Powerfully nilpotent groups of maximal powerful class

In this paper we continue the study of powerfully nilpotent groups started in Traustason and Williams (J Algebra 522:80–100, 2019). These are powerful p-groups possessing a central series of a special kind. To each such group one can attach a powerful class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. The focus here is on powerfully nilpotent groups of maximal powerful class but these can be seen as the analogs of groups of maximal class in the class of all finite p-groups. We show that for any given positive integer r and prime $$p>r$$, there exists a powerfully nilpotent group of maximal powerful class and we analyse the structure of these groups. The construction uses the Lazard correspondence and thus we construct first a powerfully nilpotent Lie ring of maximal powerful class and then lift this to a corresponding group of maximal powerful class. We also develop the theory of powerfully nilpotent Lie rings that is analogous to the theory of powerfully nilpotent groups.     

Quotients of maximal class of thin lie algebras. the odd characteristic case

Among thin graded Lie algebras, which are particular instances of Lie algebras of finite width, there are many interesting objects, such as the graded Lie algebra associated to the Nottingham group. Among the factors of a thin algebra with respect to the terms of the lower central series, there is a greatest factor which is of maximal class. In thin Lie algebras associated to groups, this factor is metabelian.

In this paper we show that the same holds in general, provided the characteristic of the underlying field is odd. In another paper by the second author it is shown that this is not the case for characteristic two.     

Central extensions of Lie algebras graded by finite root systems

Lie algebras graded by finite irreducible reduced root systems have been classified up to central extensions by Berman and Moody, Benkart and Zelmanov, and Neher. In this paper we determine the central extensions of these Lie algebras and hence describe them completely up to isomorphism. Received: 22 May 1997 / in final form: 13 January 1999     

Homogeneous nilradicals over semigroup graded rings

In this paper we study the homogeneity of radicals defined by nilpotence or primality conditions, in rings graded by a semigroup S. When S is a unique product semigroup, we show that the right (and left) strongly prime and uniformly strongly prime radicals are homogeneous, and an even stronger result holds for the generalized nilradical. We further prove that rings graded by torsion-free, nilpotent groups have homogeneous upper nilradical. We conclude by showing that non-semiprime rings graded by a large class of semigroups must always contain nonzero homogeneous nilpotent ideals.     

带有限性条件Abel群的自同态环和自同构群

给出了带极大或极小条件的Abel群A的自同构群以及自同态环的相伴Lie环是可解或幂零的充要条件.同时也给出了群A=Q_(π1)⊕Q_(π2)⊕…⊕Q_(πr)的自同构群是可解或幂零的充要条件,以及群A的自同态环的相伴Lie环是可解或幂零的充要条件.     

The complexity of central series in nilpotent computable groups

The terms of the upper and lower central series of a nilpotent computable group have computably enumerable Turing degree. We show that the Turing degrees of these terms are independent even when restricted to groups which admit computable orders.     

Polynome de Hilbert-Samuel dans les algebres enveloppantes et les algebres de groupe

An analogue of the Hilbert-Samuel polynomial is considered with respect to the augmentation ideal of the enveloping algebra of a finite dimensional nilpotent Lie algebra and the group ring of a finitely generated, torsion-free nilpotent group. Then, the Hilbert series of finitely generated modules are rational.      

On Anosov automorphisms of nilmanifolds

The only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds, and the existence of such automorphisms is a really strong condition on the rational nilpotent Lie algebra determined by the lattice, so called an Anosov Lie algebra. We prove that n⊕?⊕n (s times, s≥2) has an Anosov rational form for any graded real nilpotent Lie algebra n having a rational form. We also obtain some obstructions for the types of nilpotent Lie algebras allowed, and use the fact that the eigenvalues of the automorphism are algebraic integers (even units) to show that the types (5,3) and (3,3,2) are not possible for Anosov Lie algebras.     

The commutator width of some relatively free lie algebras and nilpotent groups

We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free ?-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.     

Some nilpotent and locally nilpotent matrix groups

Say a division ring D is special if for every finite subset X of D there is a homomorphism of the subring of D generated by X into a division ring of finite Schur index a power of its positive characteristic. (D is not assumed to have positive characteristic.) We make a detailed study of nilpotent and locally nilpotent matrix groups over special division rings.

This has been done previously for a number of ‘special’ division rings arising from group algebras and Lie algebras, particularly by A.I. Lichtman. The present paper therefore presents single proofs of all these results. It also covers many division rings not considered before and produces some new results for those that have been considered before.

In view of the definition of ‘special’ it is not surprising that the proofs depend on a detailed analysis of the finite-dimensional case.     

Homology of multiplicative Lie rings

Two homology theories of multiplicative Lie rings are constructed, studied and compared with the usual homology theories of groups and Lie rings. Central extensions of multiplicative Lie rings are introduced. It is shown that the Steinberg multiplicative Lie ring of a ring is the direct product of the Steinberg group (viewed as a multiplicative Lie ring under the commutator bracket) and the Steinberg Lie ring.     

Rational forms of nilpotent Lie algebras and Anosov diffeomorphisms

We compute the set of all rational forms up to isomorphism for some real nilpotent Lie algebras of dimension 8. This is part of the classification of nilmanifolds admitting an Anosov diffeomorphism in dimension at most 8. Author’s address: FaMAF and CIEM, Universidad Nacional de Córdoba, Córdoba, Argentina     

无单位元的群分次环、Smash积及其一个应用

本文对于具有局部单位元的群分次环Ｒ证明了左Ｒ＃Ｃ^*－模范畴与分次左Ｒ－模范畴是同构的，并给出Ｒ是分次右完全环的一些充要条件．     

量子环面上一类导子李代数的结构和自同构群

本文研究量子环面上的一类导子李代数,它包含了Virasoro-Like代数及其q类似．首先证明了这 类导子李代数之间的同构一定是分次同构,并进一步给出了代数同构的充要条件及同构映射的具体表达 式,最后确定了该类李代数的自同构群．     

On the second cohomology group in semi-abelian categories

We develop some new aspects of cohomology in the context of semi-abelian categories: we establish a Hochschild-Serre 5-term exact sequence extending the classical one for groups and Lie algebras; we prove that an object is perfect if and only if it admits a universal central extension; we show how the second Barr-Beck cohomology group classifies isomorphism classes of central extensions; we prove a universal coefficient theorem to explain the relationship with homology.     

Representations and cohomology for Frobenius-Lusztig kernels

Let U_ζ be the quantum group (Lusztig form) associated to the simple Lie algebra g, with parameter ζ specialized to an ?-th root of unity in a field of characteristic p>0. In this paper we study certain finite-dimensional normal Hopf subalgebras U_ζ(G_r) of U_ζ, called Frobenius-Lusztig kernels, which generalize the Frobenius kernels G_r of an algebraic group G. When r=0, the algebras studied here reduce to the small quantum group introduced by Lusztig. We classify the irreducible U_ζ(G_r)-modules and discuss their characters. We then study the cohomology rings for the Frobenius-Lusztig kernels and for certain nilpotent and Borel subalgebras corresponding to unipotent and Borel subgroups of G. We prove that the cohomology ring for the first Frobenius-Lusztig kernel is finitely-generated when g has type A or D, and that the cohomology rings for the nilpotent and Borel subalgebras are finitely-generated in general.     

On the Hochschild cohomology ring of tensor products of algebras

We prove that, as Gerstenhaber algebras, the Hochschild cohomology ring of the tensor product of two algebras is isomorphic to the tensor product of the respective Hochschild cohomology rings of these two algebras, when at least one of them is finite dimensional. In case of finite dimensional symmetric algebras, this isomorphism is an isomorphism of Batalin–Vilkovisky algebras. As an application, we explain by examples how to compute the Batalin–Vilkovisky structure, in particular, the Gerstenhaber Lie bracket, over the Hochschild cohomology ring of the group algebra of a finite abelian group.     

On Solvability of Lie Rings with an Automorphism of Finite Order

A new criterion for a Lie ring with a semisimple automorphism of finite order to be solvable is proved. It generalizes the effective version of Winter's criterion obtained earlier by Khukhro and Shumyatsky and by Bergen and Grzeszczuk in replacing the ideal generated by a certain set by the subring generated by this set. The proof is inspired by the original theorem of Kreknin on solvability of Lie rings with regular automorphisms of finite order and is conducted mostly in terms of Lie rings graded by a finite cyclic group.     

Isomorphism of a Ring to the Endomorphism Ring of an Abelian Group

This paper presents necessary and sufficient conditions under which isomorphism of endomorphism rings of additive groups of arbitrary associative rings with 1 implies isomorphism of these rings. For a certain class of Abelian groups, we present a criterion which shows when isomorphism of their endomorphism rings implies isomorphism of these groups. We demonstrate necessary and sufficient conditions under which an arbitrary ring is the endomorphism ring of an Abelian group. This solves Problem 84 in L. Fuchs’ “Infinite Abelian Groups.”__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 231–234, 2003.