# Example of a commutative algebra object in a braded monoidal category C

Hi,

I am looking for an example of a commutative algebra object in a braided monoidal category C which it can also be turned into a commutative Frobenius algebra. If you have any examples could you also tell me what the multiplication and unit are?

Thank you

Dimtris

-
I have a feeling that your question is strongly related to this one: mathoverflow.net/questions/71077/… –  Scott Carter Oct 25 '11 at 12:44
Thanks but it didn't help much. I need a concrete example of a commutative algebra object. I have something in mind but I am not sure: The algebra of polynomials C[x], divided by the ideal <x^d>. If correct, what is its dimension, multiplication and unit in this case? –  user18768 Oct 25 '11 at 13:12

The standard example here is where the braided tensor category is the Drinfeld center Z(C) and the algebra object is the induction of the trivial object from C to Z(C). If C is semsimple over an algebraically closed field then this can be written explicitly as $\sum_x x \otimes x^*$ with half braiding given by Theorem 2.3 of Kirillov-Balsam.
The group ring of any group $G$ yields a special case of Noah's answer, where $C$ is the monoidal category of $G$-graded vector spaces. I wrote this up in a blog post a few years ago.