# Examples of basic coalgebras

For an algebraically closed field $$k$$, let $$C$$ be a $$k$$-coalgebra. Given a minimal injective cogenerator $$E$$, there is a so-called basic coalgebra $$B_C=coend^C(E)$$, s.t. the comodule categories $$Mod^C$$ and $$Mod^{B_C}$$ are equivalent. I would like to understand this object $$B_C$$ better, but I didn't find any elaborated examples.

I am particularly interested in the case when $$C$$ is a coquasitriangular Hopf algebra. In this case, is $$B_C$$ still a (coquasitriangular) Hopf algebra, so that we have a (braided) monoidal equivalence of categories?

A good starting point for me would be the polynomial ring $$k[x_{ij}]$$ with coproduct $$\Delta(x_{ij})=\sum_{k=0}^n\,x_{ik}\otimes x_{kj},$$ but I am happy with all kinds of examples/references.

In general this coend is not a Hopf algebra. To convince yourself, think of a finite dimensional example, let $$C=H^*$$, so that you look for a minimal projective generator $$H$$-module $$P$$ and then look at the basic algebra $$End_H(P)$$. Take $$H=k[G]$$ with $$G$$ finite and k algebraically closed and ch(k)=0. Then by Weddebunn $$k[G]$$ is isomorphic to a product of (say r copies of) Matrix algebras (of eventually different sizes). $$k[G]=M_{n-1}\times\cdots M_{n_r}$$. The basic algebra associated is $$k\times \cdots\times k$$ r-times. (by the way, $$r$$=number of conjugacy classes of $$G$$). This is a commutative algebra, if it is Hopf, it must be of the form $$k^{\tilde G}$$ for some group $$\tilde G$$. There is no obvious candidate of natural group of order $$r$$ associated to $$G$$. Isn't it?
About your example $$k[x_{ij}]$$, this is not a Hopf algebra, you should localize by the determinant. But assume you do that. Then $$O(GL(n(k))=k[x_{ij}][det^{-1}]$$ is a reductive group, you can use "Peter-Weyl" to write $$H=O(GL(n,k))=\oplus_{\mu}V_\mu\otimes V_{\mu}^*$$, it is an infinite sum of comatrix coalgebras. The basic coalgebra associated to it is an infinite direct sum of copies of $$k$$, one for each $$\mu$$. It is not a very explicit construction.. it is more or less equivalent to know the (co)representation theory of $$H$$.