Re: [Mupad-combinat-users] combinat: *-algebras

[Nicolas M. T. <Nic...@u-...>]

	Dear Vladimir,

> after a while I address you with a question about mupad once 
> again. 

You are welcome!

> Are there by chance ready routines that allow to work with
> *-algebras (at least compute something in free *-algebras) - that
> is, associative algebras with an involution * - so in addition to
> associativity, we have the reduction rule (ab)^*=b^* a^* for all
> a,b. (A free *-algebra with n generators as an associative algebra
> is the free algebra with 2n generators a_i, a_i^*, and the obvious
> involution [which is uniquely extended from generators to the whole
> algebra].)

We have a few finite dimensional Kac algebra implemented, and some
generic functions (mostly in the context of involutive Hopf algebras)
that take advantage of the existence of the involution (isUnitary,
is2Cocycle, coboundary, is2PseudoCocycle). That's all. But its easy to
define your own involution. For example:


domain FreeAlgebraWithInvolution(R = Dom::ExpressionField())
    category Cat::AlgebraWithBasis(R);
    inherits Dom::FreeModule(combinat::words, R);

    oneBasis        := [];
    mult2Basis      := dom::term @ _concat;
    involutionBasis := word -> dom::term(revert(map(word, conjugate)));

end_domain:

>> F := FreeAlgebraWithInvolution()
>> f := F[a,b,c] + (2*x*y+1) * F[c,d] 

                     B([a, b, c]) + (2 x y + 1) B([c, d])

>> f::involution()
                        _ _         _  _        _  _  _
                     (2 x y + 1) B([d, c]) + B([c, b, a])

>> (f*f::involution())

                     _  _  _                 _  _  _
(2 x y + 1) B([c, d, c, b, a]) + B([a, b, c, c, b, a]) +

                  _ _               _  _        _ _                  _  _
   (2 x y + 1) (2 x y + 1) B([c, d, d, c]) + (2 x y + 1) B([a, b, c, d, c])

This assumes that the ground field knows how to conjugate its elements
appropriately; here notice that x,y are in the field of expressions
which is commutative.

Note also that you can use conjugate(f) instead of f::involution(). I
don't know if this is a desirable feature from a naming point of view;
what do you think?

Btw: your point of view on
http://mupad-combinat.sourceforge.net/Wiki/Sage is very much welcome!

Best regards,
				Nicolas
-- 
Nicolas M. Thiery "Isil" <nt...@us...>
http://Nicolas.Thiery.name/