]*_Y[3]*`&t`(e1we2we3)+_X[3]*_Y[1]*`&t`(e2)+_X[2]*_Y[1]*`&t`(e1)+_X[6]*_Y[1]*`&t`(e1we3)+_X[5]*_Y[4]*`&t`(e1we2we3)+_X[1]*_Y[8]*`&t`(e1we2we3)+_X[2]*_Y[7]*`&t`(e1we2we3)+_X[7]*_Y[1]*`&t`(e2we3)+_X[1]*_Y[5]*`&t`(e1we2)+_X[4]*_Y[5]*`&t`(e1we2we3)+_X[1]*_Y[6]*`&t`(e1we3)+_X[1]*_Y[7]*`&t`(e2we3)+_X[1]*_Y[4]*`&t`(e3)+_X[5]*_Y[1]*`&t`(e1we2)-_X[4]*_Y[2]*`&t`(e1we3)-_X[4]*_Y[3]*`&t`(e2we3)+_X[2]*_Y[3]*`&t`(e1we2)+_X[2]*_Y[4]*`&t`(e1we3)+_X[3]*_Y[4]*`&t`(e2we3)

#####

In dimension 3 the equation `gsw=agsw` is `true`

#####

gsw=_X[4]*_Y[1]*`&t`(e3)+_X[1]*_Y[2]*`&t`(e1)+_X[1]*_Y[1]*`&t`(Id)+_X[1]*_Y[3]*`&t`(e2)-_X[3]*_Y[2]*`&t`(e1we2)+_X[3]*_Y[1]*`&t`(e2)+_X[2]*_Y[1]*`&t`(e1)+_X[1]*_Y[4]*`&t`(e3)-_X[4]*_Y[2]*`&t`(e1we3)-_X[4]*_Y[3]*`&t`(e2we3)+_X[2]*_Y[3]*`&t`(e1we2)+_X[2]*_Y[4]*`&t`(e1we3)+_X[3]*_Y[4]*`&t`(e2we3)+_X[2]*_Y[10]*`&t`(e1we2we4)+_X[9]*_Y[8]*`&t`(e1we2we3we4)+_X[1]*_Y[14]*`&t`(e1we3we4)+_X[6]*_Y[4]*`&t`(e1we2we3)+_X[9]*_Y[1]*`&t`(e2we3)+_X[6]*_Y[1]*`&t`(e1we2)+_X[5]*_Y[1]*`&t`(e4)-_X[5]*_Y[2]*`&t`(e1we4)+_X[15]*_Y[1]*`&t`(e2we3we4)-_X[4]*_Y[8]*`&t`(e1we3we4)+_X[1]*_Y[7]*`&t`(e1we3)+_X[2]*_Y[9]*`&t`(e1we2we3)+_X[4]*_Y[6]*`&t`(e1we2we3)+_X[10]*_Y[1]*`&t`(e2we4)+_X[5]*_Y[9]*`&t`(e2we3we4)+_X[1]*_Y[12]*`&t`(e1we2we3)+_X[6]*_Y[11]*`&t`(e1we2we3we4)+_X[1]*_Y[8]*`&t`(e1we4)+_X[4]*_Y[5]*`&t`(e3we4)+_X[1]*_Y[5]*`&t`(e4)+_X[14]*_Y[1]*`&t`(e1we3we4)+_X[1]*_Y[16]*`&t`(e1we2we3we4)+_X[5]*_Y[6]*`&t`(e1we2we4)+_X[1]*_Y[6]*`&t`(e1we2)-_X[7]*_Y[10]*`&t`(e1we2we3we4)+_X[9]*_Y[2]*`&t`(e1we2we3)-_X[13]*_Y[4]*`&t`(e1we2we3we4)-_X[8]*_Y[3]*`&t`(e1we2we4)+_X[13]*_Y[1]*`&t`(e1we2we4)+_X[11]*_Y[3]*`&t`(e2we3we4)+_X[2]*_Y[5]*`&t`(e1we4)+_X[3]*_Y[5]*`&t`(e2we4)-_X[3]*_Y[7]*`&t`(e1we2we3)-_X[10]*_Y[4]*`&t`(e2we3we4)-_X[15]*_Y[2]*`&t`(e1we2we3we4)-_X[10]*_Y[7]*`&t`(e1we2we3we4)+_X[8]*_Y[1]*`&t`(e1we4)-_X[4]*_Y[10]*`&t`(e2we3we4)-_X[5]*_Y[4]*`&t`(e3we4)+_X[11]*_Y[1]*`&t`(e3we4)+_X[2]*_Y[11]*`&t`(e1we3we4)+_X[9]*_Y[5]*`&t`(e2we3we4)+_X[3]*_Y[11]*`&t`(e2we3we4)+_X[2]*_Y[15]*`&t`(e1we2we3we4)+_X[11]*_Y[2]*`&t`(e1we3we4)+_X[16]*_Y[1]*`&t`(e1we2we3we4)+_X[7]*_Y[5]*`&t`(e1we3we4)+_X[12]*_Y[1]*`&t`(e1we2we3)+_X[1]*_Y[15]*`&t`(e2we3we4)-_X[3]*_Y[8]*`&t`(e1we2we4)-_X[8]*_Y[4]*`&t`(e1we3we4)+_X[6]*_Y[5]*`&t`(e1we2we4)-_X[5]*_Y[3]*`&t`(e2we4)+_X[1]*_Y[10]*`&t`(e2we4)-_X[3]*_Y[14]*`&t`(e1we2we3we4)+_X[10]*_Y[2]*`&t`(e1we2we4)+_X[14]*_Y[3]*`&t`(e1we2we3we4)+_X[4]*_Y[13]*`&t`(e1we2we3we4)+_X[1]*_Y[11]*`&t`(e3we4)+_X[1]*_Y[9]*`&t`(e2we3)-_X[5]*_Y[12]*`&t`(e1we2we3we4)+_X[7]*_Y[1]*`&t`(e1we3)+_X[1]*_Y[13]*`&t`(e1we2we4)-_X[7]*_Y[3]*`&t`(e1we2we3)+_X[11]*_Y[6]*`&t`(e1we2we3we4)+_X[8]*_Y[9]*`&t`(e1we2we3we4)+_X[5]*_Y[7]*`&t`(e1we3we4)+_X[12]*_Y[5]*`&t`(e1we2we3we4)

agsw=_X[4]*_Y[1]*`&t`(e3)+_X[1]*_Y[2]*`&t`(e1)+_X[1]*_Y[1]*`&t`(Id)+_X[1]*_Y[3]*`&t`(e2)-_X[3]*_Y[2]*`&t`(e1we2)+_X[3]*_Y[1]*`&t`(e2)+_X[2]*_Y[1]*`&t`(e1)+_X[1]*_Y[4]*`&t`(e3)-_X[4]*_Y[2]*`&t`(e1we3)-_X[4]*_Y[3]*`&t`(e2we3)+_X[2]*_Y[3]*`&t`(e1we2)+_X[2]*_Y[4]*`&t`(e1we3)+_X[3]*_Y[4]*`&t`(e2we3)+_X[2]*_Y[10]*`&t`(e1we2we4)+_X[9]*_Y[8]*`&t`(e1we2we3we4)+_X[1]*_Y[14]*`&t`(e1we3we4)+_X[6]*_Y[4]*`&t`(e1we2we3)+_X[9]*_Y[1]*`&t`(e2we3)+_X[6]*_Y[1]*`&t`(e1we2)+_X[5]*_Y[1]*`&t`(e4)-_X[5]*_Y[2]*`&t`(e1we4)+_X[15]*_Y[1]*`&t`(e2we3we4)-_X[4]*_Y[8]*`&t`(e1we3we4)+_X[1]*_Y[7]*`&t`(e1we3)+_X[2]*_Y[9]*`&t`(e1we2we3)+_X[4]*_Y[6]*`&t`(e1we2we3)+_X[10]*_Y[1]*`&t`(e2we4)+_X[5]*_Y[9]*`&t`(e2we3we4)+_X[1]*_Y[12]*`&t`(e1we2we3)+_X[6]*_Y[11]*`&t`(e1we2we3we4)+_X[1]*_Y[8]*`&t`(e1we4)+_X[4]*_Y[5]*`&t`(e3we4)+_X[1]*_Y[5]*`&t`(e4)+_X[14]*_Y[1]*`&t`(e1we3we4)+_X[1]*_Y[16]*`&t`(e1we2we3we4)+_X[5]*_Y[6]*`&t`(e1we2we4)+_X[1]*_Y[6]*`&t`(e1we2)-_X[7]*_Y[10]*`&t`(e1we2we3we4)+_X[9]*_Y[2]*`&t`(e1we2we3)-_X[13]*_Y[4]*`&t`(e1we2we3we4)-_X[8]*_Y[3]*`&t`(e1we2we4)+_X[13]*_Y[1]*`&t`(e1we2we4)+_X[11]*_Y[3]*`&t`(e2we3we4)+_X[2]*_Y[5]*`&t`(e1we4)+_X[3]*_Y[5]*`&t`(e2we4)-_X[3]*_Y[7]*`&t`(e1we2we3)-_X[10]*_Y[4]*`&t`(e2we3we4)-_X[15]*_Y[2]*`&t`(e1we2we3we4)-_X[10]*_Y[7]*`&t`(e1we2we3we4)+_X[8]*_Y[1]*`&t`(e1we4)-_X[4]*_Y[10]*`&t`(e2we3we4)-_X[5]*_Y[4]*`&t`(e3we4)+_X[11]*_Y[1]*`&t`(e3we4)+_X[2]*_Y[11]*`&t`(e1we3we4)+_X[9]*_Y[5]*`&t`(e2we3we4)+_X[3]*_Y[11]*`&t`(e2we3we4)+_X[2]*_Y[15]*`&t`(e1we2we3we4)+_X[11]*_Y[2]*`&t`(e1we3we4)+_X[16]*_Y[1]*`&t`(e1we2we3we4)+_X[7]*_Y[5]*`&t`(e1we3we4)+_X[12]*_Y[1]*`&t`(e1we2we3)+_X[1]*_Y[15]*`&t`(e2we3we4)-_X[3]*_Y[8]*`&t`(e1we2we4)-_X[8]*_Y[4]*`&t`(e1we3we4)+_X[6]*_Y[5]*`&t`(e1we2we4)-_X[5]*_Y[3]*`&t`(e2we4)+_X[1]*_Y[10]*`&t`(e2we4)-_X[3]*_Y[14]*`&t`(e1we2we3we4)+_X[10]*_Y[2]*`&t`(e1we2we4)+_X[14]*_Y[3]*`&t`(e1we2we3we4)+_X[4]*_Y[13]*`&t`(e1we2we3we4)+_X[1]*_Y[11]*`&t`(e3we4)+_X[1]*_Y[9]*`&t`(e2we3)-_X[5]*_Y[12]*`&t`(e1we2we3we4)+_X[7]*_Y[1]*`&t`(e1we3)+_X[1]*_Y[13]*`&t`(e1we2we4)-_X[7]*_Y[3]*`&t`(e1we2we3)+_X[11]*_Y[6]*`&t`(e1we2we3we4)+_X[8]*_Y[9]*`&t`(e1we2we3we4)+_X[5]*_Y[7]*`&t`(e1we3we4)+_X[12]*_Y[5]*`&t`(e1we2we3we4)

#####

In dimension 4 the equation `gsw=agsw` is `true`

#####

However, be aware that the reversion is the Clifford reversion which introduces possibly additional terms! If one thinks about a Graßmann reversion one would have to set B to be a diagonal bilinear form (their values do not matter and could even be zero), since only in this setting one has identities like e1 &c e2 = e1 &w e2 etc. i.e. the Clifford and Graßmann bases coincide (since only off diagonal B[i,j] terms occure in reorderings).

> reversion(e2we1);
subs(B[2,1]=-F[1,2],B[1,2]=F[1,2],%); ## antisymmetric part
reversion(reversion(e1we2)); ## reversion is involutive !!

> B:=linalg[diag](1$2);
reversion(e2we1); ## works out (as expected?) in this case

The graded switch is involutive:

> &t(e1,e2);
gswitch(%,1);
gswitch(%,1);

> &t(e1,a*e2+b*e2we3,e1we4-sin(x)*e5);
gswitch(%,1);
gswitch(%%,2);

If the index is not in the range of the tensor slots, an error occurs, so the user has to account for that.

> gswitch(&t(e1,e2),3);

Error, (in gswitch) invalid subscript selector

>

See Also:

Bigebra[`&t`] , Bigebra[switch] , Bigebra[`&gco`]