Free Nilpotent Research Articles

Poincaré series of character varieties for nilpotent groups

Abstract For any compact, connected Lie group G and any finitely generated nilpotent group Γ, we determine the cohomology of the path component of the trivial representation of the group character variety (representation space) Rep ⁢ ( Γ , G ) 1 {{\rm Rep}(\Gamma,G)_{1}} , with coefficients in a field 𝔽 {{\mathbb{F}}} with characteristic 0 or relatively prime to the order of the Weyl group W. We give explicit formulas for the Poincaré series. In addition, we study G-equivariant stable decompositions of subspaces X ⁢ ( q , G ) {{\rm X}(q,G)} of the free monoid J ⁢ ( G ) {J(G)} generated by the Lie group G, obtained from representations of finitely generated free nilpotent groups.

The Axiomatic Rank of Levi Classes

A Levi class L(ℳ) generated by a class ℳ of groups is a class of all groups in which the normal closure of each element belongs to ℳ. It is stated that there exist finite groups G such that a Levi class L(qG), where qG is a quasivariety generated by a group G, has infinite axiomatic rank. This is a solution for [The Kourovka Notebook, Quest. 15.36]. Moreover, it is proved that a Levi class L(ℳ), where ℳ is a quasivariety generated by a relatively free 2-step nilpotent group of exponent ps with a commutator subgroup of order p, p is a prime, p ≠ 2, s ≥ 2, is finitely axiomatizable.

Almost inner derivations of Lie algebras

We study almost inner derivations of Lie algebras, which were introduced by Gordon and Wilson in their work on isospectral deformations of compact solvmanifolds. We compute all almost inner derivations for low-dimensional Lie algebras, and introduce the concept of fixed basis vectors for proving that all almost inner derivations are inner for [Formula: see text]-step nilpotent Lie algebras determined by graphs, free [Formula: see text] and [Formula: see text]-step nilpotent Lie algebras, free metabelian nilpotent Lie algebras on two generators, almost abelian Lie algebras and triangular Lie algebras. On the other hand, we also exhibit families of nilpotent Lie algebras having an arbitrary large space of non-inner almost inner derivations.

Universal input of knapsack problem for groups

In this work we investigate the group version of the well known knapsack problem (KP) in class of nilpotent groups. We show that there exists the universal input for this problem. In other word it mean that there is the two step nilpotent group Gu and parametric input In, where n is natural parameter, such that for any torsion free nilpotent group G and input I for KP there exists a natural number m such that KP for group Gu on input Im is equivalent to KP for group G on input I.

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.

Lie bialgebra structures on 2-step nilpotent graph algebras

We generalize a result on the Heisenberg Lie algebra that gives restrictions to possible Lie bialgebra cobrackets on 2-step nilpotent algebras with some additional properties. For the class of 2-step nilpotent Lie algebras coming from graphs, we describe these extra properties in a very easy graph-combinatorial way. We exhibit applications for fn, the free 2-step nilpotent Lie algebra.

Free algebras in division rings with an involution

Some general criteria to produce explicit free algebras inside the division ring of fractions of skew polynomial rings are presented. These criteria are applied to some special cases of division rings with natural involutions, yielding, for instance, free subalgebras generated by symmetric elements both in the division ring of fractions of the group algebra of a torsion free nilpotent group and in the division ring of fractions of the first Weyl algebra.

Free 2-step nilpotent Lie algebras and indecomposable representations

ABSTRACTGiven an algebraically closed field F of characteristic 0 and an F-vector space V, let L(V) = V⊕Λ2(V) denote the free 2-step nilpotent Lie algebra associated to V. In this paper, we classify all uniserial representations of the solvable Lie algebra 𝔤 = ⟨x⟩⋉L(V), where x acts on V via an arbitrary invertible Jordan block.

Central automorphisms of free nilpotent Lie algebras

Let [Formula: see text] be the free nilpotent Lie algebra of finite rank [Formula: see text] [Formula: see text] and nilpotency class [Formula: see text] over a field of characteristic zero. We give a characterization of central automorphisms of [Formula: see text] and we find sufficient conditions for an automorphism of [Formula: see text] to be a central automorphism.

The q-tensor square of finitely generated nilpotent groups, q odd

In the present paper, the authors extend to the [Formula: see text]-tensor square [Formula: see text] of a group [Formula: see text], [Formula: see text] an odd positive integer, some structural results due to Blyth, Fumagalli and Morigi concerning the non-abelian tensor square [Formula: see text] ([Formula: see text]). The results are applied to the computation of [Formula: see text] for finitely generated nilpotent groups [Formula: see text], specially for free nilpotent groups of finite rank. We also generalize to all [Formula: see text] results of Bacon regarding an upper bound to the minimal number of generators of the non-abelian tensor square [Formula: see text] when [Formula: see text] is a [Formula: see text]-generator nilpotent group of class 2. We end by computing the [Formula: see text]-tensor squares of the free [Formula: see text]-generator nilpotent group of class 2, [Formula: see text]. This shows that the above mentioned upper bound is also achieved for these groups when [Formula: see text] odd.

Local Functional Equations for Submodule Zeta Functions Associated to Nilpotent Algebras of Endomorphisms

We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on such functional equations for ideal zeta functions of nilpotent Lie lattices. Via the Mal'cev correspondence, these results have corollaries pertaining to zeta functions enumerating normal subgroups of finite index in finitely generated nilpotent groups, most notably finitely generated free nilpotent groups of any given class.

Random walks and L\xe9vy processes as rough paths

We consider random walks and Lévy processes in a homogeneous group G. For all p > 0, we completely characterise (almost) all G-valued Lévy processes whose sample paths have finite p-variation, and give sufficient conditions under which a sequence of G-valued random walks converges in law to a Lévy process in p-variation topology. In the case that G is the free nilpotent Lie group over mathbb {R}^d, so that processes of finite p-variation are identified with rough paths, we demonstrate applications of our results to weak convergence of stochastic flows and provide a Lévy–Khintchine formula for the characteristic function of the signature of a Lévy process. At the heart of our analysis is a criterion for tightness of p-variation for a collection of càdlàg strong Markov processes.

Finite dimensional characteristic functions of Brownian rough path

The Brownian rough path is the canonical lifting of Brownian motion to the free nilpotent Lie group of order 2: Equivalently, it is a process taking values in the algebra of Lie polynomials of degree 2, which is described explicitly by the Brownian motion coupled with its area process. The aim of this article is to compute the finite dimensional characteristic functions of the Brownian rough path in ℝd and obtain an explicit formula for the case when d = 2

Algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups

We study algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups. A special attention is paid to free nilpotent groups and the groups UTn(Z) of unitriangular (n×n)-matrices over the ring Z of integers for arbitrary n. We observe that the sets of retracts of finitely generated nilpotent groups coincides with the sets of their algebraically closed subgroups. We give an example showing that a verbally closed subgroup in a finitely generated nilpotent group may fail to be a retract (in the case under consideration, equivalently, fail to be an algebraically closed subgroup). Another example shows that the intersection of retracts (algebraically closed subgroups) in a free nilpotent group may fail to be a retract (an algebraically closed subgroup) in this group. We establish necessary conditions fulfilled on retracts of arbitrary finitely generated nilpotent groups. We obtain sufficient conditions for the property of being a retract in a finitely generated nilpotent group. An algorithm is presented determining the property of being a retract for a subgroup in free nilpotent group of finite rank (a solution of a problem of Myasnikov). We also obtain a general result on existentially closed subgroups in finitely generated torsion-free nilpotent with cyclic center, which in particular implies that for each n the group UTn(Z) has no proper existentially closed subgroups.

Random Nilpotent Groups I

We study random nilpotent groups in the well-established style of random groups, by choosing relators uniformly among freely reduced words of (nearly) equal length and letting the length tend to infinity. Whereas random groups are quotients of a free group by such a random set of relators, random nilpotent groups are formed as corresponding quotients of a free nilpotent group. This model reveals new phenomena because nilpotent groups are not visible in the standard model of random groups (due to the sharp phase transition from infinite hyperbolic to trivial groups).

On the dimension of matrix embeddings of torsion-free nilpotent groups

Since the work of Jennings (1955), it is well-known that any finitely generated torsion-free nilpotent group can be embedded into unitriangular integer matrices UTN(Z) for some N. In 2006, Nickel proposed an algorithm to calculate such embeddings. In this work, we show that if UTn(Z) is embedded into UTN(Z) using Nickel's algorithm, then N≥2n/2−2 if the standard ordering of the Mal'cev basis (as in Nickel's original paper) is used. In particular, we establish an exponential worst-case running time of Nickel's algorithm.On the other hand, we also prove a general exponential upper bound on the dimension of the embedding by showing that for any torsion free, finitely generated nilpotent group the matrix representation produced by Nickel's algorithm has never larger dimension than Jennings' embedding. Moreover, when starting with a special Mal'cev basis, Nickel's embedding for UTn(Z) has only quadratic size. Finally, we consider some special cases like free nilpotent groups and Heisenberg groups and compare the sizes of the embeddings.

The isomorphism problem for torsion free nilpotent groups of Hirsch length at most 5

AbstractWe consider the isomorphism problem for the finitely generated torsion free nilpotent groups of Hirsch length at most five. We show how this problem translates to solving an explicitly given set of polynomial equations. Based on this, we introduce a canonical form for each isomorphism type of finitely generated torsion free nilpotent group of Hirsch length at most 5 and, using a variation of our methods, we give an explicit description of its automorphisms.

Free weak nilpotent minimum algebras

We give a combinatorial description of the finitely generated free weak nilpotent minimum algebras and provide explicit constructions of normal forms.

On the influence of fixed point free nilpotent automorphism groups

A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that \( [F,h]=F\) for all nonidentity elements \(h\in H\). Let FH be a Frobenius-like group with complement H of prime order such that \(C_F(H)\) is of prime order. Suppose that FH acts on a finite group G by automorphisms where \( (|G|,|H|)=1\) in such a way that \(C_G(F)=1.\) In the present paper we prove that the Fitting series of \(C_G(H)\) coincides with the intersections of \( C_G(H)\) with the Fitting series of G, and the nilpotent length of G exceeds the nilpotent length of \(C_G(H)\) by at most one. As a corollary, we also prove that for any set of primes \(\pi \), the upper \(\pi \)-series of \( C_G(H)\) coincides with the intersections of \(C_G(H)\) with the upper \(\pi \) -series of G, and the \(\pi \)- length of G exceeds the \(\pi \)-length of \( C_G(H)\) by at most one.

Extensions and automorphisms of Lie algebras

Let [Formula: see text] be a short exact sequence of Lie algebras over a field [Formula: see text], where [Formula: see text] is abelian. We show that the obstruction for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text] lies in the Lie algebra cohomology [Formula: see text]. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text], where [Formula: see text] is a free nilpotent Lie algebra of rank [Formula: see text] and step [Formula: see text].