Diagrams of Hopf algebras with the Chevalley property

In this paper, we study non-cosemisimple Hopf algebras through their underlying coalgebra structure. We introduce the concept of the maximal pointed subcoalgebra/Hopf subalgebra. For a non-cosemisimple Hopf algebra A with the Chevalley property, if its diagram is a Nichols algebra, then the diagram of its maximal pointed Hopf subalgebra is also a Nichols algebra. When A is of finite dimension, we provide a necessary and sufficient condition for A’s diagram equaling the diagram of its maximal pointed Hopf subalgebra.     

Most algebras have the interpolation property

We show that for "almost all" algebras on a given set A, every function on A can be interpolated by a polynomial on any set of cardinality (for reasonable ). Received August 27, 1996; accepted in final form July 3, 1997.     

On the antipode of Hopf algebras with the dual Chevalley property

In this paper, we study the antipode of a finite-dimensional Hopf algebra H with the dual Chevalley property and obtain an annihilation polynomial for the antipode. This generalizes an old result given by Taft and Wilson in 1974. As consequences, we show that 1) the quasi-exponent of H is the same as the exponent of its coradical, that is, $\mathrm{qexp}(H)=\exp ?(H_0)$; 2) $\mathrm{qexp}(H?\mathbb{k}〈S^2〉)=\mathrm{qexp}(H)$.     

Confluent operator algebras and the closability property

Certain operator algebras A on a Hilbert space have the property that every densely defined linear transformation commuting with A is closable. Such algebras are said to have the closability property. They are important in the study of the transitive algebra problem. More precisely, if A is a two-transitive algebra with the closability property, then A is dense in the algebra of all bounded operators, in the weak operator topology. In this paper we focus on algebras generated by a completely nonunitary contraction, and produce several new classes of algebras with the closability property. We show that this property follows from a certain strict cyclicity property, and we give very detailed information on the class of completely nonunitary contractions satisfying this property, as well as a stronger property which we call confluence.     

On the Weak-Lefschetz property for Artinian Gorenstein algebras

We deal with the weak Lefschetz property (WLP) for Artinian standard graded Gorenstein algebras of codimension 3. We prove that many Gorenstein sequences force the WLP for such algebras. Moreover for every Gorenstein sequence \(H\) of codimension 3 we found several Gorenstein Betti sequences compatible with \(H\) which again force the WLP. Finally we show that for every Gorenstein Betti sequence the general Artinian standard graded Gorenstein algebra with such Betti sequence has the WLP.     

A Hamiltonian property for nilpotent algebras

In this paper we show that any finite algebra A satisfying a weak left nilpotence condition has the property that all maximal subuniverses are congruence blocks. Conversely, if every subalgebra of A ² has the property that all maximal subuniverses are congruence blocks, then A satisfies the aforementioned nilpotence condition. Received August 3, 1994; accepted in final form June 27, 1996.     

The maximality of the core model

Our main results are: 1) every countably certified extender that coheres with the core model is on the extender sequence of , 2) computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of , 4) (joint with W. J. Mitchell) is universal for mice of height whenever , 5) if there is a such that is either a singular countably closed cardinal or a weakly compact cardinal, and fails, then there are inner models with Woodin cardinals, and 6) an -Erdös cardinal suffices to develop the basic theory of .

     

Coefficient ideals and the Cohen-Macaulay property of Rees algebras

Let be a local ring and let be an ideal of positive height. If is a reduction of , then the coefficient ideal is by definition the largest ideal such that . In this article we study the ideal when the Rees algebra is Cohen-Macaulay.

     

On the weak Freese-Nation property of complete Boolean algebras

The following results are proved:

(a) In a model obtained by adding ₂ Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □_μ and cof([μ]^₀,)=μ⁺ hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of is equivalent to the weak Freese-Nation property of any of or for uncountable κ. (d) Modulo the consistency of (_ω+1,_ω)(₁,₀), it is consistent with GCH that does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding _ω Cohen reals destroys the weak Freese-Nation property of .

These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.     

The Noetherian property in some quadratic algebras

We introduce a new class of noncommutative rings called pseudopolynomial rings and give sufficient conditions for such a ring to be Noetherian. Pseudopolynomial rings are standard finitely presented algebras over a field with some additional restrictions on their defining relations--namely that the polynomials in a Gröbner basis for the ideal of relations must be homogeneous of degree 2--and on the Ufnarovskii graph . The class of pseudopolynomial rings properly includes the generalized skew polynomial rings introduced by M. Artin and W. Schelter. We use the graph to define a weaker notion of almost commutative, which we call almost commutative on cycles. We show as our main result that a pseudopolynomial ring which is almost commutative on cycles is Noetherian. A counterexample shows that a Noetherian pseudopolynomial ring need not be almost commutative on cycles.

     

Koszul property of a class of graded algebras with nonpure resolutions

Given any integers a, b, c, and d with a > 1, c ≥ 0, b ≥ a + c, and d ≥ b + c, the notion of (a, b, c, d)-Koszul algebra is introduced, which is another class of standard graded algebras with “nonpure” resolutions, and includes many Artin-Schelter regular algebras of low global dimension as specific examples. Some basic properties of (a, b, c, d)-Koszul algebras/modules are given, and several criteria for a standard graded algebra to be (a, b, c, d)-Koszul are provided.