Hyperbolic unit groups

In this paper we study the groups whose integral group rings have hyperbolic unit groups . We classify completely the torsion subgroups of and the polycyclic-by-finite subgroups of the group . Finally, we classify the groups for which the boundary of has dimension zero.

     

Group algebras of Kleinian type and groups of units

The algebras of Kleinian type are finite-dimensional semisimple rational algebras A such that the group of units of an order in A is commensurable with a direct product of Kleinian groups. We classify the Schur algebras of Kleinian type and the group algebras of Kleinian type. As an application, we characterize the group rings RG, with R an order in a number field and G a finite group, such that the group of units of RG is virtually a direct product of free-by-free groups.     

Commutative group algebras of -summable abelian groups

In this note we study the commutative modular and semisimple group rings of -summable abelian -groups, which group class was introduced by R. Linton and Ch. Megibben. It is proved that is -summable if and only if is -summable, provided is an abelian group and is a commutative ring with 1 of prime characteristic , having a trivial nilradical. If is a -summable -group and the group algebras and over a field of characteristic are -isomorphic, then is a -summable -group, too. In particular provided is totally projective of a countable length.

Moreover, when is a first kind field with respect to and is -torsion, is -summable if and only if is a direct sum of cyclic groups.

     

Construction of free subgroups in the group of units of modular group algebras

Let KGbe the group algebra of a p¹ -group Gover a field Kof characteristic p > 0, and let U(KG)be its group of units. If KGcontains a nontrivial bicyclic unit and if Kis not algebraic over its prime field, then we prove that the free product Z_p? Z_p? Z_pcan be embedded in U(KG).     

Automorphism groups of pointed Hopf algebras

The group of Hopf algebra automorphisms for a finite-dimensional semisimple cosemisimple Hopf algebra over a field k was considered by Radford and Waterhouse. In this paper, the groups of Hopf algebra automorphisms for two classes of pointed Hopf algebras are determined. Note that the Hopf algebras we consider are not semisimple Hopf algebras.      

Smallest graphs with given generalized quaternion automorphism group

For , a smallest graph whose automorphism group is isomorphic to the generalized quaternion group is constructed. If , then such a graph has vertices and edges. In the special case when , a smallest graph has 16 vertices but 44 edges.     

Group algebras with units satisfying a group identity II

We classify group algebras of torsion groups over a field of characteristic with units satisfying a group identity.

     

Lie algebras and separable morphisms in pro-affine algebraic groups

Let be an algebraically closed field of arbitrary characteristic, and let be a surjective morphism of connected pro-affine algebraic groups over . We show that if is bijective and separable, then is an isomorphism of pro-affine algebraic groups. Moreover, is separable if and only if (its differential) is surjective. Furthermore, if is separable, then .

     

On finite-dimensional copointed Hopf algebras over dihedral groups

We classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero such that its coradical is isomorphic to the algebra of functions ${\mathbb{k}}^{{\mathbb{D}}_m}$ over a dihedral group ${\mathbb{D}}_m$, with $m=4a\ge 12$. We obtain this classification by means of the lifting method, where we use cohomology theory to determine all possible deformations. Our result provides an infinite family of new examples of finite-dimensional copointed Hopf algebras over dihedral groups.     

Basic subgroups in modular abelian group algebras

Suppose F is a perfect field of char F = p ≠ 0 and G is an arbitrary abelian multiplicative group with a p-basic subgroup B and p-component G _p . Let FG be the group algebra with normed group of all units V(FG) and its Sylow p-subgroup S(FG), and let I _p (FG; B) be the nilradical of the relative augmentation ideal I(FG; B) of FG with respect to B. The main results that motivate this article are that 1 + I _p (FG; B) is basic in S(FG), and B(1 + I _p (FG; B)) is p-basic in V(FG) provided G is p-mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston J. Math. when G is p-primary. Thus the problem of obtaining a (p-)basic subgroup in FG is completely resolved provided that the field F is perfect. Moreover, it is shown that G _p (1 + I _p (FG; B))/G _p is basic in S(FG)/G _p , and G(1 + I _p (FG; B))/G is basic in V(FG)/G provided G is p-mixed. As consequences, S(FG) and S(FG)/G _p are both starred or divisible groups. All of the listed assertions enlarge in a new aspect affirmations established by us in Czechoslovak Math. J. (2002), Math. Bohemica (2004) and Math. Slovaca (2005) as well.     

Positive definite Minkowski Lie algebras and bi-invariant Finsler metrics on Lie groups

In this paper, we introduce the notion of a Minkowski Lie algebra, which is the natural generalization of the notion of a real quadratic Lie algebra (metric Lie algebra). We then study the positive definite Minkowski Lie algebras and obtain a complete classification of the simple ones. Finally, we present some applications of our results to Finsler geometry and give a classification of bi-invariant Finsler metrics on Lie groups. This work was supported by NSFC (No.10671096) and NCET of China.     

Fibonacci代数的Hochschild同调群

设Λd是Fibonacci代数,基于对Bardzell极小投射双模分解的细致分析,用组合的方法清晰地计算了Fibonacci代数Λd的各阶Hochschild同调群的维数.     

Group rings with metabelian unit groups

Let F be a field of odd characteristic and G a group. In 1991 Shalev established necessary and sufficient conditions so that the unit group of the group ring FG is metabelian when G is finite. Here, in the modular case, we do the same without restrictions on G. In particular, new cases emerge when G contains elements of infinite order.     

Sofic groups and direct finiteness

We construct an analogue of von Neumann's affiliated algebras for sofic group algebras over arbitrary fields. Consequently, we settle Kaplansky's direct finiteness conjecture for sofic groups.     

Nontriviality of the first Hochschild cohomology of some block algebras of finite groups

We show that for some finite group block algebras, with nontrivial defect groups, the first Hochschild cohomology is nontrivial. Along the way we obtain methods to investigate the nontriviality of the first Hochschild cohomology of some twisted group algebras.     

Semisimple Banach algebras generated by compact groups of operators on a Banach space

We prove that if τ is a strongly continuous representation of a compact group G on a Banach space X, then the weakly closed Banach algebra generated by the Fourier transforms with μ∈M(G) is a semisimple Banach algebra.