Volume 20, Issue 1

A q -analogue ζ q ( s ) of the Riemann zeta function ζ( s ) was studied in [Kaneko M., Kurokawa N. and Wakayama M.: A variation of Euler's approach to values of the Riemann zeta function. Kyushu J. Math. 57 (2003), 175–192] via a certain q -series of two variables. We introduce in a similar way a q -analogue of the Dirichlet L -functions and make a detailed study of them, including some issues concerning the classical limit of ζ q ( s ) left open in [Kaneko M., Kurokawa N. and Wakayama M.: A variation of Euler's approach to values of the Riemann zeta function. Kyushu J. Math. 57 (2003), 175–192]. We also examine a “crystal” limit (i.e. q ↓ 0) behavior of ζ q ( s ). The q -trajectories of the trivial and essential zeros of ζ( s ) are investigated numerically when q moves in (0, 1]. Moreover, conjectures for the crystal limit behavior of zeros of ζ q ( s ), which predict an interesting distribution of “trivial zeros” and an analogue of the Riemann hypothesis for a crystal zeta function, are given. 2000 Mathematics Subject Classification: 11M06.

Let G be a subgroup of the group GL ( V, F ) of automorphisms of a vector space V over a field F . By definition, the augmentation dimension of G , augdim F G for short, is the F -dimension of the vector subspace V (ω FG ), where ω FG is the augmentation ideal of the group ring FG . It is well known that G is finitary if and only if every finitely generated subgroup of G has finite augmentation dimension. In the antipodes of this concept, we define G to be antifinitary , if every proper subgroup of G of infinite augmentation dimension is finitely generated. In this paper we describe some generalized soluble antifinitary groups. 2000 Mathematics Subject Classification: 20F22; 20H20.

Isomorphisms are constructed between generalized Schur algebras in different degrees. The construction covers both the classical case (of general linear groups over infinite fields of arbitrary characteristic) and the quantized case (in type A , for any non-zero value of the quantum parameter q ). The construction does not depend on the characteristic of the underlying field or the choice of q ≠ 0. The proof combines a combinatorial construction with comodule structures and Ringel duality. Applications range from equivalences of categories to results on the structure and cohomology of Schur algebras to identities of decomposition numbers and also of p -Kostka numbers, in both cases reproving and generalizing row and column removal rules. 2000 Mathematics Subject Classification: 16G10, 16W30, 20G05, 20G15.

In this paper we define a class of stable equivalences, namely, the stable equivalences of adjoint type, and study the Hochschild cohomology groups of algebras that are linked by a stable equivalence of adjoint type. This notion of adjoint type is a special case of Morita type, covers the stable equivalence of Morita type for self-injective algebras, and thus includes the case where Broué's conjecture was made (see for instance [Broué M.: Equivalences of blocks of group algberas. In: Finite dimensional algebras and related topics (V. Dlab and L. L. Scott eds.). Kluwer, 1994, 1–26]). The main results in this paper are: Let A and B be two artin k -algebras such that A and B are projective over k , and let H n ( A ) and H n ( B ) be the n -th Hochschild cohomology groups of A and B , respectively. (1) If A and B are stably equivalent of adjoint type, then H n ( A ) ≃ H n ( B ) for all n ≥ 1. (2) If A and B are stably equivalent of Morita type, then the absolute values of Cartan determinants of A and B are equal. In particular, two cellular algebras over a field have the same Cartan determinant if they are stably equivalent of Morita type. 2000 Mathematics Subject Classification: 16G10, 16E30; 16G70, 18G05, 20J05.

We provide a solution to the isoperimetric problem in the Heisenberg group ℍ n when the competing sets belong to a restricted class of C 2 graphs. Within this restricted class we characterize the isoperimetric profiles as the bubble sets (1.5) (modulo nonisotropic dilations and left-translations). We also compute the isoperimetric constant. 2000 Mathematics Subject Classification: 53C17, 49K20.

In this paper we investigate Brill-Noether problems for stable sheaves on higher dimensional varieties. By means of the extension method, we deal with the non-emptiness and the birational types of Brill-Noether loci. We illustrate our result by explicit examples for certain projective bundles. 2000 Mathematics Subject Classification: 14J60, 14D20.

In this article, we show a truncated Second Main Theorem of meromorphic mappings from ℂ n into ℙ N (ℂ) for moving targets. The moving targets are only assumed to be nondegenerate. 2000 Mathematics Subject Classification: 32H30, 32A22; 30D35.

The purpose of this article is to extend the results in [AlAli M.; Hering C.; Neumann A. and Rawashdeh A.: On the existence of semifields of prime power order admitting free automorphism groups, to appear in J. Geom.] and to describe all semifields of order q 4 , q is an odd prime power greater than 3, admitting free automorphism groups isomorphic to Z 2 × Z 2 . 2000 Mathematics Subject Classification: 12K10, 51A40.