The focus of this paper is actions of finite dimensional pointed Hopf algebras \(H\) on a quantum polynomial algebra \(\Lambda=k_Q[X_1^{\pm 1},\dots,X_r^{\pm 1},X_{r+1},\dots,X_n]\), where \(Q=(q_{ij})\) is a multiplicatively antisymmetric matrix over a field \(k\) and \(X_iX_j=q_{ij}X_jX_i\) for all \(i,j\). Assume that \(n\geq 3\), and that the subgroup \(\langle q_{ij}\rangle\subseteq k^\times\) is free Abelian of rank \(n(n-1)/2\). The author shows that the given action extends to an action of \(H\) on the localization \(k_Q[X_1^{\pm 1},\dots,X_n^{\pm 1}]\), which allows him to reduce to the case \(r=n\). He then proves that there are linear forms \(\chi_v,\omega_v\in H^*\), for each multi-index \(v\in\mathbb{Z}^n\), such that \(h\cdot X^v=\chi_v(h)X^v+\omega_v(h)X^{-v}\) for \(h\in H\). These forms span a commutative sub-Hopf-algebra \(F^*\subseteq H^*\), and the action of \(H\) on \(\Lambda\) factors naturally through an action of \(F=F^{**}\) on \(\Lambda\). In case some \(\omega_w\) is nonzero, \(F\) is a smash product of the dual of the Hopf algebra spanned by the \(\chi_v\) with the group algebra \(k\mathbb{Z}_2\). The most complete result is obtained for the case that \(k\) is perfect and its characteristic does not divide either \(\dim H\) or the order of the group of grouplikes in \(H\). Then the action of \(H\) on \(\Lambda\) factors through an epimorphism of Hopf algebras \(H\to k\Gamma\) where \(\Gamma\) is a finite group of automorphisms of \(\Lambda\).
As for the subalgebra \(\Lambda^H\) of \(H\)-invariants, it is proved that there exist a subgroup \(U\subseteq\mathbb{Z}^n\) of finite index and scalars \(\zeta_u\in k\) such that \(X^u+\zeta_u X^{-u}\in\Lambda^H\) for \(u\in U\). Consequently, \(\Lambda\) is left and right Noetherian as a \(\Lambda^H\)-module.
The development of the above results also allows the author to determine the Lie algebra \(\text{Der}_k\Lambda\): it is the tensor product of the Lie algebra of inner derivations on \(\Lambda\) with an \(n\)-dimensional Abelian Lie algebra spanned by derivations \(\partial_j\) such that \(\partial_j(X_i)=\delta_{ij}X_j\).