Invariants of simple algebras

We determine the group of invariants with values in Galois cohomology with coefficients of central simple algebras of degree at most 8 and exponent dividing 2. The work of A. S. Merkurjev has been supported by the NSF grant DMS #0652316.     

Levels and sublevels of composition algebras over \mathfrak{p}-adic function fields

In [7], the level and sublevel of composition algebras are studied, wherein these quantities are determined for those algebras defined over local fields. In this paper, the level and sublevel of composition algebras, of dimension 4 and 8 over rational function fields over local non-dyadic fields, are determined completely in terms of the local ramification data of the algebras. The proofs are based on the “classification” of quadratic forms over such fields, as is given in [8]. The first author gratefully acknowledges financial support provided through the European Community’s Human Potential Programme, under contract HPRN-CT-2002-00287 KTAGS, which made possible an enjoyable stay at Ghent University.     

On the socles of fully invariant subgroups of Abelian p-groups

The classification of the fully invariant subgroups of a reduced Abelian p-group is a difficult long-standing problem when one moves outside of the class of fully transitive groups. In this work we restrict attention to the socles of fully invariant subgroups and introduce a new class of groups which we term socle-regular groups; this class is shown to be large and strictly contains the class of fully transitive groups. The basic properties of such groups are investigated but it is shown that the classification of even this simplified class of groups, seems extremely difficult. Received: 4 September 2008     

Group gradings on diagonal algebras

We describe all group gradings on the diagonal algebra k ⁿ , where k is an arbitrary field. Received: 21 January 2008     

Hermitian modular forms congruent to 1 modulo p

For any natural number and any prime (mod 4) not dividing there is a Hermitian modular form of arbitrary genus n over that is congruent to 1 modulo p which is a Hermitian theta series of an O_L-lattice of rank p − 1 admitting a fixed point free automorphism of order p. It is shown that also for non-free lattices such theta series are modular forms. Received: 29 October 2008     

Endomorphism algebras of transitive permutation modules for p-groups

Let G be a finite p-group with subgroup H and k a field of characteristic p. We study the endomorphism algebra E = End_kG(k_H ↑^G), showing that it is a split extension of a nilpotent ideal by the group algebra kN_G(H)/H. We identify the space of endomorphisms that factor through a projective kG-module and hence the endomorphism ring of k_H ↑^G in the stable module category, and determine the Loewy structure of E when G has nilpotency class 2 and [G, H] is cyclic. Received: 3 November 2008     

Undecidability of the centers of groups and group algebras

Let k ≧ 3 be an integer or k = ∞ and let K be a field. There is a recursive family of finitely presented groups G_n over a fixed finite alphabet with solvable word problem such that

On SU(1,D)/[U(1,D),U(1,D)] for a quaternion division algebra D

Let D be a quaternion division algebra with an involution of the second kind. We show that the quotient group SU(1,D)/[U(1,D),U(1,D)] is nontrivial in general. For global fields, we completely determine this group. Received: 11 May 2007, Revised: 15 December 2007     

Noetherian Automorphisms of Groups

An automorphism α of a group G is called a noetherian automorphism if for each ascending chain

Algorithms for Singleton Attractor Detection in Planar and Nonplanar AND/OR Boolean Networks

Singleton attractor (also called fixed point) detection is known to be NP-hard even for AND/OR Boolean networks (AND/OR BNs in short, i.e., BNs consisting of AND/OR nodes), where BN is a mathematical model of genetic networks and singleton attractors correspond to steady states. In our recent paper, we developed an O(1.787ⁿ) time algorithm for detecting a singleton attractor of a given AND/OR BN where n is the number of nodes. In this paper, we present an O(1.757ⁿ) time algorithm with which we succeeded in improving the above algorithm. We also show that this problem can be solved in time, which is less than O((1 + ∈)ⁿ) for any positive constant ∈, when a BN is planar. A preliminary version of this paper has appeared in Proc. 3rd International Conference on Algebraic Biology (AB2008) [27].     

On linear operators preserving the set of positive polynomials

Following the classical approach of Pólya-Schur theory [13] we initiate in this paper the study of linear operators acting on and preserving either the set of positive univariate polynomials or similar sets of non-negative and elliptic polynomials. To Vladimir Igorevich Arnold with admiration     

Representation of nonnegative convex polynomials

We provide a specific representation of convex polynomials nonnegative on a convex (not necessarily compact) basic closed semi-algebraic set . Namely, they belong to a specific subset of the quadratic module generated by the concave polynomials that define . Received: 15 December 2007     

The automorphism group of a nonsplit metacyclic p-group

This note considers a finite group G = HK, which is a product of a subgroup H and a normal subgroup K, and determines subgroups of Aut G. The special case when G is a nonsplit metacyclic p-group, where p is odd, is then considered and the structure of its automorphism group Aut G is given. Received: 13 September 2007, Revised: 22 November 2007     

Notes on the minimal number of ramified primes in some l-extensions of Q

For a finite l-group G, let ram ^t (G) denote the minimal integer such that G can be realized as the Galois group of a tamely ramified extension of Q ramified only at ram ^t (G) finite primes. We study the upper bound of ram ^t (G) and give an improvement of the result of Plans. We also give the best bound of ram ^t (G) for all 3-groups G of order less than or equal to 3⁵. Received: 14 September 2007     

A simple example of two p-groups with the same automorphism group

It is an open problem whether there exist two different p-groups with the same automorphism group. In this paper, we give a positive answer to this problem by constructing a simple example. Received: 30 September 2008     

Automorphism groups of semidirect products

This paper shows that if is a semidirect product of finite groups, then if and only if and for all . As an application, we investigate the automorphism group of a split metacyclic p-group for odd p. The second author is supported by the Natural Science Foundation of China (10671058).     

Non-linear Lie conformal algebras with three generators

We classify certain non-linear Lie conformal algebras with three generators, which can be viewed as deformations of the current Lie conformal algebra of sℓ ₂. In doing so we discover an interesting 1-parameter family of non-linear Lie conformal algebras and the corresponding freely generated vertex algebras , which includes for d = 1 the affine vertex algebra of sℓ ₂ at the critical level k = –2. We construct free-field realizations of the algebras extending the Wakimoto realization of at the critical level, and we compute their Zhu algebras. Dedicated to our teacher Victor Kac on the occasion of his 65th birthday     

Bifunctional-elementary relation algebras

A relation algebra is bifunctional-elementary if it is atomic and for any atom a, the element a;1;a is the join of at most two atoms, and one of these atoms is bifunctional (an element x is bifunctional if ’). We show that bifunctional-elementary relation algebras are representable. Our proof combines the representation theorems for: pair-dense relation algebras given by R. Maddux; relation algebras generated by equivalence elements provided corresponding relativizations are representable by S. Givant; and strong-elementary relation algebras dealt with in our earlier work. It turns out that atomic pair-dense relation algebras are bifunctional elementary, showing that our theorem generalizes the representation theorem of atomic pair-dense relation algebras. The problem is still open whether the related classes of rather elementary, functional-elementary, and strong functional-elementary relation algebras are representable. Received July 15, 2007; accepted in final form March 17, 2008.