# Characters on Hopf algebras

For any algebra $$A$$, a character for $$A$$ is a non-zero algebra map $$c:A \to \mathbb{C}$$. For $$H$$ be a Hopf algebra, a character is given by $$\epsilon:H \to \mathbb{C}$$ the counit of $$H$$. I am looking for other examples of (co-semi-simple) Hopf algebras with characters distinct of the counit.

I am really interested in noncommutative, noncocommutative examples.

• Would you consider Larson's character? Jan 8, 2020 at 21:51
• What is Larson's character? Jan 8, 2020 at 22:35

I think that a general example is the so-called Larson's character, which in a sense ties together the trace and determinant functions.

To make the long story short: Let $$C$$ be a cocommutative bialgebra, $$V$$ a vector space and $$EV$$, the exterior algebra. Then, it has been shown that:

If $$C\otimes V\rightarrow V$$ is an action which makes $$V$$ a $$C$$-module, then there is a unique measuring $$C\otimes EV\rightarrow EV$$, extending the action on $$V$$.

In this sense, $$EV$$ becomes a $$C$$-module, with $$C\cdot E^kV\subset E^kV$$. If we furthermore assume that $$\dim V=n$$ then $$E^nV$$ is 1-dim. Let it be spanned by $$\{z\}$$. For any $$c\in C$$, let $$\chi(c)$$ be defined by $$c\cdot z=\chi(c)z$$. In this way, a linear map $$\chi:C\rightarrow k$$ is defined. It can be easily shown that this is an algebra map. It is called the Larson's character.
It can furthermore be shown that, if $$g$$ is a grouplike element of $$C$$ then $$\chi(g)=\det T_g$$, where $$T_g:V\rightarrow V$$ is explicitly given by $$v\mapsto g\cdot v$$; and that if $$g$$ is a primitive element then $$\chi(g)=Trace(T_g)$$.

For a detailed presentation of the above, you can see ch. VII, sect. 7.1, p.146-153, from Sweedler's book on Hopf algebras.

Furthermore, you can also take a look at Larson's paper on Characters of Hopf algebras. However, the presentation there looks quite different:
Larson adopts a dual point of view (to the usual notion of characters in group/algebra representation theory) and develops a theory of characters based on comodules of Hopf algebras. He actually considers characters as elements of the hopf algebra (instead of functionals on it) which are associated with comodules over the hopf algebra rather than modules over the hopf agebra. Furthermore, for the case of cosemisimple hopf algebras, an orthogonality relation for characters is proved.
Edit: Although i have not studied Larson's paper in detail, from what i can understand, i think that his approach is more general than Sweedler's approach (in the sense that it is not limited to the cocommutative case). In the cocommutative case, i think is essentially equivalent to the one followed in Sweedler's book; Sweedler's presentation can be recovered if we adopt Larson's approach and start from comodules of the finite dual $$C^{\circ}$$ hopf algebra.

• What does 'measuring' mean in "there is a unique measuring $E V \otimes V \to E V$"? Jan 10, 2020 at 4:44
• @LSpice, if $A$ is an algebra and $H$ is a bialgebra, and we have a bilinear map (not necessarily a $H$-action), $\triangleright: H \times A \to A$ satisfying $h \triangleright(ac) = (h_{1} \triangleright a)(h_{2}\triangleright c)$ and $h\triangleright 1_A=\varepsilon(h)1_A$, then we say that the bilinear map $\triangleright$ is a measuring or that $(\triangleright,H)$ measures $A$ to $A$. Jan 10, 2020 at 21:10
• [Sorry for what may be an ignorant question but...] Is there any relation between Larson's character and the Fredholm determinant? I know that the latter can be a trace and determinant connection (e.g. on MSE); but, I don't know if there is anything more to say in this direction. Jan 10, 2020 at 23:17
• @Benjamin Dickman, to speak the truth, i do not know. In fact i am not very familiar with Fredholm determinant. However, your remark seems very interesting to me. I will try to study a little and to think about it. Meanwhile, maybe it would be interesting to post this as a question (either here or on MSE). Jan 10, 2020 at 23:30
• I don't think I have the necessary background to parse an answer around their connection; so, I am not intending to post such a question. But please ping me if you ask any such thing on either site - now or in the future! Jan 11, 2020 at 4:17

Take an example of a finite dimensional Hopf algebra $$A$$, presented by generators and relations, generated by grouplikes and primitives. There are a lot of non-commutative non-cocommutative examples in the literature. Compute the group $$G(A)$$ of group like elements (from the presentation this should be very easy). Now the example is $$H=A^*$$, the group likes in $$A=A^{**}$$ are the characters of $$H$$.

• Trying to "rephrase" your description: if $A$ is a finite dimensional $k$-algebra and $A^*$ its dual $k$-coalgebra, then $G(A^*)=\mathcal{A}lg(A,k)$, i.e. the grouplikes of the dual coalgebra are exactly the algebra maps from $A$ to $k$, that is the characters of $A$. Jan 10, 2020 at 3:17
• And in the case that $A$ is a fin dim cocommutative bialgebra then Larson's character (mentioned in my post) is one of them. Jan 11, 2020 at 3:24