Actions of cocommutative Hopf algebras (Q2285309)

Let \(H\) be a cocommutative Hopf algebra acting on an algebra \(A\). In these notes, the authors study the prime and semiprime spectrum of \(A\) in terms of \(H\)-stable ideals, under the assumptions that the field is algebraically closed and the \(H\)-action is induced by the coaction of a Hopf subalgebra of the finite (or Sweedler) dual \(H^{\circ}\) of \(H\). The paradigmatic examples of cocommutative Hopf algebras are group algebras and universal enveloping algebras of Lie algebras. Thus, the results contained in this paper (partially) generalize the work of the first author on rational actions of algebraic groups, see [\textit{M. Lorenz}, Algebra Number Theory 2, No. 4, 467--499 (2008; Zbl 1166.16015); Transform. Groups 14, No. 3, 649--675 (2009; Zbl 1184.16043); Proc. Am. Math. Soc. 142, No. 9, 3013--3017 (2014; Zbl 1306.16039)]. In the last years there was an increasing interest in the study of actions of Hopf algebras on different families of algebras, see for example [\textit{P. Etingof} and \textit{C. Walton}, Adv. Math. 251, 47--61 (2014; Zbl 1297.16029); \textit{J. Cuadra} et al., Adv. Math. 282, 47--55 (2015; Zbl 1369.16026); \textit{P. Etingof} and \textit{C. Walton}, Algebra Number Theory 10, No. 10, 2287--2310 (2016; Zbl 1355.16030)]. Let us describe shortly the main result of the paper. An ideal \(I\) of \(A\) that is also an \(H\)-submodule of \(A\) is called an \(H\)-ideal. Given an \(H\)-ideal \(I\), then the quotient \(A/I\) is also an \(H\)-module algebra. For an arbitrary ideal \(I\) one may consider the \(H\)-ideal \(I\!:\!H\) given by the sum of all \(H\)-ideals that are contained in \(I\). This is called the \(H\)-core of \(I\) and equals \(I\!:\!H = \{a \in A\, |\, H.a \subseteq I\}\). If \(A\neq 0\) and the product of any two nonzero \(H\)-ideals of \(A\) is again nonzero, then \(A\) is said to be \(H\)-prime. An \(H\)-ideal \(I\) of \(A\) is called \(H\)-prime if the quotient \(A/I\) is so. It holds that \(H\)-cores of prime ideals are \(H\)-prime. Write \(H-\text{Spec}\;A\) for the collection of all \(H\)-primes of \(A\). Then there exists a map \(\text{Spec }A \longrightarrow H-\text{Spec}\;A\), given by \(P\mapsto P\!:\!H\). The fibers \[ \text{Spec}_{I}\, A = \{P \in \text{Spec}\; A \, |\, P\!:\!H = I\} \] are called the \(H\)-strata of \(\text{Spec}\;A\). This yields an stratification \(\text{Spec}\;A = \bigsqcup_{I\in H-\text{Spec}\; A } \text{Spec}_{I}\, A\) which was first studied by Goodearl and Letzter for the case of actions of group algebras. The main goal of this paper is to generalize this result to more general cocommutative Hopf algebras over an algebraically closed field. This is done by describing the \(H\)-strata \(\text{Spec}_{I}\, A\) in terms of the prime spectrum of certain commutative algebras \(\mathcal{C}_{I}\) under the assumption that the induced \(H\)-coaction on \(A\) has coefficients in a Hopf subalgebra \(\mathcal{O}\) of \(H^{\circ}\); that is, the \(H\)-action is \textit{integral}. In such a case, the commutative algebras are given by the tensor product \(\mathcal{C}_{I}= \mathcal{C}(A/I)\otimes \mathcal{O}\) of the center of the symmetric ring of quotients of \(A/I\) and \(\mathcal{O}\). The article is well written and carefully organized. It contains all the preliminaries that are needed through the paper and ends with a section devoted to study the question of semiprimeness, where it is proved that the \(H\)-ideal \(I\!:\!H\) is semiprime for every semiprime ideal \(I\) of \(A\), in case \(H\) is cocommutative and the characteristic of the field is zero.