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If $a$ and $b$ are elements of a Hopf algebra over a field $k$ and $\alpha, \beta \in k$, then what is $\Delta(\alpha a+\beta b)$?

Is it $\alpha\Delta(a)+\beta\Delta(b)$?

For example if $\Delta(x)=x \otimes 1$, $\Delta(p)=p \otimes 1 + 1 \otimes p$, $\Delta(g) = g \otimes g$ and $\alpha, \beta, \gamma \in k$, then what is $\Delta(\alpha x + \beta p + \gamma g)$?

This property is nowhere to be found in the literature I've looked into (mostly Majid). Also, what would be answer for analogous question for counit?

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the structure maps of a bialgebra $H$ over a field $k$ are defined to be $k$-linear.

This means, exactly, that if $a,b\in H$ and $\alpha,\beta \in k$, then $\Delta(\alpha a + \beta b) = \alpha \Delta(a) + \beta \Delta(b)$ The counit is also a $k$-linear map.

This fact is glossed over in most of Majid's literature; he starts by assuming that a coalgebra $C$ (or whatever) is a vector space, and requires that structure maps (like $\Delta$) are "maps" - it's implicit that a map between vector spaces is linear. In the category of vector spaces, the morphisms (or maps) are linear maps.

you can define algebras, coalgebras, bialgebras, et cetera, over a ring $R$ as well, in which case you require these maps to be $R$-linear.

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