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
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.
There are plenty of trivial examples when the category is allowed to be symmetric (which presumably you don't want), for example any ordinary commutative algebra is an algebra object in the symmetric (and hence braided) tensor category Vec.
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.
As an example of a commutative algebra object which can also be turned into a Frobenius algebra, consider the algebra of polynomials $\mathbb{C}[X]$ in one indeterminate $X$ over the field $\mathbb{C}$, divided by the ideal $\langle X^d \rangle$, i.e. $A=\mathbb{C}[X]/\langle X^d \rangle$. This is a commutative algebra object in the category of finite dimensional complex vector spaces $\mathbf{Vect}_{\mathbb{C}}$. The reason one divides by the ideal $\langle X^d \rangle$ is to make the algebra object (viewed as a vector space over $\mathbb{C}$) finite dimensional, in order to be able to turn it into a Frobenius algebra. Thus in the case at hand $\dim A=d$. The tensor product bifunctor $\otimes_{\mathbb{C}} \colon \mathbf{Vect}_{\mathbb{C}} \times \mathbf{Vect}_{\mathbb{C}} \to \mathbf{Vect}_{\mathbb{C}}$ is given by $\cdot \colon A \times A \to A$. More explicitly the multiplication is $$ \left( \sum_{i=0}^{d-1} a_i X^i \right)\cdot \left( \sum_{j=0}^{d-1} b_j X^j \right) = \sum_{k=0}^{d-1}\left( \sum_{i+j=k} a_ib_j \right)X^k ~, $$ where $a_i,b_j\in \mathbb{C}$ and $X^i\in A$. The monoidal unit is given by the underlying field $\mathbb{C}$.
