Test elements for endomorphisms of free groups and algebras

Shirshov published six papers on Lie algebras in which he found the following results (in order of publication. 1953–1962):

Let F(x,y) be a relatively free algebra of rank 2 in some variety of algebras over a field K of characteristic 0. In this paper we consider the problem whether p(x,y) ∈ F(x,y) is a primitive element (i.e. an automorphic image of x): (i) If F(x,y)/(p(x,y)) ≅ F(z), the relatively free algebra of rank 1 (ii) If p(f,g) is primitive for some injective endomorphism (f,g) of F(x,y) (iii) If p(x,y) is primitive in a relatively free algebra of larger rank. These problems have positive solutions for polynomial algebras in two variables. We give the complete answer for the free metabelian associative and Lie algebras and some partial results for free associative algebras.

Bases, filtrations and module decompositions of free Lie algebras

The additive primitive length of an element [Formula: see text] of a relatively free algebra [Formula: see text] in a variety of algebras [Formula: see text] is equal to the minimal number [Formula: see text] such that [Formula: see text] can be presented as a sum of [Formula: see text] primitive elements. We give an upper bound for the additive primitive length of the elements in the [Formula: see text]-generated polynomial algebra over a field of characteristic 0, [Formula: see text]. The bound depends on [Formula: see text] and on the degree of the element. We show that if the field has more than two elements, then the additive primitive length in free [Formula: see text]-generated nilpotent-by-abelian Lie algebras is bounded by 5 for [Formula: see text] and by 6 for [Formula: see text]. If the field has two elements only, then our bounds are 6 for [Formula: see text] and 7 for [Formula: see text]. This generalizes a recent result of Ela Aydın for two-generated free metabelian Lie algebras. In all cases considered in the paper, the presentation of the elements as sums of primitive elements can be found effectively in polynomial time.

The Nowicki conjecture for relatively free algebras

We exhibit a class of Lie algebras (which contains, in particular, free polynilpotent Lie algebras) that have wild automorphisms, i.e., automorphisms which are not induced by automorphisms of a free Lie algebra.

Let R be an associative ring with identity, X a set of noncommuting variables, = {αx} x ∈ X a set of automorphisms αx of R and R {X} the -twisted free associative algebra on X over R. Let Y be another set of noncommuting variables, ℬ = {βy}y∈Y a set of automorphisms βy of R {X} and S = (R{X})ℬ {Y} the ℬ-twisted free associative algebra on Y over R{X}. Next, let X1 be a set of noncommuting variables, for each l = 1,2,…. We form the free associative algebra S1 = S{X1}on Xl over S and inductively, we form the free associative algebra Sl+1 = Sl{Xl+1} on Xl+1 over Sl, l = 1,2,….

Deformations and rigidity in varieties of Lie algebras

According to Racinet’s work, the scheme of double shuffle and regularization relations between cyclotomic analogs of multiple zeta values has the structure of a torsor over a pro-unipotent |$\mathbb Q$|-algebraic group |$\sf{DMR}_0$|⁠, which is an algebraic subgroup of a pro-unipotent |$\mathbb Q$|-algebraic group of outer automorphisms of a free Lie algebra. We show that the harmonic (stuffle) coproduct of double shuffle theory may be viewed as an element of a module over the above group, and that |$\sf{DMR}_0$| identifies with the stabilizer of this element. We identify the tangent space at origin of |$\sf{DMR}_0$| with the stabilizer Lie algebra of the harmonic coproduct, thereby obtaining an alternative proof of Racinet’s result stating that this space is a Lie algebra (the double shuffle Lie algebra).

The various characterizations of Lie polynomials obtained in Chapter 1 are extended to Lie series, in a straightforward way. These results become really interesting when one considers exponentials of Lie series. This leads naturally to the Campbell–Baker–Hausdorff formula. Section 3.2 is devoted to the canonical projections of the free associative algebra, especially the first one, whose image is the free Lie algebra and which is obtained by a logarithm. Section 3.3 is devoted to the computation of the coefficients of the Hausdorff series. A generalization of this series leads again to the canonical projection. The computation is done using descent numbers of permutations and shuffle algebra. Several properties of the coefficients are established: symmetries, generating functions, and recursions. The last section presents the original methods of Campbell, Baker, and Hausdorff, which explains the introduction of Bernoulli numbers. A closed formula is also given for the Hausdorff series.

In the work of M. Hall [1], a certain way of fixing a basis of a free Lie algebra is indicated. However, the concrete bases which one needs to construct, in order to solve certain problems, do not always fall into Hall's scheme. For instance, the basis of a free Lie algebra considered in the work [2] cannot be constructed using Hall's method. For this reason, in each such case it is necessary to reprove that a certain subset of a free Lie algebra is a basis. Below, we give a method that generalizes Hall's method for choosing a basis in a free Lie algebra.

In this work on g=Fn,2 , free 2-step nilpotent Lie algebra on n generators, we use the group of automorphisms to give a basis-free description of the Fourier Inversion Formula, thereby generalizing and strengthening an example discussed by Corwin & Greenleaf. In the Introduction we discuss Example 4.3.14 in Corwin & Greenleaf's book. It demonstrates how different bases of F3,2 lead to different inversion formulas. But the third "more" invariant formula describes Plancherel measure on a support expressed in terms of rotations, dilations, and translations. Actually it is not canonical since it still depends on choices of bases for F3,2 . Our goal is to redescribe Plancherel measure on a support expressed in terms of Aut*( g ). We accomplish this in the following two chapters. Chapter 2 provides a procedure for reparametrizing the family of generic orbits by establishing a 1-1 correspondence between the maximum-dimensional orbits and Ad*G \Aut*( g )/Stab(l0). Chapter 3 provides background material about relatively invariant measures. Then we prove that Plancherel measure, modeled on Ad*G \Aut*( g )/Stab(l0), is the essentially unique relatively invariant measure corresponding to a specific homomorphism. Chapter 4 demonstrates that there does not exist an Aut*g -invariant measure on Ad*G \Aut*( g )/Stab(l0) for the example g=F3,2 . Our explicit calculations for F3,2 are done in Chapter 5 and are in agreement with the results of the first four chapters. We start by finding an almost global coordinate patch for AdG\AdG · Stab*( Z*3 ); use this patch to construct a right and left Haar measure on this quotient space. Thus we get its modular function delta. A similar process applies to AdG\Aut( g ), thus obtaining its modular function Delta. Hence the ratio of Delta to delta is the restriction of any modular function for any relatively invariant measure on AdG\Aut( g )/Stab*( Z*3 ). Furthermore, we find a cross-section X for Ad G\Aut( g )/Stab*(l0) with general l0∈g*Max . Then we use X to verify the relative invariance for the measure corresponding to the example of Corwin and Greenleaf for g=F3,2 . Chapter 6 illustrates three general properties for Fn,2 and one additional result for linear algebra, which are used in Chapter 5.

Let R be a commutative integral unital domain and L a free noncommutative Lie algebra over R. In this article we show that the ring R and its action on L are 0-interpretable in L, viewed as a ring with the standard ring language $+ , \cdot ,0$. Furthermore, if R has characteristic zero then we prove that the elementary theory $Th\left( L \right)$ of L in the standard ring language is undecidable. To do so we show that the arithmetic ${\Bbb N} = \langle {\Bbb N}, + , \cdot ,0\rangle $ is 0-interpretable in L. This implies that the theory of $Th\left( L \right)$ has the independence property. These results answer some old questions on model theory of free Lie algebras.

Necklace rings and logarithmic functions

Equivalent constructions of nilpotent quadratic Lie algebras