On arithmetic macaulayfication of certain local rings

Let A be a Noetherian ring.We consider the existence of Cohen-Macaulay Rees algebras of A. If the non-Cohen-Macaulay locus of A is of dimension 0, then we already know that such a Rees algebra exists. In the present paper, we show that such a Rees algebra also exists when the non-Cohen-Macaulay locus of A is of dimension 1.     

Hilbert Module Realization of the Square of White Noise and Finite Difference Algebras

We develop an approach to the representations theory of the algebra of the square of white noise based on the construction of Hilbert modules. We find the unique Fock representation and show that the representation space is the usual symmetric Fock space. Although we started with one degree of freedom we end up with countably many degrees of freedom. Surprisingly, our representation turns out to have a close relation to Feinsilver's finite difference algebra. In fact, there exists a holomorphic image of the finite difference algebra in the algebra of square of white noise. Our representation restricted to this image is the Boukas representation on the finite difference Fock space. Thus we extend the Boukas representation to a bigger algebra, which is generated by creators, annihilators, and number operators.     

On existence of finite universal Korovkin sets in Segal algebras

We prove that there exists a finite universal Korovkin set w.r.t positive operators for the centre of a Segal algebra on a compact groupG if and only ifG is metrizable. As a consequence it follows that a Segal algebra on a compact abelian group admits a finite universal Korovkin set w.r.t. positive operators iff the group is metrizable.     

The Structure Theorem of Weak Comodule Algebras

This article mainly gives the structure theorem of weak comodule algebras, that is, assume that H is a weak Hopf algebra, and B a weak right H-comodule algebra, if there exists a morphism φ: H → B of a weak right H-comodule algebras, then there exists an algebra isomorphism: B ? B ^coH #H, where B ^coH denotes the coinvariant subalgebra of B, and B ^coH #H denotes the weak smash product.     

An algebraic slice in the coadjoint space of the Borel and the Coxeter element

Let g be a complex simple Lie algebra and b a Borel subalgebra. The algebra Y of polynomial semi-invariants on the dual b^? of b is a polynomial algebra on rank g generators (Grothendieck and Dieudonné (1965–1967)) [16]. The analogy with the semisimple case suggests there exists an algebraic slice to coadjoint action, that is an affine translate y+V of a vector subspace of b^? such that the restriction map induces an isomorphism of Y onto the algebra R[y+V] of regular functions on y+V. This holds in type A and even extends to all biparabolic subalgebras (Joseph (2007)) [20]; but the construction fails in general even with respect to the Borel. Moreover already in type C(2) no algebraic slice exists.Very surprisingly the exception of type C(2) is itself an exception. Indeed an algebraic slice for the coadjoint action of the Borel subalgebra is constructed for all simple Lie algebras except those of types B(2m), C(n) and F(4).Outside type A, the slice obtained meets an open dense subset of regular orbits, even though the special point y of the slice is not itself regular. This explains the failure of our previous construction.     

Actions of Ore extensions and growth of polynomial H-identities

We show that if A is a finite-dimensional associative H-module algebra for an arbitrary Hopf algebra H, then the proof of the analog of Amitsur’s conjecture for H-codimensions of A can be reduced to the case when A is H-simple. (Here we do not require that the Jacobson radical of A is an H-submodule.) As an application, we prove that if A is a finite-dimensional associative H-module algebra where H is a Hopf algebra H over a field of characteristic 0 such that H is constructed by an iterated Ore extension of a finite-dimensional semisimple Hopf algebra by skew-primitive elements (e.g., H is a Taft algebra), then there exists integer PIexp^H(A). In order to prove this, we study the structure of algebras simple with respect to an action of an Ore extension.     

Sesquitransitive and Localizing Operator Algebras

An algebra of operators on a Banach space X is said to be transitive if X has no nontrivial closed subspaces invariant under every member of the algebra. In this paper we investigate a number of conditions which guarantee that a transitive algebra of operators is “large” in various senses. Among these are the conditions of algebras being localizing or sesquitransitive. An algebra is localizing if there exists a closed ball B ∌ 0 such that for every sequence (x _n ) in B there exists a subsequence and a bounded sequence (A _k ) in the algebra such that converges to a non-zero vector. An algebra is sesquitransitive if for every non-zero z ∈ X there exists C > 0 such that for every x linearly independent of z, for every non-zero y ∈ X, and every there exists A in the algebra such that and ||Az|| ≤ C||z||. We give an algebraic version of this definition as well, and extend Jacobson’s density theorem to algebraically sesquitransitive rings. The second and the third authors were supported by NSERC.     

Isotopy and invariants of Albert algebras

Let k be a field with characteristic different from 2 and 3. Let B be a central simple algebra of degree 3 over a quadratic extension K/k, which admits involutions of second kind. In this paper, we prove that if the Albert algebras and have same and invariants, then they are isotopic. We prove that for a given Albert algebra J, there exists an Albert algebra J' with , and . We conclude with a construction of Albert division algebras, which are pure second Tits' constructions. Received: December 9, 1997.     

Structure Theorems for Bicomodule Algebras Over Quasi-Hopf Algebras,Weak Hopf Algebras,and Braided Hopf Algebras

Let H be a quasi-Hopf algebra, a weak Hopf algebra, or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v: H → B. Then we can define an object B^co(H), which is a left-left Yetter–Drinfeld module over H, having extra properties that allow to make a smash product B^co(H)#H, which is an H-bicomodule algebra, isomorphic to B.     

Identities in the varieties generated by the algebras of upper triangular matrices

Consider the algebra UT _s of upper triangular matrices of size s over an arbitrary field. Petrogradsky proved that the exponent of an arbitrary subvariety in var(UT _s ) exists and is an integer. We strengthen the estimates for the growth of these varieties and provide equivalent conditions for finding these exponents. Kemer showed that in the case of a ground field of characteristic zero there exists no varieties of associative algebras with growth intermediate between polynomial and exponential. We prove that this property extends to the case of the fields of arbitrary characteristic distinct from 2.     

Derivation Simple Color Algebras and Semisimple Lie Color Algebras

Let K be an algebraically closed field of arbitrary characteristic and Γ an abelian multiplicative group equipped with a bicharacter ε: Γ × Γ → K*. It is proved that, for any finite-dimensional derivation simple color algebra A over K, there exists a simple color algebra S and a color vector space V such that A? S? S^ε(V), where S^ε(V) is the ε-symmetric algebra of V. As an application of this result, a necessary and sufficient condition such that a Lie color algebra is semisimple is obtained.     

Invariant Differential Operators on Certain Nilpotent Homogeneous Spaces

 Let be a nilpotent connected and simply connected Lie group, and an analytic subgroup of G. Let , be a unitary character of H and let . Suppose that the multiplicities of all the irreducible components of τ are finite. Corwin and Greenleaf conjectured that the algebra of the differential operators on the Schwartz-space of τ which commute with τ is isomorphic to the algebra of H-invariant polynomials on the affine space . We prove in this paper this conjecture under the condition that there exists a subalgebra which polarizes all generic elements in . We prove also that if is an ideal of , then the finite multiplicities of τ is equivalent to the fact that the algebra is commutative.     

The formal theory of Hopf algebras Part II: The case of Hopf algebras

Abstract

The category Hopf_R of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category Hopf_R is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then Hopf_R is reflective in the category of bialgebras as well, and there exists a free Hopf algebra over every R-coalgebra. Similar results are obtained for relevant subcategories of Hopf_R. Moreover it is shown that, for every commutative unital ring R, the so-called “dual algebra functor” has a left adjoint and that, more generally, universal measuring coalgebras exist.     

Order topology in vector lattices with closure by one step

An order topology in vector lattices and Boolean algebras is studied under the additional condition of “closure by one step” that generalizes the well-known “regularity” property of Boolean algebras and K-spaces. It is proved that in a vector lattice or a Boolean algebra possessing such a property there exists a basis of solid neighborhoods of zero with respect to an order topology. An example of a Boolean algebra without basis of solid neighborhoods of zero (an algebra of regular open subsets of the interval (0, 1)) is given. Bibliography: 3 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15 1995, pp. 213–220.     

Simple Twisted Group Algebras of Dimension p4 and their Semi-Centers

For a simple twisted group algebra over a group G, if G^∣ is Hall subgroup of G, then the semi-center is simple. Simple twisted group algebras correspond to groups of central type. We classify all groups of central type of order p⁴ where p is prime and use this to show that for odd primes p there exists a unique group G of order p⁴, such that there exists simple twisted group algebra over G with a commutative semi-center. Moreover, if 1 < |G| <64, then the semi-center of simple twisted group algebras over G is noncommutative and this bounds are strict.     

Invariant Differential Operators on Certain Nilpotent Homogeneous Spaces

 Let be a nilpotent connected and simply connected Lie group, and an analytic subgroup of G. Let , be a unitary character of H and let . Suppose that the multiplicities of all the irreducible components of τ are finite. Corwin and Greenleaf conjectured that the algebra of the differential operators on the Schwartz-space of τ which commute with τ is isomorphic to the algebra of H-invariant polynomials on the affine space . We prove in this paper this conjecture under the condition that there exists a subalgebra which polarizes all generic elements in . We prove also that if is an ideal of , then the finite multiplicities of τ is equivalent to the fact that the algebra is commutative. (Received 15 November 2000)     

Non-surjectivity of the Clifford invariant map

The question whether there exists a commutative ringA for which there is an element in the 2-torsion of the Brauer group not represented by a Clifford algebra was raised by Alex Hahn. Such an example is constructed in this paper and is arrived at using certain results of Parimala-Sridharan and Parimala-Scharlau which are also reviewed here. Dedicated to the memory of Professor K G Ramanathan     

On universal Korovkin sets w.r.t. positive spectral contractions

We prove that there exists a finite universal Korovkin set with respect to positive spectral contraction operators for a Segal algebra on a locally compact abelian groupG if and only ifĜ is finite dimensional separable metrizable space.     

On the codimension growth of almost nilpotent Lie algebras

We study codimension growth of infinite dimensional Lie algebras over a field of characteristic zero. We prove that if a Lie algebra L is an extension of a nilpotent algebra by a finite dimensional semisimple algebra then the PI-exponent of L exists and is a positive integer.     

Equivariant Cyclic Cohomology of H-Algebras

We define an equivariant K ₀-theory for Yetter–Drinfeld algebras over a Hopf algebra with an invertible antipode. We then show that this definition can be generalized to all Hopf-module algebras. We show that there exists a pairing, generalizing Connes pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory.