Local power series quotients of commutative Banach and Fréchet algebras

We consider the relationship between derivations and local power series quotients for a locally multiplicatively convex Fréchet algebra. In §2 we derive necessary conditions for a commutative Fréchet algebra to have a local power series quotient. Our main result here is Proposition 2.6, which shows that if the generating element has finite closed descent, the algebra cannot be simply a radical algebra with identity adjoined--it must have nontrivial representation theory; if the generating element does not have finite closed descent, then the algebra cannot be a Banach algebra, and the generating element must be locally nilpotent (but non-nilpotent) in an associated quotient algebra. In §3 we consider a fundamental situation which leads to local power series quotients. Let be a derivation on a commutative radical Fréchet algebra with identity adjoined. We show in Theorem 3.10 that if the discontinuity of is not concentrated in the (Jacobson) radical, then has a local power series quotient. The question of whether such a derivation can have a separating ideal so large it actually contains the identity element has been recently settled in the affirmative by C. J. Read.

     

A simple proof of some generalized principal ideal theorems

Using symmetric algebras we simplify and slightly strengthen the Bruns-Eisenbud-Evans ``generalized principal ideal theorem' on the height of order ideals of nonminimal generators in a module. We also obtain a simple proof and an extension of a result by Kwiecinski, which estimates the height of certain Fitting ideals of modules having an equidimensional symmetric algebra.

     

Some characterizations of the automorphisms of

We present some nonlinear characterizations of the automorphisms of the operator algebra and the function algebra by means of their spectrum preserving properties.

     

Cuntz-Krieger algebras and endomorphisms of finite direct sums of type I factors

A correspondence between algebra endomorphisms of a finite sum of copies of the algebra of all bounded operators on a Hilbert space and representations of certain norm closed -subalgebras of bounded operators generated by a finite collection of partial isometries is introduced. Basic properties of this correspondence are investigated after developing some operations on bipartite graphs that usefully describe aspects of this relationship.

     

The universal central extension of the three-point loop algebra

We consider the three-point loop algebra,

where denotes a field of characteristic 0 and is an indeterminate. The universal central extension of was determined by Bremner. In this note, we give a presentation for via generators and relations, which highlights a certain symmetry over the alternating group . To obtain our presentation of , we use the realization of as the tetrahedron Lie algebra.

     

Moufang loops and alternative algebras

Let be the algebra of classical real octonions or the (split) algebra of octonions over the finite field 2$">. Then the quotient loop of the Moufang loop of all invertible elements of the algebra modulo its center is not embedded into a loop of invertible elements of any alternative algebra.

     

Minimal projective resolutions

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the -algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the ``no loop' conjecture.

     

Crystal bases for the quantum superalgebra

A crystal base theory is introduced for the quantized enveloping algebra of the general linear Lie superalgebra , and an explicit realization of the crystal base is given in terms of semistandard tableaux.     

Real equivariant bordism and stable transversality obstructions for

In this paper we compute homotopical equivariant bordism for the group , namely , geometric equivariant bordism , and their quotient as modules over geometric bordism. This quotient is a module of stable transversality obstructions. We construct these rings from knowledge of their localizations.

     

Quiver varieties and finite dimensional representations of quantum affine algebras

We study finite dimensional representations of the quantum affine algebra using geometry of quiver varieties introduced by the author.

As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.

     

Heegner divisors in the moduli space of genus three curves

S. Kondo used periods of surfaces to prove that the moduli space of genus three curves is birational to an arithmetic quotient of a complex 6-ball. In this paper we study Heegner divisors in the ball quotient, given by arithmetically defined hyperplane sections of the ball. We show that the corresponding loci of genus three curves are given by hyperelliptic curves, singular plane quartics and plane quartics admitting certain rational ``splitting curves'.

     

Exemples séparant certaines classes d'algèbres topologiques

Esterle (1979) and Zelazko (1996 and 1990) gave an example of an algebra which cannot be topologized as a topological algebra (with jointly continuous multiplication), and Müller (in 1991) showed that there exists a topological algebra, which is indeed a locally bounded algebra, which cannot be topologized as a locally convex algebra.

To complete this study on the separation between the classes of algebras, we construct for every a locally bounded algebra which is -normed for every which cannot be topologized as a locally -convex algebra, and we deduce an example of a locally pseudoconvex algebra which cannot be topologized as a locally -convex algebra for every . We also show the existence of a topological algebra which cannot be topologized as a locally pseudoconvex algebra.

     

Koszul duality and equivalences of categories

Let and be dual Koszul algebras. By Positselski a filtered algebra with gr is Koszul dual to a differential graded algebra . We relate the module categories of this dual pair by a Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg, and Soergel.

     

Star ring homomorphisms between commutative Banach algebras

We consider a -ring homomorphism from a commutative Banach algebra with an involution to a commutative Banach algebra with a symmetric involution. We give the Gelfand transform of the -ring homomorphism image.

     

Operator weak amenability of the Fourier algebra

We show that for any locally compact group , the Fourier algebra is operator weakly amenable.

     

Uniformly complete quotient space

The uniformly complete quotient space of a locally compact group is introduced. It is shown that the operator space dual is a completely contractive Banach algebra, which contains the completely bounded Fourier multiplier algebra as a completely contractively complemented Banach subalgebra. A natural completely isometric representation of on is studied and some equivalent amenability conditions associated with are proved.

     

Wavelet-based filters for accurate computation of derivatives

Let be a smooth function whose derivative of a given order must be computed. The signal is affected by two kinds of perturbation. The perturbation caused by the presence of the machine epsilon of the computer may be considered to be an extremely high-frequency noise of very small amplitude. The way to minimize its effect consists of choosing an appropriate value for the step size of the difference quotient.

The second perturbation, caused by the presence of noise, requires first the signal to be treated in some way. It is the purpose of this work to construct a wavelet-based band-pass filter that deals with the two cited perturbations simultaneously. In effect our wavelet acts like a ``smoothed difference quotient" whose stepsize is of the same order as that of the usual difference quotient. Moreover the wavelet effectively removes the noise and computes the derivative with an accuracy equal to the one obtained by the corresponding difference quotient in the absence of noise.

     

The Hochschild cohomology ring of a selfinjective algebra of finite representation type

This paper describes the Hochschild cohomology ring of a selfinjective algebra of finite representation type over an algebraically closed field , showing that the quotient of the Hochschild cohomology ring by the ideal generated by all homogeneous nilpotent elements is isomorphic to either or , and is thus finitely generated as an algebra. We also consider more generally the property of a finite dimensional algebra being selfinjective, and as a consequence show that if all simple -modules are -periodic, then is selfinjective.

     

A categorification of integral Specht modules

We suggest a simple definition for categorification of modules over rings and illustrate it by categorifying integral Specht modules over the symmetric group and its Hecke algebra via the action of translation functors on some subcategories of category for the Lie algebra .