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
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In my lecture notes it says that there are cocommutative, non-unimodular Hopf algebras. What would be an example?
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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
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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} $).
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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 $$I_l=I_r=\mathbb k \cdot \sum\limits_{g\in G} g$$ What about infinite groups?
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Regarding the universal enveloping algebra, tensor algebra, symmetric algebra, alternating algebra I am not sure. What can be said here?
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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. Here, the modular element $a$ is the unique linear form such that $t\cdot h = t a(h)$ for all $h\in H, t\in I_l(H)$. It exists because $t\cdot h \in I_l(H)$ and $I_l(H)$ is one dimensional. It can be shown to be a morphism of algebras, hence a group-like element in $H^*$.
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