# dimension of induced comodule

Let $\pi : G \to H$ be epimorphism of Hopf superalgebras, where $G$ be an quantum super group of function on $GL(m|n)$, $H$ be an quantum group of function on $GL(m) \otimes GL(n)$; $W$ an finite dimensional $H$-comodule. Using induce, A. Huffmann in "On Representations of super coalgebras" showed that $\dim Ind_H^G(W)= 2^{2mn} \dim W$. Could you explain to me this formular?

I give some remark of this problem as the following: Let $C(m,n)$ be the bialgebra generated by a supermatrix $((c_i^j))$ of dimension $(m+n)\times (m+n)$ with structure $\Delta(c_i^j)= \sum c_i^k \otimes c_k^j, \epsilon(c_i^j)= \sigma_i^j.$ Matrix $(c_i^j)$ can write $\matrix{A & B\\ C & D}$. To get a Hopf algebra $GL(m,n)$ with involution antipode $C(m,n)$ has to be localized at the monoid which is generated by $det(A)$ and $det(D)$. The structure of $GL(m,n)$ can be analysed with the help of induced modules with respect to the coalgebras $GL_0(m,n) \cong GL(m,0) \otimes GL(0,n)$. The matrix of generators of $GL(m,0)$ will be denoted by $A_0$;those by $GL(0,n)$ by $D_0$.The projection $\pi: GL(m,n)--> GL_0(m,n)$ is defined by $\pi(A)=A_0, \pi(B)=0, \pi(C)=0, \pi(D)=D_0$. Let $\beta: V ->V \otimes GL_0(m,n)$ be a $GL_0(m,n)$ comodule and ${e_i} \subset V$ a basis such that $\beta(e_i)= e_j \otimes \mathcal{D}_i^j(A_0,D_0)$ . It is easy to check that $x= \sum e_i \otimes x^i \in Ind_{GL_0(m,n)}^{GL(m,n)}(W)$ means $$(\pi \otimes id) \Delta(x_i) = \sum_j \mathcal{D}_j^i(A_0,D_0) \otimes x^j \;\; (1)$$ A. Huffmann said that this equation is straightforward to solve with the result $$x^i(A,B,C,D)= \sum_j \mathcal{D}_j^i(A,D) F^j(A^{-1} B, D^{-1} C) \;\;\;(2)$$ I have a trouble with this assert:

1) Which could be solved (1) straightforwardly?

2) In (2), what is $F^j(A^{-1}B,D^{-1}C)$?

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I think you need to say a bit more in order for people to help. Where exactly in the proof are you having trouble? – David White Sep 3 '11 at 12:41
Oh! thank you very much for helping me. – Nguyen Sep 3 '11 at 17:15
Thanks for editing, the question now seems much clearer. I don't have an answer, but I voted the question up. Hopefully someone else will be able to answer. – David White Sep 4 '11 at 23:01
Beware: if you keep editing this it'll become community wiki. I believe it takes 8 edits and then the change is automatic – David White Sep 6 '11 at 15:32