# Example of a cocommutative, non-unimodular Hopf algebra?

Mathematics Asked by M.C. on August 30, 2020

1. Definitions: Unimodularity and cocommutativity
Let $$H$$ be a Hopf algebra over a field $$mathbb k$$.

• We call $$H$$ unimodular if the space of left integrals $$I_l(H)$$ is equal to the space of right integrals $$I_r(H)$$.
• We call $$H$$ cocommutative if $$tau_{H,H} circ Delta = Delta$$. Here, $$Delta$$ denotes the coproduct of $$H$$, while $$tau: H otimes H rightarrow H otimes H; v otimes w mapsto w otimes v$$ is the twist map.

2. Question

• In my lecture notes it says that there are cocommutative, non-unimodular Hopf algebras. What would be an example?

• Apparently, an example is given in Hopf algebras and their action on rings by Susan Montgomery. However, due to the pandemic I am unable to get it from the library. If you have a copy and could write down the relevant section, that would be very much appreciated.

3. My ideas so far

• The Taft-Hopf algebra $$H$$ over a field $$mathbb k$$ is not an example:
If $$H$$ is commutative (i.e. root of unity $$zeta =1_{mathbb k}$$), then $$H$$ is unimodular. In this case, it is even isomorphic to the boring group algebra of the zero group. Otherwise, $$H$$ is not cocommutative (even though it is non-unimodular then). Non-cocommutativity follows easily from the observation that the square of the antipode is not the identity (if $$zeta neq 1_{mathbb k}$$).

• Group algebras:
As the coproduct of a group algebra is given by the diagonal map any group algebra is cocommutative. However, any group algebra $$mathbb k[G]$$ over a finite group $$G$$ is unimodular, since
$$I_l=I_r=mathbb k cdot sumlimits_{gin G} g$$ What about infinite groups?

• Regarding the universal enveloping algebra, tensor algebra, symmetric algebra, alternating algebra I am not sure. What can be said here?

• Maybe the following proposition turns out to be useful: A finite dimensional Hopf algebra $$H$$ is unimodular iff its distinguished group-like element/modular element $$a in G(H^*)$$ is equal to the counit $$epsilon_H$$. Here, the modular element $$a$$ is the unique linear form such that $$tcdot h = t a(h)$$ for all $$hin H, tin I_l(H)$$. It exists because $$tcdot h in I_l(H)$$ and $$I_l(H)$$ is one dimensional. It can be shown to be a morphism of algebras, hence a group-like element in $$H^*$$.