Thank You for the new version. I looked into it
and this time found it almost unreadable and not 
well formulated. To be specific,

1) For example, it is written that the indices 
   {i_1,...,i_k;j_1,...,j_l} are from {1,...,k+l}. 

   How this could be, if each index i_? and j_? 
   have its value in {1,...,n} (with n dimension 
   of V)?

2) I started to read+understand the text from the
   middle page 4 up to page 7.

   It is not explained, why there appears some 
   four cases or possibilities. What they are, 
   where they come from?

   For example, page 5, Cases 3 and 4. The second
   coefficients A' with (j_r first, then after ";"
   i_l) have the value

  - (-q)^{(r-1)(l-1)+r+t(s-l)}  for i_l<j_r

  - (-q)^{(r-1)l+t(s-l)}        for j_r<i_l

   but I can write the first value as

(-q)^{(r-1)(l-1)+r+t(s-l)}=
(-q)^{(r-1)(l-1)+(r-1)+1+t(s-l)}=
(-q)^{(r-1)(l)+1+t(s-l)}   for i_l<j_r

 and this differs by (-q) from the second term
for j_r<i_l. Of course, the difference is (-q),
coming just from the switch of two indices i_l
and j_r.

The whole description (of all possibilities) is
not appropriate, redundant and can be shortened
to just one (maybe two, I do not know) elementary
cases from which the rest has to follow. 

There should be an elementary observation and then 
there should be a "uniform" elementary description 
(formula) expressing the values of A (depending on
the order of indices.)

As it looks, it is mess in which nobody will be able
to logically orient in its structure.  

3) I looked into Proposition 3.8. : there are again 
   some cases, dependent on the fact whether or not
   "i" belongs to the set {1,...,k+l}. However, all
   i's are indices from the set {1,...,n} (dim V = n)
   and there are k+l of them, if You wish. This is 
   confusing.

 I hope Proposition 3.7. could be fine, but "indexes"
 like "x" or "y" have to be replaced as in the first 
 part of the proof.

 Best,

Petr Somberg  

================================================

I looked on the writing. 

1/ Lemma 3.4. is (probably correct), but is not
   sufficient to conclude the required result. In
   particular, Remark 3.5. claims that the result
   can be proved for general (k,l) (and not just
   those with one of the arguments being 1 or 2),
   and so it HAS TO be proved in this generality.

   In principle, Definition 3.1. is a non-logical
   statement: is it a definition of what, of A's?
   The coproduct is defined to be compatible with
   product, and this produces ad hoc some recursion 
   relations for A's (take the product of forms, 
   and apply the coproduct - one gets recursion 
   relation for A's.)

   Then (for example) the co-associativity relations
   have to be the consequences of the initial defining
   relations. In fact, the coassociativity conditions
   as in Lemma 3.6. are quite special, because special
   component of the coproduct was chosen. In general,
   the relations are quadratic (because coassociativity
   relations are), and one has to prove if they are 
   compatible (i.e., follow from) with the initial 
   defining relations for A's.

2/ Theorem 3.7. is unclear - is the condition 1. on
   A's true or not? The properties of A's have to be 
   deduced from the compatibility among structural 
   relations for operations in graded Hopf algebra,
   and equivariance will follow from them. The equivalence
   in Theorem 3.7. is useless as stated.

 My preference is to "build" recursive relation for 
A's out of the compatibility of the product and coproduct
(I do not see a computation or consideration of this 
defining property), and then to use them to prove 
the remaining structural compatibilities of the 
coproduct with other operations. This is just the way 
how the coefficients A's are recursively introduced! 
One can not write a "Definition" 3.1. without knowing 
that there is a family of A's which is not contradictory. 
For example, the relations 1. in Theorem 3.7. are either 
true, or are not true - and this follows from the structural 
properties of the Hopf algebra, and then they can be
used to prove the invariance. 

Best,

Petr Somberg