gantipode.html

Function: Bigebra[gantipode] - the antipode map for Graßmann Hopf gebra

Calling Sequence:

t1 := gantipode(t1,i)
c1 := gantipode(c1)

Parameters:

t2 : a tensor polynom

c2 : a Clifford polynom

Output:

t1 : a tensor polynom

c1 : a Clifford polynom

Description:

The Graßmann antipode is the convolutive inverse of the identity map. It fulfills the antipode axioms (e.g. Sweedler)

S(x_(1)) x_(2) = eta\circ\epsilon(x) = x_(1) S(x_(2)).

The Graßmann antipode is closely related to the grade involution of the Graßmann algebra, see examples below. This involution constitues a Z_2 grading which is also present in Clifford algebras.

The Graßmann antipode can be obtained by direct computation (e.g. using tsolev1 ) from the unital Graßmann bi-convolution.

Examples:

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

Warning, new definition for drop_t

Warning, new definition for gco_d_monom

Warning, new definition for gco_monom

Warning, new definition for init

On a Graßmann basis we compute the antipode map, which shows it to be equivalent to the grade involution, this is not an accident but can be proved:

> #_CLIENV[_QDEF_PREFACTOR]:=-q;
dim_V:=4:bas:=cbasis(dim_V);
Sbas:=map(gantipode,bas);
Cbas:=map(gradeinv,bas);
printf("It is `%a` that the two lists Sbas and Cbas containe the same elements",evalb({0}={op(zip((i,j)->i-j,Sbas,Cbas))}));

It is `true` that the two lists Sbas and Cbas containe the same elements

Now we give some examples where the Graßmann antipode acts on slots of tensor products:

> gantipode(&t(e1,e2we3,e4),2);
gantipode(&t(e1,e2we3,e4),3);

On special elements we find:

> gantipode(0),gantipode(&t(e1,0,e3),2);

On inhomogenous elements we find:

> gantipode(&t(a*Id-b*e1-e1we2,d*e2we3we4,Id+e2we3-4*sin(x)*e1we2),1);

>

See Also: Bigebra[`&t`] , Bigebra[`type/tensorpolynom`] , Bigebra help page