Lie Polynomial Research Articles

The image of Lie polynomials on real Lie algebras of dimension up to 3

Let Fn be the free Lie algebra over R of rank n generated by y1,…,yn, and let f∈Fn′ be a multilinear Lie polynomial contained in the commutator ideal Fn′ of Fn. In this paper, we determine the imageImf={f(w1,…,wn)|wi∈L,i=1,…,n}⊂L, for Lie algebras L of dimension ≤3, and of the Lie algebra of dimension 4 stated in a paper of Baker dating back to 1901. In all the cases studied, the L'vov-Kaplansky Conjecture has a positive answer.

Almost multiplicity free subgroups of compact Lie groups and polynomial integrability of sub-Riemannian geodesic flows

Almost multiplicity free subgroups of compact Lie groups and polynomial integrability of sub-Riemannian geodesic flows

Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra

Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $ a a^+-qa^+ a =1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^+$ and $ a $ is the linear span of $a^+$ and $ a $, together with all operators of the form ${a^+}^l{\left[a,a^+\right]}^k$, and ${\left[a,a^+\right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $ a ^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the commutator algebra generated by $ a $ and $a^+$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^+$ and $ a $. In this work, we extend the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $ a $, $a^+$, and the operator $e^{\omega N}$ for some nonzero real parameter $\omega$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$.

A Lie Bracket for the Momentum Kernel.

We prove results for the study of the double copy and tree-level colour/kinematics duality for tree-level scattering amplitudes using the properties of Lie polynomials. We show that the 'S-map' that was defined to simplify super-Yang-Mills multiparticle superfields is in fact a Lie bracket. A generalized KLT map from Lie polynomials to their dual is obtained by studying our new Lie bracket; the matrix elements of this map yield a recently proposed 'generalized KLT matrix', and this reduces to the usual KLT matrix when its entries are restricted to a basis. Using this, we give an algebraic proof for the cancellation of double poles in the KLT formula for gravity amplitudes. We further study Berends-Giele recursion for biadjoint scalar tree amplitudes that take values in Lie polynomials. Field theory amplitudes are obtained from these 'Lie polynomial amplitudes' using numerators characterized as homomorphisms from the free Lie algebra to kinematic data. Examples are presented for the biadjoint scalar, Yang-Mills theory and the nonlinear sigma model. That these theories satisfy the Bern-Carrasco-Johansson amplitude relations follows from the structural properties of Lie polynomial amplitudes that we prove.

Image of Lie polynomial of degree 2 evaluated on nilpotent lie algebra

We delineate the image of multilinear Lie polynomial of degree 2 evaluated on L where L is a finite-dimensional nilpotent Lie algebra over field k with

Maximal polynomial modulations of singular integrals

Let K be a standard Hölder continuous Calderón–Zygmund kernel on Rd whose truncations define L2 bounded operators. We show that the maximal operator obtained by modulating K by polynomial phases of a fixed degree is bounded on Lp(Rd) for 1<p<∞. This extends Sjölin's multidimensional Carleson theorem and Lie's polynomial Carleson theorem.

Lie polynomials in an algebra defined by a linearly twisted commutation relation

We present an elementary approach to characterizing Lie polynomials on the generators [Formula: see text] of an algebra with a defining relation in the form of a twisted commutation relation [Formula: see text]. Here, the twisting map [Formula: see text] is a linear polynomial with a slope parameter, which is not a root of unity. The class of algebras defined as such encompasses [Formula: see text]-deformed Heisenberg algebras, rotation algebras, and some types of [Formula: see text]-oscillator algebras, the deformation parameters of which, are not roots of unity. Thus, we have a general solution for the Lie polynomial characterization problem for these algebras.

Lie identities and images of Lie polynomials for the skew-symmetric elements of UTm

ABSTRACT Let UT m be the algebra of all m × m upper triangular matrices over a field whose characteristic is different from 2. Given an involution * of the first kind on UT m , we will obtain the minimal integer t such that the Lie polynomial [[z 1, z 2], …, [z 2t−1, z 2t ]] is an identity for . Afterwards, under a mild technical restriction on , we will describe all multilinear Lie polynomials whose image evaluated on K(m, *) is the space formed by all strictly upper triangular matrices of K(m, *).

A relatively short self-contained proof of the Baker–Campbell–Hausdorff theorem

We give a new purely algebraic proof of the Baker–Campbell–Hausdorff theorem, which states that the homogeneous components of the formal expansion of log(eAeB) are Lie polynomials. Our proof is based on a recurrence formula for these components and a lemma that states that if under certain conditions a commutator of a non-commuting variable and a given polynomial is a Lie polynomial, then the given polynomial itself is a Lie polynomial.

The images of Lie polynomials evaluated on matrices

ABSTRACTKaplansky asked about the possible images of a polynomial f in several noncommuting variables. In this paper, we consider the case of f a Lie polynomial. We describe all the possible images of f in M2(K) and provide an example of f whose image is the set of non-nilpotent trace zero matrices, together with 0. We provide an arithmetic criterion for this case. We also show that the standard polynomial sk is not a Lie polynomial, for k>2.

On twin and anti-twin words in the support of the free Lie algebra

Let ${\mathcal{L}}_{K}(A)$ be the free Lie algebra on a finite alphabet A over a commutative ring K with unity. For a word u in the free monoid A ? let $\tilde{u}$ denote its reversal. Two words in A ? are called twin (resp. anti-twin) if they appear with equal (resp. opposite) coefficients in each Lie polynomial. Let l denote the left-normed Lie bracketing and ? be its adjoint map with respect to the canonical scalar product on the free associative algebra K?A?. Studying the kernel of ? and using several techniques from combinatorics on words and the shuffle algebra , we show that, when K is of characteristic zero, two words u and v of common length n that lie in the support of ${\mathcal{L}}_{K}(A)$ --i.e., they are neither powers a n of letters a?A with exponent n>1 nor palindromes of even length--are twin (resp. anti-twin) if and only if u=v or $u = \tilde{v}$ and n is odd (resp. $u =\tilde{v}$ and n is even).

On the support of the free Lie algebra: the Schutzenberger problems

Combinatorics M.-P. Schutzenberger asked to determine the support of the free Lie algebra L(Zm) (A) on a finite alphabet A over the ring Z(m) of integers mod m and all pairs of twin and anti-twin words, i.e., words that appear with equal (resp. opposite) coefficients in each Lie polynomial. We characterize the complement of the support of L(Zm) (A) in A* as the set of all words w such that m divides all the coefficients appearing in the monomials of l* (w), where l* is the adjoint endomorphism of the left normed Lie bracketing l of the free Lie ring. Calculating l* (w) via the shuffle product, we recover the well known result of Duchamp and Thibon (Discrete Math. 76 (1989) 123-132) for the support of the free Lie ring in a much more natural way. We conjecture that two words u and v of common length n, which lie in the support of the free Lie ring, are twin (resp. anti-twin) if and only if either u = v or n is odd and u = (v) over tilde (resp. if n is even and u = (v) over tilde), where (v) over tilde denotes the reversal of v and we prove that it suffices to show this for a two-lettered alphabet. These problems can be rephrased, for words of length n, in terms of the action of the Dynkin operator l(n) on lambda-tabloids, where lambda is a partition of n. Representing a word w in two letters by the subset I of [n] = \1, 2, ... , n\ that consists of all positions that one of the letters occurs in w, the computation of l* (w) leads us to the notion of the Pascal descent polynomial p(n)(I), a particular commutative multi-linear polynomial which is equal to the signed binomial coefficient when vertical bar I vertical bar = 1. We provide a recursion formula for p(n) (I) and show that if m inverted iota Sigma(i is an element of I)(1)(i-1) (n - 1 i - 1), then w lies in the support of L(Zm) (A).

A note on the Lie polynomial $ \mbf{(x+y)^{p}-x^{p}-y^{p}} $

Using the techniques of group action on a set, we will give an elementary ane complete proof in this paper that the Lie polynomial $ (x+y)^{p}-x^{p}-y^{p} $ over prime field $ F_{p} $ is a sum of Lie brackets, with $ p $ a rational prime.

Semi-simple splittings for solvable Lie groups and polynomial structures

Semi-simple splittings for solvable Lie groups and polynomial structures

An algorithm for analysis of the structure of finitely presented Lie algebras

We consider the following problem: what is the most general Lie algebra satisfying a given set of Lie polynomial equations? The presentation of Lie algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. That problem is of great practical importance, covering applications ranging from mathematical physics to combinatorial algebra. Some particular applications are constructionof prolongation algebras in the Wahlquist-Estabrook method for integrability analysis of nonlinear partial differential equations and investigation of Lie algebras arising in different physical models. The finite presentations also indicate a way to q-quantize Lie algebras. To solve this problem, one should perform a large volume of algebraic transformations which is sharply increased with growth of the number of generators and relations. For this reason, in practice one needs to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented Lie algebra and its commutator table, and its implementation in the C language. Some computer results illustrating our algorithmand its actual implementation are also presented.

Lie Algebras and Superalgebras Defined by a Finite Number of Relations: Computer Analysis

The presentation of Lie (super)algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. It is very important, for instance, for investigation of the particular Lie (super)algebras arising in dierent (super)symmetric physical models. Generally, one can put the following question: what is the most general Lie algebra or superalgebra satisfying to the given set of Lie polynomial equations? To solve this problem, one has to perform a large volume of algebraic transformations which sharply increases with growth of the number of generators and relations. By this reason, in practice, one needs to use a computer algebra tool. We describe here an algorithm and its implementation in C for constructing the bases of finitely presented Lie (super)algebras and their commutator tables.