Irreducible actions and faithful actions of hopf algebras

LetH be a Hopf algebra acting on an algebraA. We will examine the relationship betweenA, the ring of invariantsA ^H, and the smash productA # H. We begin by studying the situation whereA is an irreducibleA # H module and, as an application of our first main theorem, show that ifD is a division ring then [D : D ^H]≦dimH. We next show that prime rings with central rings of invariants satisfy a polynomial identity under the action of certain Hopf algebras. Finally, we show that the primeness ofA # H is strongly related to the faithfulness of the left and right actions ofA # H onA. The first author was supported by the Faculty Research and Development Fund of the College of Liberal Arts and Sciences and the University Research Council at DePaul University and NSF Grant No. 8521704. He also wishes to thank Ben Gurion University for its hospitality, where much of this work was done. The second and third authors were supported by the Fund for Basic Research administered by the Israel Academy of Sciences and Humanities. Part of this author’s contribution is included in her Ph.D. thesis at Ben Gurion University under the supervision of the second author.     

The hopf algebra of linearly recursive sequences

We explain how the space of linearly recursive sequences over a field can be considered as a Hopf algebra. The algebra structure is that of divided-power sequences, so we concentrate on the perhaps lesser-known coalgebra (diagonalization) structure. Such a sequence satisfies a minimal recursive relation, whose solution space is the subcoalgebra generated by the sequence. We discuss possible bases for the solution space from the point of view of diagonalization. In particular, we give an algorithm for diagonalizing a sequence in terms of the basis of the coalgebra it generates formed by its images under the difference-operator shift. The computation involves inverting the Hankel matrix of the sequence. We stress the classical connection (say over the real or complex numbers) with formal power series and the theory of linear homogeneous ordinary differential equations. It is hoped that this exposition will encourage the use of Hopf algebraic ideas in the study of certain combinatorial areas of mathematics.     

Semi-invariants of actions of multiplier hopf algebras

Let A be a multiplier Hopf algebra which acts on an algebra R. In this paper we study semi-invariants of this action. This idea has proved interesting in the case thatA is a Hopf algebra.     

Hopf algebra actions

Finite groups acting on rings by automorphisms, and group-graded rings are instances of Hopf algebras H acting on H-module algebras A. We study such actions. Let A^H = {a ε A¦h · A = (h) a, all h ε H}, the ring of H-invariants, and form the smash product A # H. We study the ring extensions A^H A A # H. We prove a Maschke-type theorem for A # H-modules. We form an associated Morita context [A^H, A, A, A # H] and use these to get connections between the various rings.     

Cotorsion dimensions and Hopf algebra actions

Let H be a finite-dimensional Hopf algebra over a field k, and let A be an H-module algebra. In this paper, we discuss the cotorsion dimension of the smash product A # H. We prove that $$l.\cot .D\left( {A\# H} \right) \leqslant l.\cot .D\left( A \right) + r.D\left( H \right),$$ which generalizes the result of group rings. Moreover, we give some sufficient conditions for which $$l.\cot .D\left( {A\# H} \right) = l.\cot .D\left( A \right).$$ As applications, we study the invariants of IF properties and Gorenstein global dimensions.     

Quasi-hopf algebra actions and smash products

A classical result of Magnus asserts that a free group F has a faithful representation in the group of units of a ring of non-commuting formal power series with integral coefficients. We generalize this result to the category of A-groups, where A is an associative ring or an Abelian group. We say that a free A-group F^A has a faithful Magnus representation if there is a ring B containing A as an additive subgroup (or a subring) such that F^A is faithfully represented (exactly as in Magnus' classical result in the case A = Z)in the group of units of the ring of formal power series in non-communting indeterminater over B.The three principal results are: (I) If A is a torsion free Abelian group and F^A is a free A-groupp of Lyndon' type, then F^A has a faithful Magnus representation; (II) If A is an ω‐residually Z ring, then F^A has a faithful Magnus representation;(III) for every nontrivial torsion-free Abelian group A, F^A has a faithful Magnus representation in B[[Y]] for a suitable ring B in and only if F^Q has a faithful Magnus representation in Q[[Y]].     

0-Hecke algebra actions on coinvariants and flags

The 0-Hecke algebra H _n (0) is a deformation of the group algebra of the symmetric group \(\mathfrak{S}_{n}\) . We show that its coinvariant algebra naturally carries the regular representation of H _n (0), giving an analogue of the well-known result for \(\mathfrak{S}_{n}\) by Chevalley–Shephard–Todd. By investigating the action of H _n (0) on coinvariants and flag varieties, we interpret the generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. We also study the action of H _n (0) on the cohomology rings of the Springer fibers, and similarly interpret the (non-commutative) Hall–Littlewood symmetric functions indexed by hook shapes.     

Uncertainty inequalities for hopf algebras

The uncertainty principle was recently formulated for locally compact Abelian groups and for finite groups. In this paper similar inequalities are found for certain Hopf algebras. Some cases of equality are characterized, showing a connection with right coideal subalgebras that are Forbenius algebras.