Galois Stability, Integrality and Realization Fields for Representations of Finite Abelian Groups

For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups GGL _n (E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E=F(G) that are obtained via adjoining all matrix coefficients of all matrices gG to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E=F(G) for prescribed integers n and t or prescribed n and d.     

On ＂Problems on von Neumann Algebras by R. Kadison, 1967＂

A brief summary of the development on Kadison‘s famous problems (1967) is given. A new set of problems in von Neumann algebras is listed.     

Centralizers of Iwahori-Hecke algebras

To date, integral bases for the centre of the Iwahori-Hecke algebra of a finite Coxeter group have relied on character theoretical results and the isomorphism between the Iwahori-Hecke algebra when semisimple and the group algebra of the finite Coxeter group. In this paper, we generalize the minimal basis approach of an earlier paper, to provide a way of describing and calculating elements of the minimal basis for the centre of an Iwahori-Hecke algebra which is entirely combinatorial in nature, and independent of both the above mentioned theories.

This opens the door to further generalization of the minimal basis approach to other cases. In particular, we show that generalizing it to centralizers of parabolic subalgebras requires only certain properties in the Coxeter group. We show here that these properties hold for groups of type and , giving us the minimal basis theory for centralizers of any parabolic subalgebra in these types of Iwahori-Hecke algebra.

     

Tame and Wild Coordinates of

Let be a field of characteristic zero. We characterize coordinates and tame coordinates in , i.e. the images of respectively under all automorphisms and under the tame automorphisms of . We also construct a new large class of wild automorphisms of which maps to a concrete family of nice looking polynomials. We show that a subclass of this class is stably tame, i.e. becomes tame when we extend its automorphisms to automorphisms of .

     

Homological Dimension of Coalgebras and Crossed Coproducts

Let H be a Hopf k-algebra. We study the global homological dimension of the underlying coalgebra structure of H. We show that gl.dim(H) is equal to the injective dimension of the trivial right H-comodule k. We also prove that if D = C H is a crossed coproduct with invertible , then gl.dim(D) gl.dim(C) + gl.dim(H). Some applications of this result are obtained. Moreover, if C is a cocommutative coalgebra such that C ^* is noetherian, then the global dimension of the coalgebra C coincides with the global dimension of the algebra C ^*.     

Open subgroups of

Let be a locally compact group and let denote the -algebra generated by left translation operators on . Let and be the spaces of almost periodic and weakly almost periodic functionals on the Fourier algebra , respectively. It is shown that if contains an open abelian subgroup, then (1) if and only if is norm dense in ; (2) is a -algebra if is norm dense in , where denotes the set of elements in with compact support. In particular, for any amenable locally compact group which contains an open abelian subgroup, has the dual Bohr approximation property and is a -algebra.

     

Extensions of Modules over Hopf Algebras Arising from Lie Algebras of Cartan Type

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( ) containing u( ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G _r ) u( ) have also been constructed, and the results are stated in this setting.     

Decomposition Numbers and Canonical Bases

We obtain some simple relations between decomposition numbers of quantized Schur algebras at an nth root of unity (over a field of characteristic 0). These relations imply that every decomposition number for such an algebra occurs as a decomposition number for some Hecke algebra of type A. We prove similar relations between coefficients of the canonical basis of the q-deformed Fock space representation of . It follows that these coefficients can all be expressed in terms of those of the global crystal basis of the irreducible subrepresentation generated by the vacuum vector. As a consequence, using the works of Ariki and Varagnolo and Vasserot, it is possible to give a new proof of Lusztig"s character formula for the simple U _v (sl _r )-modules at roots of unity, which does not involve representations of of negative level.     

On the Classification of q-Algebras

The problem is the classification of the ideals of free differential algebras, or the associated quotient algebras, the q-algebras; being finitely generated, unital C-algebras with homogeneous relations and a q-differential structure. This family of algebras includes the quantum groups, or at least those that are based on simple (super) Lie or Kac–Moody algebras. Their classification would encompass the so far incompleted classification of quantized (super) Kac–Moody algebras and of the (super) Kac–Moody algebras themselves. These can be defined as singular limits of q-algebras, and it is evident that to deal with the q-algebras in their full generality is more rational than the examination of each singular limit separately. This is not just because quantization unifies algebras and superalgebras, but also because the points q=1 and q=–1 are the most singular points in parameter space. In this Letter, one of two major hurdles in this classification program has been overcome. Fix a set of integers n ₁,...,n _k, and consider the space of homogeneous polynomials of degree n ₁ in the generator e ₁, and so on. Assume that there are no constants among the polynomials of lower degree, in any one of the generators; in this case all constants in the space have been classified. The task that remains, the more formidable one, is to remove the stipulation that there are no constants of lower degree.