Free Lie Algebra Research Articles

The hypercentral structure of the group of unitriangular automorphisms of a free metabelian Lie algebra

We describe the hypercentral structure of the group of unitriangular automorphisms of a free metabelian Lie algebra over an arbitrary field. Using it, we prove that this group admits no faithful representation by matrices over a field provided that the algebra rank is at least four.

An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications

We provide a new algorithm for generating the Baker–Campbell–Hausdorff (BCH) series Z=log(eXeY) in an arbitrary generalized Hall basis of the free Lie algebra L(X,Y) generated by X and Y. It is based on the close relationship of L(X,Y) with a Lie algebraic structure of labeled rooted trees. With this algorithm, the computation of the BCH series up to degree of 20 [111 013 independent elements in L(X,Y)] takes less than 15min on a personal computer and requires 1.5Gbytes of memory. We also address the issue of the convergence of the series, providing an optimal convergence domain when X and Y are real or complex matrices.

Anti-commutative Gröbner-Shirshov basis of a free Lie algebra

The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct construction, that is, first he started with a linear space spanned by Hall words, then defined the Lie product of Hall words and finally checked that the product yields the Lie identities. In this paper, we give a Gröbner-Shirshov basis for a free Lie algebra. As an application, by using the Composition-Diamond lemma established by Shirshov in 1962 for free anti-commutative (non-associative) algebras, we provide another method different from that of M. Hall to construct a basis of a free Lie algebra.

Strictly nontame primitive elements of the free metabelian Lie algebra of rank 3

We prove that the free metabelian Lie algebra M 3 of rank 3 over an arbitrary field K admits strictly nontame primitive elements.

Computing the test rank of a free solvable Lie algebra

Computing the test rank of a free solvable Lie algebra

Enumeration of maximal subalgebras in free restricted lie algebras

Given a finitely generated restricted Lie algebra L over the finite field \( \mathbb{F}_q \), and n ≥ 0, denote by an(L) the number of restricted subalgebras H ⊆ L with \( \dim _{\mathbb{F} _q} \)L/H = n. Denote by an(L) the number of the subalgebras satisfying the maximality condition as well. Considering the free restricted Lie algebra L = Fd of rank d ≥ 2, we find the asymptotics of an(Fd) and show that it coincides with the asymptotics of an(Fd) which was found previously by the first author. Our approach is based on studying the actions of restricted algebras by derivations on the truncated polynomial rings. We establish that the maximal subalgebras correspond to the so-called primitive actions. This means that “almost all” restricted subalgebras H ⊂ Fd of finite codimension are maximal, which is analogous to the corresponding results for free groups and free associative algebras.

Noetherianness of wreath products of Abelian Lie algebras with respect to equations of universal enveloping algebra

It is proved that a wreath product of two Abelian finite-dimensional Lie algebras over a field of characteristic zero is Noetherian w.r.t. equations of a universal enveloping algebra. This implies that an index 2 soluble free Lie algebra of finite rank, too, has this property.

Schwinger–Dyson operators as invariant vector fields on a matrix model analog of the group of loops

For a class of large-N multimatrix models, we identify a group G that plays the same role as the group of loops on space-time does for Yang–Mills theory. G is the spectrum of a commutative shuffle-deconcatenation Hopf algebra that we associate with correlations. G is the exponential of the free Lie algebra. The generating series of correlations is a function on G and satisfies quadratic equations in convolution. These factorized Schwinger–Dyson or loop equations involve a collection of Schwinger–Dyson operators, which are shown to be right-invariant vector fields on G, one for each linearly independent primitive of the Hopf algebra. A large class of formal matrix models satisfying these properties are identified, including as special cases, the zero momentum limits of the Gaussian, Chern–Simons, and Yang–Mills field theories. Moreover, the Schwinger–Dyson operators of the continuum Yang–Mills action are shown to be right-invariant derivations of the shuffle-deconcatenation Hopf algebra generated by sources labeled by position and polarization.

Lie theory for Hopf operads

The present article takes advantage of the properties of algebras in the category of S -modules (twisted algebras) to investigate further the fine algebraic structure of Hopf operads. We prove that any Hopf operad P carries naturally the structure of a comonoid in the category of twisted P -algebras. Many properties of Hopf algebraic structures are then shown to be encapsulated in this remarquable structure. In particular, various classical theorems of the theory of free Lie algebras are lifted to arbitrary Hopf operads.

Automorphisms of the endomorphism semigroups of free linear algebras of homogeneous varieties

We consider homogeneous varieties of linear algebras over an associative–commutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F = F ( x 1 , … , x n ) be a free algebra of some variety Θ of linear algebras over K freely generated by a set X = { x 1 , … , x n } , End F be the semigroup of endomorphisms of F, and Aut End F be the group of automorphisms of the semigroup End F. We investigate the structure of the group Aut End F and its relation to the algebraic and categorical equivalence of algebras from Θ. We define a wide class of R 1 MF -domains containing, in particular, Bezout domains, unique factorization domains, and some other domains. We show that every automorphism Φ of semigroup End F, where F is a free finitely generated Lie algebra over an R 1 MF -domain, is semi-inner. This solves the Problem 5.1 left open in [G. Mashevitzky, B. Plotkin, E. Plotkin, Automorphisms of the category of free Lie algebras, J. Algebra 282 (2004) 490–512]. As a corollary, semi-inner character of all automorphisms of the category of free Lie algebras over R 1 MF -domains is obtained. Relations between categorical and geometrical equivalence of Lie algebras over R 1 MF -domains are clarified. The group Aut End F for the variety of m-nilpotent associative algebras over R 1 MF -domains is described. As a consequence, a complete description of the group of automorphisms of the full matrix semigroup of n × n matrices over R 1 MF -domains is obtained. We give an example of the variety Θ of linear algebras over a Dedekind domain such that not all automorphisms of Aut End F are quasi-inner. The results obtained generalize the previous studies of various special cases of varieties of linear algebras over infinite fields.

Novikov algebras and Novikov structures on Lie algebras

We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov structure. On the other hand we show that any free three-step nilpotent Lie algebra admits a Novikov structure. We study the existence question also for Lie algebras of triangular matrices. Finally we show that there are families of Lie algebras of arbitrary high solvability class which admit Novikov structures.

THE FOX PROBLEM FOR FREE RESTRICTED LIE ALGEBRAS

Let L be a free restricted Lie algebra and R a restricted ideal of L. Denote by u(L) the restricted enveloping algebra of L and by ω(L) the associative ideal of u(L) generated by L. The purpose of this paper is to identify the subalgebra R ∩ ωn(L)ω(R) in terms of R only. This problem is the analogue of the Fox problem for free groups.

ON THE AUTOMORPHISM GROUP OF A FREE METABELIAN LIE ALGEBRA

Let K be a field of any characteristic. We prove that a free metabelian Lie algebra M3of rank 3 over K admits wild automorphisms. Moreover, the subgroup I Aut M3of all automorphisms identical modulo the derived subalgebra [Formula: see text] cannot be generated by any finite set of IA-automorphisms together with the sets of all inner and all tame IA-automorphisms. In the case if K is finite the group Aut M3cannot be generated by any finite set of automorphisms together with the sets of all tame, all inner automorphisms and all one-row automorphisms. We present an infinite set of wild IA-automorphisms of M3which generates a free subgroup F∞modulo normal subgroup generated by all tame, all inner and all one-row automorphisms of M3.

On the [formula omitted]-module structure of the noncommutative harmonics

Using a noncommutative analog of Chevalley's decomposition of polynomials into symmetric polynomials times coinvariants due to Bergeron, Reutenauer, Rosas, and Zabrocki we compute the graded Frobenius characteristic for their two sets of noncommutative harmonics with respect to the left action of the symmetric group (acting on variables). We use these results to derive the Frobenius series for the enveloping algebra of the derived free Lie algebra in n variables.

The homotopy Lie algebra of the complements of subspace arrangements with geometric lattices

Let A be a geometric arrangement such that codim(x) > 1 for every x in A. We prove that, if the complement space M(A) is rationally hyperbolic, then there exists an injective from a free Lie algebra L(u,v) to the homotopy Lie algebra of M(A).

Some subalgebras of indefinite type Kac–Moody Lie algebras

We show that every Kac–Moody Lie algebra of indefinite type contains a subalgebra with a Dynkin diagram having two adjacent vertices whose edge labels multiply to a number greater than or equal to five. Consequently, every Kac–Moody algebra of indefinite type contains a subalgebra of strictly hyperbolic type, and a free Lie algebra of rank two.

Tanaka prolongation of free Lie algebras

With the exception of the three step real free Lie algebra on two generators, all real free Lie algebras of step at least three are shown to have trivial Tanaka prolongation. This result, together with the known results concerning the step two real free Lie algebras and the step three real free Lie algebra on two generators, gives a complete list of Tanaka prolongations for real free Lie algebras.

Bases, filtrations and module decompositions of free Lie algebras

We use the technique known as elimination to devise some new bases of the free Lie algebra which (like classical Hall bases) consist of Lie products of left normed basic Lie monomials. Our bases yield direct decompositions of the homogeneous components of the free Lie algebra with direct summands that are particularly easy to describe: they are tensor products of metabelian Lie powers. They also give rise to new filtrations and decompositions of free Lie algebras as modules for groups of graded algebra automorphisms. In particular, we obtain some new decompositions for free Lie algebras and free restricted Lie algebras over fields of positive characteristic.

A Lie Theoretic Approach to Renormalization

Motivated by recent work of Connes and Marcolli, based on the Connes–Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and on properties of Hopf algebras encapsulated in the notion of associated descent algebras. Besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures, the new techniques give rise to an algebraic approach to the Galois theory of renormalization. In particular, they do not depend on the geometry underlying the case of dimensional regularization and the Riemann–Hilbert correspondence. This is illustrated with a discussion of the BPHZ renormalization scheme.

THE EQUATION [x,u] + [y,v] = 0 IN FREE LIE ALGEBRAS

We investigate equations of the form [x,u] + [y,v] = 0 over a free Lie algebra L. In the case where u and v are free generators of L, we exhibit two series of solutions, we work out the dimensions of the homogeneous components of the solution space, and we determine its radical. In the general case we show that the results on free generator coefficients are sufficient to obtain the solution space up to finite codimension. As an application we determine the radical of the bilinear equation [x1,x2] + [x3,x4] = 0.