# Hopf algebra with a non-grouplike invertible element

What is an example of a Hopf algebra $$(H,\Delta,\epsilon)$$ containing an invertible element $$h$$ which is not grouplike: An element $$h \in H$$ such that $$\Delta(h) \neq h \otimes h\qquad\text{(not grouplike)}$$ and such that there exists a $$h^{-1} \in H$$ with $$hh^{-1} = h^{-1}h = 1$$ (invertible).

• Hae you looked at the group ring of a (finite) group? Feb 22, 2021 at 23:30
• Related (though clearly not the same question—this one seems much quicker to address, per @Tyrone's comment): mathoverflow.net/questions/86197/… . Feb 22, 2021 at 23:45
• Trivial example: scalar multiples of grouplikes are generally not grouplike. Feb 23, 2021 at 6:13
• @Tyrone: But in the group algebra every element is grouplike? Feb 23, 2021 at 13:23
• You seem to have just changed your question... Feb 23, 2021 at 13:34

Let $$L$$ be a finite-dimensional $$p$$-nilpotent restricted Lie algebra over a field of characteristic $$p>0$$ and consider its restricted enveloping algebra $$u(L)$$. Then the only group-like element of the Hopf algebra $$u(L)$$ is 1. On the other hand, every element of $$u(L)$$ that is not in the kernel of the counit is invertible.
Let $$G$$ be any finite group. Then the group algebra $$\mathbb{C}[G]$$ is, as an algebra, isomorphic to $$\bigoplus_V \mathrm{End}(V)$$, where the direct sum is over irreducible representations of $$V$$ of $$G$$. So the group of units of $$\mathbb{C}[G]$$ is isomorphic to $$\prod_V \mathrm{GL}(V)$$. Only finitely many elements of this product come from the original group $$G$$. Any element of $$\prod_V \mathrm{GL}(V)$$ not coming from $$G$$ is invertible but not group like.
To be concrete, let $$G$$ be the cyclic group $$C_2$$, with nontrivial element $$g$$. Then $$a+bg$$ is invertible as long as $$(a+b)(a-b) \neq 0$$, but $$a+bg$$ is group like only for $$(a,b) = (1,0)$$ and $$(a,b) = (0,1)$$.