Representations Of Lie Algebras Research Articles

Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials

Sparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithm to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function.

Modelling Oil Price with Lie Algebras and Long Short-Term Memory Networks

In this paper, we propose hybrid models for modelling the daily oil price during the period from 2 January 1986 to 5 April 2021. The models on S2 manifolds that we consider, including the reference ones, employ matrix representations rather than differential operator representations of Lie algebras. Firstly, the performance of LieNLS model is examined in comparison to the Lie-OLS model. Then, both of these reference models are improved by integrating them with a recurrent neural network model used in deep learning. Thirdly, the forecasting performance of these two proposed hybrid models on the S2 manifold, namely Lie-LSTMOLS and Lie-LSTMNLS, are compared with those of the reference LieOLS and LieNLS models. The in-sample and out-of-sample results show that our proposed methods can achieve improved performance over LieOLS and LieNLS models in terms of RMSE and MAE metrics and hence can be more reliably used to assess volatility of time-series data.

GroupMath: A Mathematica package for group theory calculations

GroupMath is a Mathematica package which performs several calculations related to semi-simple Lie algebras and the permutation groups, both of which are important in various areas of physics. Having in mind the specific needs of theoretical particle physicists, the program computes several basis-independent quantities (such as roots, weights, decomposition of products of representations, branching rules) as well as basis-dependent ones (including explicit representation matrices and Clebsch-Gordon coefficients). Program summaryProgram Title: GroupMath: A Mathematica package for group theory calculationsCPC Library link to program files:https://doi.org/10.17632/hdkksr6v7t.1Developer's repository link:renatofonseca.net/groupmathLicensing provisions: GNU General Public License 3Programming language: MathematicaNature of problem: Computations involving semi-simple Lie algebras are commonplace in several areas of research, such as particle physics. Indeed model builders trying to extend the Standard Model often need to consider new fields which transform as irreducible representations of these algebras. It is therefore convenient to have a computer program which can perform these calculations systematically. Such code might be used directly by the user, or be incorporated in a longer chain of programs. The permutation groups Sm are also important, both by themselves and also as a tool to better describe the properties of the representations of semi-simple Lie algebras.Solution method: GroupMath performs several computations related to semi-simple Lie algebras which are important in high energy physics: Cartan matrices, roots, weights, Weyl reflections, decomposition of products of representations, subgroups, branching rules, and others. Algorithms available in the literature are used to efficiently perform some of these calculations. The program also computes basis-dependent quantities such as representation matrices and Clebsch-Gordon coefficients. Several elements of the theory of finite permutation groups have also been implemented, including Young diagrams and tableaux, the decomposition of products of representations, branching rules, and explicit representation matrices.Additional comments including restrictions and unusual features: Most of GroupMath's functions can handle arbitrary representations of any semi-simple Lie algebra and any permutation group. However, memory and time constraints are important when considering large and/or numerous representations. This is particularly true for the functions which perform basis-dependent computations.

The Weyl BMS group and Einstein\u2019s equations

We propose an extension of the BMS group, which we refer to as Weyl BMS or BMSW for short, that includes super-translations, local Weyl rescalings and arbitrary diffeomorphisms of the 2d sphere metric. After generalizing the Barnich-Troessaert bracket, we show that the Noether charges of the BMSW group provide a centerless representation of the BMSW Lie algebra at every cross section of null infinity. This result is tantamount to proving that the flux-balance laws for the Noether charges imply the validity of the asymptotic Einstein’s equations at null infinity. The extension requires a holographic renormalization procedure, which we construct without any dependence on background fields. The renormalized phase space of null infinity reveals new pairs of conjugate variables. Finally, we show that BMSW group elements label the gravitational vacua.

On the growth and zeros of polynomials attached to arithmetic functions

In this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and 0<h(n) le h(n+1). We put P_0^{g,h}(x)=1 and Png,h(x):=xh(n)∑k=1ng(k)Pn-kg,h(x).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} P_n^{g,h}(x) := \\frac{x}{h(n)} \\sum _{k=1}^{n} g(k) \\, P_{n-k}^{g,h}(x). \\end{aligned}$$\\end{document}As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind eta -function. Here, g is the sum of divisors and h the identity function. Kostant’s result on the representation of simple complex Lie algebras and Han’s results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein’s j-invariant, and Chebyshev polynomials of the second kind.

Representations of Lie algebras

Representations of Lie algebras

Quantum manipulation through finite fluctuations for a generalized parametric oscillator using a lie algebra representation

In this paper, we focus on the effects of temporally compact fluctuations on the parameters m(t) (mass), (frequency), and (spatial dilation coefficient), for the generalized Caldirola-Kanai Hamiltonian. We propose an algorithm such that given control over the timing of repeated compact fluctuations, one can produce any time-evolution associated with complete parameter control of the generalized Caldirola-Kanai Hamiltonian. Computational simplicity is achieved in this endeavor using a faithful representation of the Hamiltonian’s associated Lie algebra. Further simplification occurs as a result of using unitarity conditions for the time-evolution operator. We apply our analysis to the effects of Dirac delta fluctuations in mass and frequency, both separately and simultaneously. We also numerically demonstrate control of the generalized Caldirola-Kanai system for the case of timed Gaussian fluctuations in the mass term. Furthermore, we use the faithful Lie algebra representation to show that any unitary evolution of this generalized Caldirola-Kanai system is given by the squeeze operator multiplied with a phase factor. We also show that the classical evolution of a probabilistic Gaussian phase space distribution and the evolution of the Wigner function of a coherent state in the quantum Caldirola-Kanai system are identical.

Weight zero in tensor-decomposable irreducible representations of simple algebraic groups

Let G be a simple algebraic group in defining characteristic p > 0 , and let V be an irreducible G -module which is the tensor product of exactly two non-trivial modules. We obtain a criterion for V to have the zero weight. In addition, we provide a uniform criterion for an irreducible representation of a simple Lie algebra over the complex numbers to have a multiple of a prescribed fundamental weight.

A Construction of Representations of Loop Group and Affine Lie Algebra of $\frak {sl}_{n}$

A Construction of Representations of Loop Group and Affine Lie Algebra of $\frak {sl}_{n}$

Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras

This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of r-matrices and solutions to classical Yang–Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed.

Representations of twisted toroidal Lie algebras from twisted modules over vertex algebras

Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one defines the notion of a twisted toroidal Lie algebra. In this paper, we construct representations of twisted toroidal Lie algebras from twisted modules over affine and lattice vertex algebras.

Yang-Yang functions, monodromy and knot polynomials

We derive a structure of ℤ[t, t−1]-module bundle from a family of Yang-Yang functions. For the fundamental representation of the complex simple Lie algebra of classical type, we give explicit wall-crossing formula and prove that the monodromy representation of the ℤ[t, t−1]-module bundle is equivalent to the braid group representation induced by the universal R-matrices of Uh(g). We show that two transformations induced on the fiber by the symmetry breaking deformation and respectively the rotation of two complex parameters commute with each other.

Representations of simple Lie algebras with vectors having a zero stationary subalgebra

The paper solves the problem of describing the representations V of compact Lie groups G and vectors v ∈ V with single stationary subgroups. It builds the classification of the representation of simple Lie algebras with vectors having a zero stationary subalgebra. If the codimension of the submanifold M 0 of dimension m in an N-dimensional Euclidean space RN is less than the dimension and the group G is simple, then all these cases are described in the corresponding tables in sections 3 and 4.

Classifications of $\Gamma$-Colored $d$-Complete Posets and Upper $P$-Minuscule Borel Representations

The $\Gamma$-colored $d$-complete posets correspond to certain Borel representations that are analogous to minuscule representations of semisimple Lie algebras. We classify $\Gamma$-colored $d$-complete posets which specifies the structure of the associated representations. We show that finite $\Gamma$-colored $d$-complete posets are precisely the dominant minuscule heaps of J.R. Stembridge. These heaps are reformulations and extensions of the colored $d$-complete posets of R.A. Proctor. We also show that connected infinite $\Gamma$-colored $d$-complete posets are precisely order filters of the connected full heaps of R.M. Green.

Universal factorization algebras of polynomials represent Lie algebras of endomorphisms

The goal of this paper is to supply an explicit description of the universal factorization algebra of the generic polynomial of degree [Formula: see text] into the product of two monic polynomials, one of degree [Formula: see text], as a representation of Lie algebras of [Formula: see text] matrices with polynomial entries. This is related with the bosonic vertex representation of the Lie algebra [Formula: see text] due to Date, Jimbo, Kashiwara and Miwa.

The Kostant invariant and special ϵ-orthogonal representations for ϵ-quadratic colour Lie algebras

Let k be a field of characteristic not two or three, let g be a finite-dimensional colour Lie algebra and let V be a finite-dimensional representation of g. In this article we give various ways of constructing a colour Lie algebra g˜ whose bracket in some sense extends both the bracket of g and the action of g on V. Colour Lie algebras, originally introduced by R. Ree ([18]), generalise both Lie algebras and Lie superalgebras, and in those cases our results imply many known results ([13], [14], [5], [23]). For a class of representations arising in this context we show there are covariants satisfying identities analogous to Mathews identities for binary cubics.

A filtration on the ring of Laurent polynomials and representations of the general linear Lie algebra

We first present a filtration on the ring Ln of Laurent polynomials such that the direct sum decomposition of its associated graded ring grLn agrees with the direct sum decomposition of grLn, as a module over the complex general linear Lie algebra gl(n), into its simple submodules. Next, generalizing the simple modules occurring in the associated graded ring grLn, we give some explicit constructions of weight multiplicity-free irreducible representations of gl(n).

Jacobi structures on real two- and three-dimensional Lie groups and their Jacobi–Lie systems

Using the adjoint representations of Lie algebras, we classify all Jacobi structures on real two- and three-dimensional Lie groups. Also, we study Jacobi-Lie systems on these real low-dimensional Lie groups. Our results are illustrated through examples of Jacobi-Lie Hamiltonian systems on some real two- and three-dimensional Lie groups.

Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H2 and H3. The geometrical structures of nested polytopes are exemplified.

Minimal faithful representations of the free 2-step nilpotent Lie algebra of the rank r

Given a finite dimensional Lie algebra g, let z(g) denote the center of g and let μ(g) be the minimal possible dimension for a faithful representation of g. In this paper we obtain μ(Lr,2), where Lr,k is the free k-step nilpotent Lie algebra of rank r. In particular we prove that μ(Lr,2)=⌈2r(r−1)⌉+2 for r≥4. It turns out that μ(Lr,2)∼μ(z(Lr,2))∼2dim⁡Lr,2 (as r→∞) and we present some evidence that this could be true for Lr,k for any k. This is considerably lower than the known bounds for μ(Lr,k), which are (for fixed k) polynomial in dim⁡Lr,k.