Free Lie Research Articles

Coadjoint Orbits and Time-Optimal Problems for Step-$$2$$ Free Nilpotent Lie Groups

The coadjoint orbits and the Casimir functions are described for free nilpotent Lie groups of step $$2$$ . The symplectic foliation consists of affine subspaces in the Lie coalgebra. Left-invariant time-optimal problems are considered on Carnot groups of step $$2$$ for which the set of admissible velocities is a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. The first integrals of the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle are described. For two-dimensional coadjoint orbits, the constancy and periodicity properties of the solutions of this subsystem, as well as the phase flow, are described.

IA-automorphisms of relatively free nilpotent torsion-free groups and Lie algebras

For a free metabelian and nilpotent group G of finite rank n and class c, with and we show that for all and we extend this result to several torsion-free nilpotent varieties of groups. By our methods, which are based on a connection of automorphisms of groups with automorphisms of Lie algebras, we also show the analogous result for several relatively free nilpotent Lie algebras.

Periodic Time-Optimal Controls on Two-Step Free-Nilpotent Lie Groups

For two-step free nilpotent Lie algebras, we describe symplectic foliations and Casimir functions. A left-invariant time-optimal problem is considered in which the set of admissible controls is given by a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. We describe integrals for the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle. The properties of solutions to this system for low ranks of the Poisson bivector are described.

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.

On Marginal Automorphisms of Free Nilpotent Lie Algebras

Let L be the free nilpotent Lie algebra of finite rank over a field of characteristic zero. We define the concepts of marginal ideals and marginal automorphisms of L, and we give some results on marginal automorphisms.

Degenerations of graph Lie algebras

In this work we associate a Lie algebra to every finite simple graph and obtain a necessary and sufficient condition for the existence of degenerations between these Lie algebras. As a corollary, we obtain that all these algebras belong to the irreducible component within the variety of 2-step nilpotent Lie algebras, given by the orbit closure of the free 2-step nilpotent Lie algebra.

Prime and Homogeneous Rings and Algebras

Let ℳ be a structure of a signature Σ. For any ordered tuple $$ \overline{a}=\left({a}_1,\dots, {a}_{\mathrm{n}}\right) $$ of elements of ℳ, $$ {\mathrm{tp}}^{\mathcal{M}}\left(\overline{a}\right) $$ denotes the set of formulas θ(x1, …, xn) of a first-order language over Σ with free variables x1, . . . , xn such that $$ \mathcal{M}\left|=\theta \left({a}_1,\dots, {a}_n\right)\right. $$. A structure ℳ is said to be strongly ω-homogeneous if, for any finite ordered tuples $$ \overline{a} $$ and $$ \overline{b} $$ of elements of ℳ, the coincidence of $$ {\mathrm{tp}}^{\mathcal{M}}\left(\overline{a}\right) $$ and $$ {\mathrm{tp}}^{\mathrm{M}}\left(\overline{b}\right) $$ implies that these tuples are mapped into each other (componentwise) by some automorphism of the structure ℳ. A structure ℳ is said to be prime in its theory if it is elementarily embedded in every structure of the theory Th (ℳ). It is proved that the integral group rings of finitely generated relatively free orderable groups are prime in their theories, and that this property is shared by the following finitely generated countable structures: free nilpotent associative rings and algebras, free nilpotent rings and Lie algebras. It is also shown that finitely generated non-Abelian free nilpotent associative algebras and finitely generated non-Abelian free nilpotent Lie algebras over uncountable fields are strongly ω-homogeneous.

Controllability of affine systems on free Nilpotent Lie groups Gm,ᵣ

Controllability properties of affine control systems on free nilpotent Lie groups are examined and controllability of affine systems on thiskind of Lie groups are characterized by the help of their associated bilinear parts. In order to show this, an automorphism in the algebra level is found, authomosrpism orbit of the system is calculated and its properties are studied.

The test rank of a solvable product of free abelian Lie algebras

Let [Formula: see text] be the [Formula: see text]th solvable product of free abelian Lie algebras of finite rank. We prove that the test rank of [Formula: see text] is one less than the number of the factors. We also give a test set for endomorphisms of [Formula: see text].

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.

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

A Presentation of the Free Lie Algebra M2,m,3

Let M2,m,3 be a free solvable nilpotent Lie algebra of rank 2 and nilpotency class m - 1. We show that M2,m,3 admits a minimal presentation whose set of defining relators consists of certain types of basic commutators using techniques in Grobner-Shirshov basis theory.

Automorphisms of free metabelian Lie algebras

We give a sharpening of a result of Bryant and Drensky [R. M. Bryant and V. Drensky, Dense subgroups of the automorphism groups of free algebras, Canad. J. Math. 45(6) (1993) 1135–1154] for the automorphism group [Formula: see text] of a free metabelian Lie algebra [Formula: see text], with [Formula: see text]. In particular, we prove that the subgroup of [Formula: see text] generated by [Formula: see text] and two more IA-automorphisms is dense in [Formula: see text] and, for [Formula: see text], the subgroup generated by [Formula: see text] and one more IA-automorphism is dense in [Formula: see text].

A Note on the $\gamma$-Coefficients of the Tree Eulerian Polynomial

We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. A classical product formula shows that this polynomial factors completely over the integers. From this product formula it can be concluded that this polynomial has positive coefficients in the $\gamma$-basis and we show that a formula for these coefficients can also be derived. We discuss various combinatorial interpretations of these coefficients in terms of leaf-labeled binary trees and in terms of the Stirling permutations introduced by Gessel and Stanley. These interpretations are derived from previous results of Liu, Dotsenko-Khoroshkin, Bershtein-Dotsenko-Khoroshkin, González D'León-Wachs and Gonzláez D'León related to the free multibracketed Lie algebra and the poset of weighted partitions.

Ideal growth in metabelian Lie P-algebras

Consider a finitely generated restricted Lie algebra L over the finite field Fq and, given n ≥ 0, denote the number of restricted ideals H ⊂ L with $${\dim _{{F_q}}}$$ L/H = n by cn(L). We show for the free metabelian restricted Lie algebra L of finite rank that the ideal growth sequence grows superpolynomially; namely, there exist positive constants λ1 and λ2 such that $${q^{{\lambda _1}{n^2}}} \leqslant {c_n}\left( L \right) \leqslant {q^{{\lambda _2}{n^2}}}$$ for n large enough.

Balloons and Hoops and their Universal Finite-Type Invariant, BF Theory, and an Ultimate Alexander Invariant

Balloons are 2D spheres. Hoops are 1D loops. Knotted balloons and hoops (KBH) in 4-space behave much like the first and second homotopy groups of a topological space—hoops can be composed as in π 1, balloons as in π 2, and hoops “act” on balloons as π 1 acts on π 2. We observe that ordinary knots and tangles in 3-space map into KBH in 4-space and become amalgams of both balloons and hoops. We give an ansatz for a tree and wheel (that is, free Lie and cyclic word)-valued invariant ζ of (ribbon) KBHs in terms of the said compositions and action and we explain its relationship with finite-type invariants. We speculate that ζ is a complete evaluation of the BF topological quantum field theory in 4D. We show that a certain “reduction and repackaging” of ζ is an “ultimate Alexander invariant” that contains the Alexander polynomial (multivariable, if you wish), has extremely good composition properties, is evaluated in a topologically meaningful way, and is least wasteful in a computational sense. If you believe in categorification, that should be a wonderful playground.

On Generators of the Tame Automorphism Group of Free Metabelian Lie Algebras

We prove that the tame automorphism group TAut(M n ) of a free metabelian Lie algebra M n in n variables over a field k is generated by a single nonlinear automorphism modulo all linear automorphisms if n ≥ 4 except the case when n = 4 and char(k) ≠ 3. If char(k) = 3, then TAut(M 4) is generated by two automorphisms modulo all linear automorphisms. We also prove that the tame automorphism group TAut(M 3) cannot be generated by any finite number of automorphisms modulo all linear automorphisms.

Commutator Width of Elements in a Free Metabelian Lie Algebra

Let M(A) be a free metabelian Lie algebra with a finite generating set A over an algebraically closed field F of characteristic zero, in which the problem of there being solutions to a system of linear equations is decided algorithmically, and let M′(A) be the derived subalgebra of M(A). We present an algorithm for finding width of elements in M′(A).

Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two

AbstractLet F2n;2 be the free nilpotent Lie group of step two on 2n generators, and let P denote the affine automorphism group of F2n;2. In this article the theory of continuous wavelet transformon F2n;2 associated with P is developed, and then a type of radial wavelet is constructed. Secondly, the Radon transform on F2n;2 is studied, and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform; the others are from the group Fourier transform. By using wavelet transforms we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. In particular, if n = 1, F2;2 is the 3-dimensional Heisenberg group H1, the inversion formula of the Radon transform is valid, which is associated with the sub-Laplacian on F2;2. This result cannot be extended to the case n ≥ 2.