4
$\begingroup$

If $V$ is a faithful finite dimensional representation of an affine group scheme $G$ over a field $k$, then every finite dimensional representation of $G$ is isomorphic to a subquotient of $\otimes^n (V \oplus V^*)$, where $V^*$ is the dual representation (see for instance Milne, Basic theory of affine group schemes, VIII, Theorem 11.7).

The question is whether there is an analogue of this statement for possibly non-commutative Hopf algebras instead of affine group schemes and comodules instead of representations. The first question is what a is a good analogue of the property of a representation to be faithful. In the case of commutative Hopf algebras a condition is given in VIII, Lemma 11.6 Milnes notes. I am not sure if this is suitable in general. Please excuse this question being rather vague.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

You might look at this paper of Rieffel http://www.sciencedirect.com/science/article/pii/002186936790018X

It has the analogue for Hopf algebras.

Improve this answer
$\endgroup$
3
  • $\begingroup$ Thank you for this reference! I guess you refer to Corollary 1 in this paper. It is restricted to Hopf algebras of finite dimension. Given a (f.d.) Hopf algebra $H$, it deals with "representations of $H$", i.e. algebra homomorphisms from $H$ into the algebra of endomorphisms of a vector space $V$. I was however rather thinking about $H$-comodules. In the finite-dimensional case one can pass to the dual of a Hopf algebra and I guess the results of this paper will provide an analogue in the spirit I was thinking of. But is there anything known about $H$-comodules for $H$ not of finite dimension? $\endgroup$
    – skew41
    Commented Sep 20, 2014 at 19:02
  • 1
    $\begingroup$ My point is that the analogue of faithful is not factoring through a quotient Hopf algebra and Corollary 1 implies any finite dimensional irreducible rep is a constituent in a tensor power. For an infinite dimensional irrep it will be a subquotient of the sum of all tensor powers. $\endgroup$
    – Benjamin Steinberg
    Commented Sep 20, 2014 at 21:14
  • 1
    $\begingroup$ You might look at math.wisc.edu/~passman/burnside.pdf too. $\endgroup$
    – Benjamin Steinberg
    Commented Sep 20, 2014 at 21:16

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .