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.     

Quantum groups of dimensionpq 2

In this paper we construct two families of non-trivial self-dual semisimple Hopf algebras of dimensionpq ² and investigate closely their (quasi) triangular structures. The paper contains also general results on finite-dimensional triangular Hopf algebras, unimodularity, semisimplicity and ribbon structures of finite-dimensional semisimple Hopf algebras. Dedicated to the memory of Professor S. A. Amitsur This work was partially supported by the Basic Research Foundation administrated by the Israel Academy of Sciences and Humanities. The paper consists of part of the author’s doctoral dissertation written at Ben Gurion University under the supervision of Prof. M. Cohen and Prof. D. E. Radford whom the author wishes to thank for their valuable guidance.     

Biinvertible actions of Hopf algebras

We study actions of a Hopf algebraH on an algebraR such that the action is twisted by an invertible mapσ:H⊗H →R; the biinvertible condition means that these actions also have both an inverse and an antiinverse in Hom(H, EndR). WhenR is an ordinaryH-module algebra, the action is biinvertible if the antipose is bijective. As a new example we show that if theH-action is twisted and the coradical ofH is cocommutative, then the action is biinvertible. After studying the continuity of these actions with respect to the filter of ideals ofR with zero annihilator, we consider when the actions may be extended to the symmetric Martindale quotient ring ofR and itsH-analog. Our results can be applied to crossed productsR# _σ . Research supported by NSF Grant No. 89-01491.     

On the Derivations of Semiprime Rings and Noncommutative Banach Algebras

Abstract Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation D : A→A such that [D(x), x]D(x)[D(x),x] ∈ rad(A) for all x∈A. In this case, D(A) ⊆ rad (A). The author has been supported by Kangnung National University, Research Fund, 1998     

Randomlyn-cyclic digraphs

A digraphD is randomlyn-cyclic (n≥3) if for each vertexv ofD, every (directed) path with initial vertexv and having length at mostn−1 can be extended to av−v (directed) cycle of lengthn. Several results related to and examples of randomlyn-cyclic digraphs are presented. Also, all randomlyn-cyclic digraphs forn=3, 4, and 5 are determined. Research supported by a Western Michigan University faculty research fellowship. Research supported in part by a College of Arts and Sciences and Graduate College research assistantship from Western Michigan University.     

Semigroup actions on homogeneous spaces

LetG be a connected semi-simple Lie group with finite center andS⊄G a subsemigroup with interior points. LetG/L be a homogeneous space. There is a natural action ofS onG/L. The relationx≤y ify ∈Sx, x, y ∈G/L, is transitive but not reflexive nor symmetric. Roughly, a control set is a subsetD ⊄G/L, inside of which reflexivity and symmetry for ≤ hold. Control sets are studied inG/L whenL is the minimal parabolic subgroup. They are characterized by means of the Weyl chambers inG meeting intS. Thus, for eachw ∈W, the Weyl group ofG, there is a control setD _w .D ₁ is the only invariant control set, and the subsetW(S)={w:D _w =D ₁} turns out to be a subgroup. The control sets are determined byW(S)/W. The following consequences are derived: i)S=G ifS is transitive onG/H, i.e.Sx=G/H for allx ∈G/H. HereH is a non discrete closed subgroup different fromG andG is simple. ii)S is neither left nor right reversible ifS #G iii)S is maximal only if it is the semigroup of compressions of a subset of some minimal flag manifold. Research partially supported by CNPq grant n^o 50.13.73/91-8     

Graded rings and essential ideals

LetG be a group andA aG-graded ring. A (graded) idealI ofA is (graded) essential ifI⊃J≠0 wheneverJ is a nonzero (graded) ideal ofA. In this paper we study the relationship between graded essential ideals ofA, essential ideals of the identity componentA _e and essential ideals of the smash productA#G ^*. We apply our results to prime essential rings, irredundant subdirect sums and essentially nilpotent rings.     

Order preserving nonexpansive operators inL 1

We study the limit behaviour ofT ^k f and of Cesaro averagesA _n f of this sequence, whenT is order preserving and nonexpansive inL ₁ ⁺ . IfT contracts also theL _∞-norm, the sequenceT ⁿ f converges in distribution, andA _n f converges weakly inL _p (1<p<∞), and also inL ₁ if the measure is finite. “Speed limit” operators are introduced to show that strong convergence ofA _n f need not hold. The concept of convergence in distribution is extended to infinite measure spaces. Much of this work was done during a visit of the first author at Ben Gurion University of the Negev in Beer Sheva, supported by the Deutsche Forschungsgemeinschaft.     

A near-linear algorithm for the planar segment-center problem

LetP be a set ofn points in the plane and lete be a segment of fixed length. The segment-center problem is to find a placement ofe (allowing translation and rotation) which minimizes the maximum euclidean distance frome to the points ofP. We present an algorithm that solves the problem in timeO(n ^1+ε), for any ε>0, improving the previous solution of Agarwalet al. [3] by nearly a factor ofO(n). Work on this paper by the second author has been supported by NSF Grants CCR-91-22103 and CCR-93-11127, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

Vertical decomposition of arrangements of hyperplanes in four dimensions

We show that, for any collection ℋ ofn hyperplanes in ℜ⁴, the combinatorial complexity of thevertical decomposition of the arrangementA(ℋ) of ℋ isO(n ⁴ logn). The proof relies on properties of superimposed convex subdivisions of 3-space, and we also derive some other results concerning them. Work on this paper by Leonidas Guibas and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Leonidas Guibas was also supported by National Science Foundation Grant CCR-9215219. Work by Micha Sharir was also supported by National Science Foundation Grant CCR-91-22103, and by grants from the G.I.F.—the German Isreali Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences. Work by Jiří Matouŝek was done while he was visiting Tel Aviv University, and its was partially supported by a Humboldt Research Fellowship. Work on this paper by Dan Halperin was carried out while he was at Tel Aviv University.     

A proof that commutative artinian rings are noetherian

Abstract

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedler's equivalence is still valid for free Hopf algebras or projective Hopf algebras over integral domains. Analogously for a left H-module algebra A we study the influence of non-zero left A#H-linear maps from A to A#H on H being finitely generated as R-module. Examples and application to separability are given.     

Every null-additive set is meager-additive

It is proved that every null-additive subset of^ω2 is meager-additive. Several characterizations of the null-additive subsets of^ω2 are given, as well as a characterization of the meager additive subsets of^ω2. Under CH, an uncountable null-additive subset of^ω2 is constructed. Publ. No. 445. First version written in April 1991. Partially supported by the Basic Research Fund of the Israel Academy of Sciences and the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Linear liftings for non-complete probability spaces

We show that it is consistent with ZFC thatL ^∞(Y,B,ν) has no linear lifting for many non-complete probability spaces (Y,B,ν), in particular forY=[0,1]^A,B=Borel subsets ofY, ν=usual Radon measure onB. Research supported by UPEI Senate Grant no 602101, by the Research Institute for Mathematical Sciences at Bar-Ilan University, and by the Landau Center for Mathematical Research in Analysis (supported by the Minerva Foundation). The author would like to thank the organizers of the Winter Institute on the Set Theory of the Reals for their hospitality while part of this research was being carried out. Partially supported by the Foundation for Basic Research of the Israel Academy of Science. Publication number 437.     

Representation theory of Hopf galois extensions

LetH be a Hopf algebra over the fieldk andB ⊂A a right faithfully flat rightH-Galois extension. The aim of this paper is to study some questions of representation theory connected with the ring extensionB ⊂A, such as induction and restriction of simple or indecomposable modules. In particular, generalizations are given of classical results of Clifford, Green and Blattner on representations of groups and Lie algebras. The stabilizer of a leftB-module is introduced as a subcoalgebra ofH. Very often the stabilizer is a Hopf subalgebra. The special case whenA is a finite dimensional cocommutative Hopf algebra over an algebraically closed field,B is a normal Hopf subalgebra andH is the quotient Hopf algebra was studied before by Voigt using the language of finite group schemes.     

The minimal cardinality where the Reznichenko property fails

A topological spaceX has the Fréchet-Urysohn property if for each subsetA ofX and each elementx inĀ, there exists a countable sequence of elements ofA which converges tox. Reznichenko introduced a natural generalization of this property, where the converging sequence of elements is replaced by a sequence of disjoint finite sets which eventually intersect each neighborhood ofx. In [5], Kočinac and Scheepers conjecture: The minimal cardinality of a setX of real numbers such thatC _p(X) does not have the weak Fréchet-Urysohn property is equal to b. (b is the minimal cardinality of an unbounded family in the Baire space^Nℕ.) We prove the Kočinac-Scheepers conjecture by showing that ifC _p(X) has the Reznichenko property, then a continuous image ofX cannot be a subbase for a non-feeble filter on ℕ. The author is partially supported by the Golda Meir Fund and the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

Galois module structure of<Emphasis Type="Italic">p</Emphasis>th-power classes of extensions of degree<Emphasis Type="Italic">p</Emphasis>

For fieldF of characteristic notp containing a primitivepth root of unity, we determine the Galois module structure of the group ofpth-power classes ofK for all cyclic extensionsK/F of degreep. Research supported in part by the Natural Sciences and Engineering Research Council of Canada and by the special Dean of Science Fund at the University of Western Ontario. Supported by the Mathematical Sciences Research Institute, Berkeley.     

Extremely noncommutative elements in rings

We study rings and K-algebras in which all elements or all noncentral elements have smallest possible centralizer. Our principal result asserts that a ring R must be either finite or commutative if each noncentral element a has centralizer equal to the subring generated by a. Supported by the Natural Sciences and Engineering Research Council of Canada, Grant 3961. Authors’ addresses: Howard E. Bell, Department of Mathematics, Brock University, St. Catharines, Ontario, Canada L2S 3A1; Abraham A. Klein, Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel     

Smash products and outer derivations

LetR be a prime ring andL a Lie algebra acting onR as “Q-outer” derivations (if charR=p≠0, assume thatL is restricted). We study ideals and the center of the smash productR #U(L) (respectivelyR #u(L) ifL is restricted) and use these results to study the relationship betweenR and the ring of constantsR ^L . More generally, for any finite-dimensional Hopf algebraH acting onR such thatR #H satisfies the “ideal intersection property”, we useR #H to study the relationship betweenR and the invariant ringR ^H . The first author wishes to thank the University of Southern California for its hospitality while this work was being done. Research of the second author was partially supported by NSF Grant MCS 83-01393.     

Polar actions on Hilbert space

An isometricH-action on a Riemannian manifoldX is calledpolar if there exists a closed submanifoldS ofX that meets everyH-orbit and always meets orbits orthogonally (S is called a section). LetG be a compact Lie group equipped with a biinvariant metric,H a closed subgroup ofG ×G, and letH act onG isometrically by (h ₁,h ₂) ·x = h ₁ xh ₂ ⁻¹ · LetP(G, H) denote the group ofH ¹-pathsg: [0, 1] →G such that (g(0),g (1)) ∈H, and letP(G, H) act on the Hilbert spaceV = H ⁰([0, 1], g) isometrically byg * u = gug ⁻¹ −g′g ⁻¹. We prove that if the action ofH onG is polar with a flat section then the action ofP(G, H) onV is polar. Principal orbits of polar actions onV are isoparametric submanifolds ofV and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples ofP(G, H)-actions for suitable choice ofH andG. Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-Institut für Mathematik in Bonn.     

Computing the connes spectrum of a Hopf algebra

LetH be a finite-dimensional Hopf algebra over the fieldK and letA be anH-module algebra. In a previous paper, we defined the Connes spectrum CS(A, H) for the action ofH onA to be a certain subset of the set Irr(H) of irreducible representations ofH. In this paper, we compute a number of examples; specifically, we consider certain inner and outer actions and we take a closer look at the cocommutative situation. We discover that the information encoded in the Connes spectrum is rather subtle and elusive. Research supported in part by NSF Grant DMS-8900405.