Lie Rings Research Articles

IDEAL ZETA FUNCTIONS ASSOCIATED TO A FAMILY OF CLASS-2-NILPOTENT LIE RINGS

Abstract We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-$2$-nilpotent Lie rings, introduced in M. N. Berman, B. Klopsch and U. Onn (A family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions, Math. Z. 290 (2018), 909935), in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question.

Two remarks on Lie rings of $2\times 2$ matrices over commutative associative rings

Let $C$ be an associative commutative ring with 1. If $a\in C$, then $aC$ denotes the principal ideal generated by $a$. Let $l, m, n$ be nonzero elements of $C$ such that $mn \in lC$. The set of matrices $\left( \begin{smallmatrix} a_{11} & a_{12} a_{21} & -a_{11} \end{smallmatrix} \right) $, where $a_{11}\in lC$, $a_{12}\in mC$, $a_{21}\in nC$, forms a Lie ring under Lie multiplication and matrix addition. The paper studies properties of these Lie rings.

Homology of Free Nilpotent Lie Rings

Some results obtained by the calculations of the integral homology of free nilpotent Lie algebras Hi (L(x1, x2,. . . , xr)/γN+1) in the system for computational discrete algebra GAP are presented. The attention is focused on the occurrence of unexpected torsion in this homology, similar to one that arises for a 4-generated free nilpotent group of class 2. The main result is that even for the case of two generators, the torsion occurs in the fourth integral homology when the nilpotency class is 5. Moreover, only 7-torsion occurs, and no others. Namely, there is an isomorphism H4(L(x1, x2)/γ6) ≅ ℤ85 ⨁ ℤ /7ℤ. We also give an explicit mathematical proof of the fact that H4(L(x1, x2, x3, x4)/γ3) has nontrivial 3-torsion. Bibliography: 6 titles.

Bogomolov multiplier and the Lazard correspondence

In this paper, we extend the notion of CP covers for groups to the class of Lie algebras, and show that despite the case of groups, all CP covers of a Lie algebra are isomorphic. Moreover we show that CP covers of groups and Lie rings which are in Lazard correspondence, are in Lazard correspondence too, and the Bogomolov multipliers of the group and the Lie ring are isomorphic.

Lie automorphic loops under half-automorphisms

Automorphic loops or [Formula: see text]-loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. Given a Lie ring [Formula: see text] we can define an operation [Formula: see text] such that [Formula: see text] is an [Formula: see text]-loop. We call it Lie automorphic loop. A half-isomorphism [Formula: see text] between multiplicative systems [Formula: see text] and [Formula: see text] is a bijection from [Formula: see text] onto [Formula: see text] such that [Formula: see text] for any [Formula: see text]. It was shown by [W. R. Scott, Half-homomorphisms of groups, Proc. Amer. Math. Soc. 8 (1957) 1141–1144] that if [Formula: see text] is a group then [Formula: see text] is either an isomorphism or an anti-isomorphism. This was used to prove that a finite group is determined by its group determinant. Here, we show that every half-automorphism of a Lie automorphic loop of odd order is either an automorphism or an anti-automorphism.

The special linear group for nonassociative rings

Abstract We extend to arbitrary rings a definition of the octonion special linear group due to Baez. At the infinitesimal level, we get a Lie ring, which we describe over some large classes of rings, including all associative rings and all algebras over a field. As a corollary, we compute all the groups Baez defined.

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.

Derivations of differentially semiprime rings

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

Integrable Two-Dimensional Lattices. Characteristic Lie Rings and Classification

This paper is devoted to the problem of classification of integrable nonlinear models with three independent variables. The classification algorithm based on the notion of characteristic Lie rings is applied to a class of two-dimensional lattices of hydrodynamic type. By imposing appropriate cutting-off boundary conditions, we reduce the lattice to a system of hyperbolic equations, which is assumed to be a Darboux integrable system. As a result, we found a new integrable lattice.

On the stable Andreadakis problem

Let Fn be the free group on n generators. Consider the group IAn of automorphisms of Fn acting trivially on its abelianization. There are two canonical filtrations on IAn: the first one is its lower central series Γ⁎; the second one is the Andreadakis filtration A⁎, defined from the action on Fn. In this paper, we establish that the canonical morphism between the associated graded Lie rings L(Γ⁎) and L(A⁎) is stably surjective. We then investigate a p-restricted version of the Andreadakis problem. A calculation of the Lie algebra of the classical congruence group is also included.

Torsion table for the lie algebra

We study the Lie ring of all strictly upper-triangular matrices with entries in . Its complete homology for is computed. We prove that every pm-torsion appears in for . For , Dwyer proved that the bound is sharp, i.e., there is no p-torsion in when prime . In general, for the bound is not sharp, as we show that there is 8-torsion in . As a sideproduct, we derive the known result, that the ranks of the free part of are the Mahonian numbers (=number of permutations of with k inversions), using a different approach than Kostant. Furthermore, we determine the algebra structure (cup products) of .

The Wells exact sequence for the automorphism group of a Lie ring extension

We give an explicit description of the Wells map for the automorphism group of a Lie ring extension. Using this map, we construct an exact sequence for the automorphism group of a Lie ring extension similar to that for group extensions.

Isoclinism of pairs of Lie rings

The concept of isoclinism on the class of groups was introduced by Philip Hall in 1940. Then, in 1993, Moneyhun used this concept on Lie algebras. In this paper, we study the notion of isoclinism on pairs of Lie rings, and some results are obtained when two isoclinic pairs of Lie rings are given. In particular, we prove that every isoclinism class of pairs of Lie rings contains at least one stem pair.

A 5-Engel associative algebra whose group of units is not 5-Engel

Let R be an associative ring with unity and let [R] and U(R) denote the associated Lie ring (with [a,b]=ab−ba) and the group of units of R, respectively. In 1983 Gupta and Levin proved that if [R] is a nilpotent Lie ring of class c then U(R) is a nilpotent group of class at most c. The aim of the present note is to show that, in general, a similar statement does not hold if [R] is n-Engel. We construct an algebra R over a field of characteristic ≠2,3 such that the Lie algebra [R] is 5-Engel but the group U(R) is not.

Color Lie rings and PBW deformations of skew group algebras

We investigate color Lie rings over finite group algebras and their universal enveloping algebras. We exhibit these universal enveloping algebras as PBW deformations of skew group algebras: Every color Lie ring over a finite group algebra with a particular Yetter–Drinfeld structure has a universal enveloping algebra that is a quantum Drinfeld orbifold algebra. Conversely, every quantum Drinfeld orbifold algebra of a particular type arising from the action of an abelian group is the universal enveloping algebra of some color Lie ring over the group algebra. One consequence is that these quantum Drinfeld orbifold algebras are braided Hopf algebras.

Description of 2-local and local derivations on some Lie rings of skew-adjoint matrices

ABSTRACTIn the present paper, we prove that every 2-local inner derivation on the Lie ring of skew-symmetric matrices over a commutative ring is an inner derivation. We also apply our technique to various Lie algebras of infinite-dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. A similar technique is applied to the same Lie algebras and proved that every local spatial derivation on such algebras is a spatial derivation.

Derivations in differentially prime rings

Earlier properties of Lie rings [Formula: see text] of derivations in commutative differentially prime rings [Formula: see text] was investigated by many authors. We study Lie rings [Formula: see text] in the non-commutative case and shown that if [Formula: see text] is a [Formula: see text]-prime ring of characteristic [Formula: see text], then [Formula: see text] is a prime Lie ring or [Formula: see text] is a commutative ring.

Enumerating graded ideals in graded rings associated to free nilpotent Lie rings

We compute the zeta functions enumerating graded ideals in the graded Lie rings associated with the free d-generator Lie rings $$\mathfrak {f}_{c,d}$$ of nilpotency class c for all $$c\leqslant 2$$ and for $$(c,d)\in \{(3,3),(3,2),(4,2)\}$$ . We apply our computations to obtain information about $$\mathfrak {p}$$ -adic, reduced, and topological zeta functions, in particular pertaining to their degrees and some special values.

Rings with simple Lie rings of Lie and Jordan derivations

Let [Formula: see text] be an associative ring. We characterize rings [Formula: see text] with simple Lie ring [Formula: see text] of all Lie derivations, reduced noncommutative Noetherian ring [Formula: see text] with the simple Lie ring [Formula: see text] of all derivations and obtain some properties of [Formula: see text]-torsion-free rings [Formula: see text] with the simple Lie ring [Formula: see text] of all Jordan derivations.

On a class of Lie rings of 2 × 2 matrices over associative commutative rings

Let k be an arbitrary associative and commutative ring. If are subgroups of the additive group of k such that , then matrices with form a subring of the Lie ring of all matrices over k whose trace is 0. The paper studies certain properties of these Lie rings. As an application of the notion introduced, the description of Lie rings lying between and , K an integral quadratic extension of , is given.