Lie Polynomials 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.

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.

Lie polynomials and a twistorial correspondence for amplitudes

We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of mathcal {M}_{0,n}, the moduli space of n points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle T^*_Dmathcal {M}_{0,n}, the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and mathcal {K}_n the space of momentum invariants of n massless particles subject to momentum conservation as the analogue of space–time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain n-3-forms on mathcal {K}_n, introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral n-3-planes in mathcal {K}_n introduced by ABHY.

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, *).

Elliptic double shuffle, Grothendieck–Teichmüller and mould theory

In this article we define an elliptic double shuffle Lie algebra \(\scriptstyle {{\mathfrak {ds}}_{ell}}\) that generalizes the well-known double shuffle Lie algebra \(\scriptstyle {{\mathfrak {ds}}}\) to the elliptic situation. The double shuffle, or dimorphic, relations satisfied by elements of the Lie algebra \(\scriptstyle {{\mathfrak {ds}}}\) express two families of algebraic relations between multiple zeta values that conjecturally generate all relations. In analogy with this, elements of the elliptic double shuffle Lie algebra \(\scriptstyle {{\mathfrak {ds}}_{ell}}\) are Lie polynomials having a dimorphic property called \(\scriptstyle {\Delta }\)-bialternality that conjecturally describes the (dual of the) set of algebraic relations between elliptic multiple zeta values, which arise as coefficients of a certain elliptic generating series (constructed explicitly in Lochak et al. [15]) in On elliptic multiple zeta values 2016, in preparation) and closely related to the elliptic associator defined by Enriquez [10]. We show that one of Ecalle’s major results in mould theory can be reinterpreted as yielding the existence of an injective Lie algebra morphism \(\scriptstyle {{\mathfrak {ds}}\rightarrow {\mathfrak {ds}}_{ell}}\). Our main result is the compatibility of this map with the tangential-base-point section \(\scriptstyle {\mathrm{Lie}\,\pi _1(MTM)\rightarrow \mathrm{Lie}\,\pi _1(MEM)}\) constructed by Hain and Matsumoto [14] and with the section \(\scriptstyle {{\mathfrak {grt}}\rightarrow {\mathfrak {grt}}_{ell}}\) mapping the Grothendieck–Teichmüller Lie algebra \(\scriptstyle {{\mathfrak {grt}}}\) into the elliptic Grothendieck–Teichmüller Lie algebra \(\scriptstyle {{\mathfrak {grt}}_{ell}}\) constructed by Enriquez. This compatibility is expressed by the commutativity of the following diagram (excluding the dotted arrow, which is conjectural).

Compactness property of Lie polynomials in the creation and annihilation operators of the q-oscillator

Given a real number q such that $$0<q<1$$ , the natural setting for the mathematics of a q-oscillator is an infinite-dimensional, separable Hilbert space that is said to provide an interpolation between the Bargmann–Segal space of holomorphic functions and the Hardy–Lebesgue space of analytic functions. The traditional basis states are interrelated by the creation and annihilation operators. Since the commutation relation is q-deformed, the commutator algebra for the creation and annihilation operators is not a low-dimensional Lie algebra like that for the canonical commutation relation. In this study, a characterization of the elements of the said commutator algebra is obtained using spectral properties of the creation and annihilation operators as these faithfully represent the generators of a q-deformed Heisenberg algebra. The derived algebra of the commutator algebra is precisely the set of all compact operators, and the resulting Calkin algebra is algebraically isomorphic to the complex algebra of Laurent polynomials in one indeterminate. As for any operator that is not in the commutator algebra, the action of such an operator on an arbitrary basis state can be approximated by a Lie series of elements from the commutator algebra.

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.

Path developments and tail asymptotics of signature for pure rough paths

Solutions to linear controlled differential equations can be expressed in terms of global iterated path integrals along the driving path. This collection of iterated integrals encodes essentially all information about the underlying path. While upper bounds for iterated path integrals are well known, lower bounds are much less understood, and it is known only relatively recently that some types of asymptotics for the n-th order iterated integral can be used to recover some intrinsic quantitative properties of the path, such as the length for C1 paths.In the present paper, we investigate the simplest type of rough paths (the rough path analogue of line segments), and establish uniform upper and lower estimates for the tail asymptotics of iterated integrals in terms of the local variation of the underlying path. Our methodology, which we believe is new for this problem, involves developing paths into complex semisimple Lie algebras and using the associated representation theory to study spectral properties of Lie polynomials under the Lie algebraic development.

An extension of a [formula omitted]-deformed Heisenberg algebra and its Lie polynomials

Let F be a field, and fix a q∈F. The q-deformed Heisenberg algebra H(q) is the unital associative algebra over F with generators A, B and a relation which asserts that AB−qBA is the multiplicative identity in H(q). We extend H(q) into an algebra R(q) defined by generators A, B and a relation which asserts that AB−qBA is central in R(q). We identify all elements of R(q) that are Lie polynomials in A, B.

Lie polynomials in q-deformed Heisenberg algebras

Let F be a field, and let q∈F. The q-deformed Heisenberg algebra is the unital associative F-algebra H(q) with generators A,B and relation AB−qBA=I, where I is the multiplicative identity in H(q). The set of all Lie polynomials in A,B is the Lie subalgebra L(q) of H(q) generated by A,B. If q≠1 or the characteristic of F is not 2, then the equation AB−qBA=I cannot be expressed in terms of Lie algebra operations only, yet this equation still has consequences on the Lie algebra structure of L(q), which we investigate. We show that if q is not a root of unity, then L(q) is a Lie ideal of H(q), and the resulting quotient Lie algebra is infinite-dimensional and one-step nilpotent.

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

Schur-Weyl duality and the free Lie algebra

We prove an analogue of Schur-Weyl duality for the space of homogeneous Lie polynomials of degree r r in n n variables.

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 the images of Lie polynomials evaluated on Lie algebras

We describe the images of multilinear Lie polynomials of degrees 3 and 4 evaluated on su(n) and so(n).

Lie Invariants in Two and Three Variables

We use computer algebra to determine the Lie invariants of degree ≤ 12 in the free Lie algebra on two generators corresponding to the natural representation of the simple 3-dimensional Lie algebra 𝔰𝔩2(ℂ). We then consider the free Lie algebra on three generators, and compute the Lie invariants of degree ≤ 7 corresponding to the adjoint representation of 𝔰𝔩2(ℂ), and the Lie invariants of degree ≤ 9 corresponding to the natural representation of 𝔰𝔩3(ℂ). We represent the action of 𝔰𝔩2(ℂ) and 𝔰𝔩3(ℂ) on Lie polynomials by computing the coefficient matrix with respect to the basis of Hall words.

On the image of a noncommutative polynomial

Let F be an algebraically closed field of characteristic zero. We consider the question which subsets of Mn(F) can be images of noncommutative polynomials. We prove that a noncommutative polynomial f has only finitely many similarity orbits modulo nonzero scalar multiplication in its image if and only if f is power-central. The union of the zero matrix and a standard open set closed under conjugation by GLn(F) and nonzero scalar multiplication is shown to be the image of a noncommutative polynomial. We investigate the density of the images with respect to the Zariski topology. We also answer Lvovʼs conjecture for multilinear Lie polynomials of degree at most 4 affirmatively.

A New Proof of the Existence of Free Lie Algebras and an Application

The existence of free Lie algebras is usually derived as a consequence of the Poincaré-Birkhoff-Witt theorem. Moreover, in order to prove that (given a set and a field of characteristic zero) the Lie algebra of the Lie polynomials in the letters of (over the field ) is a free Lie algebra generated by , all available proofs use the embedding of a Lie algebra into its enveloping algebra . The aim of this paper is to give a much simpler proof of the latter fact without the aid of the cited embedding nor of the Poincaré-Birkhoff-Witt theorem. As an application of our result and of a theorem due to Cartier (1956), we show the relationships existing between the theorem of Poincaré-Birkhoff-Witt, the theorem of Campbell-Baker-Hausdorff, and the existence of free Lie algebras.

On maps preserving zeros of Lie polynomials of small degrees

We describe maps preserving zeros of multilinear Lie polynomials of degrees 3 and 4 on prime algebras and matrices over unital algebras. In particular, our theorems generalize several results related to commutativity preserving maps.

On Some Universal Algebras Associated to the Category of Lie Bialgebras

In our previous work (math/0008128), we studied the set Quant(K) of all universal quantization functors of Lie bialgebras over a field K of characteristic zero, compatible with the operations of taking duals and doubles. We showed that Quant(K) is canonically isomorphic to a product G0(K)×ш(K), where G0(K) is a universal group and ш(K) is a quotient set of a set B(K) of families of Lie polynomials by the action of a group G(K). We prove here that G0(K) is equal to the multiplicative group 1+ℏK[[ℏ]]. So Quant(K) is “as close as it can be” to ш(K). We also prove that the only universal derivations of Lie bialgebras are multiples of the composition of the bracket with the cobracket. Finally, we prove that the stabilizer of any element of B(K) is reduced to the 1-parameter subgroup generated by the corresponding “square of the antipode.”