When do counital coalgebras have a basis of grouplike elements? Question.
Under what conditions do counital coalgebras have bases consisting entirely of grouplike elements? At least in the case of finite-dimensional coalgebras, or for bialgebras  (or Hopf algebras in particular), is there a simple characterization?
Definitions.


*

*A counital (coassociative) coalgebra is a vector space $V$ over a field $K$, together with operators
$$\def\id{\mathrm{id}}
 \Delta : V \to V \otimes V \qquad\qquad \varepsilon : V \to K $$
such that the following equalities hold:
$$\begin{gather*}
(\Delta \otimes \id_V) \Delta \;=\; (\id_V \otimes \Delta) \Delta \;;\\[1ex]
 (\varepsilon \otimes \id_V) \Delta \;=\; \id_V \;=\; (\id_V \otimes \varepsilon) \Delta \;,
\end{gather*}$$
which are the coassociative property of $\Delta$ (dual to the usual associative/distributive property of multiplication) and the counital property of $\varepsilon$ (dual to the property of being a multiplicative identity).

*An element $\mathbf v \in V$ is grouplike if $\Delta(\mathbf v) = \mathbf v \otimes \mathbf v$, and $\mathbf v \ne \mathbf 0$. My question is about the conditions in which there exists a basis for $V$ consisting of such elements.
Examples.
There are simple examples with and without a basis of grouplike elements.
For instance, for an arbitrary field $K$ and $V$ a vector space generated by two basis vectors $ \def\r{\mathbf x}
\def\i{\mathbf y} \r, \i$, if we choose
 $$
\begin{align*}
\Delta(\r) &= \r \otimes \r
&\quad
\varepsilon(\r) &= 1
\\
\Delta(\i) &= \i \otimes \i 
&
\varepsilon (\i) &= 1
\end {align*}$$ 
then $\{\r,\i\}$ itself is such a basis. In particular, we then have $$\Delta(a\r + b\i) = a(\r\otimes\r) + b(\i\otimes\i) \,,$$ which is a product if and only if either $a=0$ or $b=0$, so that $\{\r,\i\}$ is uniquely a basis of grouplike elements.
On the other hand, a coalgebra need not have any grouplike elements at all: if we instead define
$$
\begin{align*}
\Delta(\r) &= \r \otimes \r - \i \otimes \i
&\quad
\varepsilon(\r) &= 1
\\
\Delta(\i) &= \r \otimes \i + \i \otimes \r
&
\varepsilon (\i) &= 0
\end {align*}$$
then $$\Delta(a\r + b\i) = a(\r \otimes \r) + b(\r \otimes \i) + b (\i \otimes \r) - a (\i \otimes \i)\,,$$ which is a product vector if and only if $a^2 = -b^2$, that is if $a = b = 0$ or $a = \pm bi$, where $i^2 = -1$.
In particular, for fields such as $\mathbb R$ in which $x^2+1$ is irreducible, there are no non-trivial solutions.
Is there a characterisation of which coalgebras have such a basis? Again, if there is a simple characterization at least for the finite-dimensional case, or for bialgebras / Hopf algebras. (Of course, in the case of a bialgebra, at least the unit $\eta$ is a grouplike element.)
 A: A bialgebra has a basis of group-like elements iff it is a monoid algebra.  A hopf algebra has a basis of group-like elements iff it is a group algebra.  The point is that in a bialgebra the group-like elements form a submonoid and are automatically linearly independent.  Thus you have a basis of group-likes iff the group-likes themselves form a basis in which case you have the monoid algebra of the set of group-likes.  In a Hopf algebra the antipode restricts to an inverse on the monoid of group-likes and so you have a group algebra.
I am not sure about what happens in the coalgebra case.
A: At the level of Hopf algebras, there is a simple characterization of such a class of hopf algebras (admitting a base consisting entirely of grouplikes): 

If $H$ is a cocommutative, fin. dim., hopf algebra over an algebraically closed field of characteristic zero, then it can be shown that 
  $$
H\cong kG(H)
$$ 
  where $G(H)$ is the group formed by the grouplikes of $H$. Thus, $H$ is a group hopf algera (and obviously satisfies your requirements). 

The above is a direct consequence of the Cartier-Kostant-Milnor-Moore theorem. You can also find alternative proofs of the above fact, in the answers of this question. 
