For a $k$-Hopf algebra $H$ and element $h \in H$ is called **grouplike** is $\Delta(h) = h \otimes h$ and $\epsilon(h)=1\_k$ ($\epsilon$ is the counit). The identity $1\_H$ is clearly grouplike, but in general non-trivial grouplike elements may fail to exist. See this question for Is there a name for a Hopf algebra whose only grouplike element if the identity?



Name for a Hopf algebra whose only grouplike element is the identity?
There is a one-to-one correspondence between the grouplike elements and the simple, $1$-dim subcoalgebras. So if the only grouplike element is $1\_H$, then there is a unique $1$-dim simple subcoalgebra (which is $k\cdot 1\_H$). In that case, the HA is - by definition- called: *connected HA*.  

Also, notice that a connected HA is the same thing as an *irreducible HA*.  

(Although if we confine ourselves at the level of coalgebras, the connected ones are a subset of the irreducibles. Actually, at the level of coalgebras, connected=pointed+irreducible, but i think that this is another story).


**Edit:** Motivated by the OP's comment, regarding the use of the term *connected* under the presence of an algebra grading, i think it would be useful to mention the following: the term connected is used in the encyclopedia link provided (<https://encyclopediaofmath.org/wiki/Hopf_algebra>) in the sense of a *connected graded algebra*. The way i am using the term *connected HA* in this post is different and is in the sense of a *connected coalgebra*. These are two different notions of connectedness with non-trivial interactions between them. I think that in contemporary literature the second use tends to be more "standard". For more details, i believe it would be interesting to take a look at: [arXiv:1601.06687v1 [math.RA]](https://arxiv.org/abs/1601.06687v1)


**Edit 2:** In my understanding, the question essentially provides another way to view the notion of a *connected coalgebra* as dual to the notion of a *[connected algebra](https://en.wikipedia.org/wiki/Connected_ring)* (in the ungraded setting):  

Since a connected algebra (ring) -in the ungraded setting- is one which has no non-trivial idempotents (other than $0$ and $1$) in the same way a connected coalgebra is one which has no non-trivial grouplikes (other than $1$).  

Here, we are essentialy viewing the notion of an idempotent ($g\cdot g=g$) element of an algebra as dual to the notion of a grouplike ($\Delta(g)=g\otimes g$) element of a coalgebra.


