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In this chapter, we demonstrate that every character Hopf algebra has a PBW basis. A Hopf algebra H is referred to as a character Hopf algebra if the group G of all group-like elements is commutative and H is generated over k [G] by skew-primitive semi-invariants, whereas a well-ordered subset \(V \subseteq H\) is a set of PBW generators of H if there exists a function \(h: V \rightarrow \mathbf{Z^{+}} \cup \{\infty \},\) called the height function, such that the set of all products $$\displaystyle{gv_{1}^{n_{1} }v_{2}^{n_{2} }\,\cdots \,v_{k}^{n_{k} },}$$ where \(g \in G,\ \ v_{1} < v_{2} <\ldots < v_{k} \in V,\ \ n_{i} < h(v_{i}),1 \leq i \leq k\) is a basis of H.
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