Non-semisimple Lie Algebras Research Articles

Integrable couplings stemming from three-dimensional unital algebras

This study aims to explore the connection between integrable couplings and three-dimensional unital algebras. Expansions in these algebras generate integrable couplings, including nonlinear integrable couplings, and their corresponding Lax pairs and hereditary recursion operators possess specific structures, associated with non-semisimple Lie algebras. Applications to the KdV equation are explicitly presented.

Application of Riemann–Hilbert method to an extended coupled nonlinear Schrödinger equations

Based on the non-semisimple Lie algebra g˜, we introduce a new integrable coupling system of the focusing nonlinear Schrödinger equation associated with a 4th-order block matrix spectral problem, which is called the extended coupled nonlinear Schrödinger (ECNLS) equations. The ECNLS equations can be reduced to many equations with physical background, including the focusing nonlinear Schrödinger equations (NLS) equation, the new integrable system of coupled NLS equations, the Manakov system, the mixed coupled NLS equations, etc. Based on the 4th-order spectral problem, we analyze the asymptotic behavior, analyticity and symmetry of the eigenfunctions and scattering coefficients. It follows that a formulation of solutions is developed for the Riemann–Hilbert problems associated with the reflectionless transforms. Furthermore, the N-soliton solutions of the ECNLS equations are generated. Additionally, the ECNLS equations are extended to a multi-component nonlinear Schrödinger system by means of a new N×N non-semisimple Lie algebra ĝ, which means that the ECNLS equations are extended to an arbitrary number of components. Actually, the coupled and multi-component equations that we obtained can enrich the existing integrable models and possibly describe new nonlinear phenomena.

Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras

We propose a systematic procedure to construct polynomial algebras from intermediate Casimir invariants arising from (semisimple or non-semisimple) Lie algebras . In this approach, we deal with explicit polynomials in the enveloping algebra of . We present explicit examples of low-dimensional Lie algebras (up to dimension six) to show how they can display different behaviours and can lead to abelian algebras, quadratic algebras or more complex structures involving higher order nested commutators. Within this framework, we also demonstrate how virtual copies of the Levi factor of a Levi decomposable Lie algebra can be used as a tool to construct ‘copies’ of polynomial algebras. Different schemes to obtain polynomial algebras associated to algebraic Hamiltonians have been proposed in the literature, among them the use of commutants of various type. The present approach is different and relies on the construction of intermediate Casimir invariants in the enveloping algebra .

A new multi-component integrable coupling and its application to isospectral and nonisospectral problems

A new non-semisimple Lie algebra ĝ is constructed. Based on the Lie algebra ĝ, we propose a method for generating ZNɛ integrable couplings. Then, we consider the application of a MKdV isospectral and an AKNS nonisospectral problem of the method respectively. By solving the zero curvature equation of the two spectral problems, we obtain a ZNɛ isospectral MKdV integrable coupling hierarchy and a ZNɛ nonisospectral ANKS integrable coupling hierarchy. It follows that the bi-Hamiltonian structures of these two hierarchies is derived based on the ZNɛ-trace identity that we constructed.

A Kind of Generalized Integrable Couplings and Their Bi-Hamiltonian Structure

We introduce a Lie algebra $\widetilde {g}$ which can be used to construct integrable couplings of some spectral problems. As two examples, the non-semisimple Lie algebra $\widetilde {g}$ is applied to enlarge the spectral problems of an extended Ablowitz-Kaup-Newell-Segur (AKNS) spectral problem and a generalized D-Kaup-Newell (D-KN) spectral problem. It follows that we obtain two generalized integrable couplings by solving these expanded zero-curvature equations. Finally, we find that the integrable hierarchies that we obtain have bi-Hamiltonian structures of combinatorial form, thereby showing their Liouville integrability.

Two integrable couplings of a generalized D-Kaup–Newell hierarchy and their Hamiltonian and bi-Hamiltonian structures

We enlarge the spectral problem of a generalized D-Kaup–Newell (D-KN) spectral problem. Solving the enlarged zero-curvature equations, we produce integrable couplings. A reduction of the spectral matrix leads to a second integrable coupling system. Next, bilinear forms that are symmetric, ad-invariant, and non-degenerate on the given non-semisimple matrix Lie algebra are computed to employ the variational identity. The variational identity is then applied to the original enlarged spectral problem of a generalized D-KN hierarchy and the reduced problem. Hamiltonian structures are presented, as well as a bi-Hamiltonian formulation of the reduced problem. Both hierarchies have infinitely many commuting high-order symmetries and conserved densities, i.e., are Liouville integrable.

Travelling waves and conservation laws of a (2+1)-dimensional coupling system with Korteweg-de Vries equation

Abstract In this paper we study a (2+1)-dimensional coupling system with the Korteweg-de Vries equation, which is associated with non-semisimple matrix Lie algebras. Its Lax-pair and bi-Hamiltonian formulation were obtained and presented in the literature. We utilize Lie symmetry analysis along with the (G′/G)–expansion method to obtain travelling wave solutions of this system. Furthermore, conservation laws are constructed using the multiplier method.

Completion of the Ablowitz-Kaup-Newell-Segur integrable coupling

Integrable couplings are associated with non-semisimple Lie algebras. In this paper, we propose a new method to generate new integrable systems through making perturbation in matrix spectral problems for integrable couplings, which is called the “completion process of integrable couplings.” As an example, the idea of construction is applied to the Ablowitz-Kaup-Newell-Segur integrable coupling. Each equation in the resulting hierarchy has a bi-Hamiltonian structure furnished by the component-trace identity.

Darboux transformations of integrable couplings and applications

A formulation of Darboux transformations is proposed for integrable couplings, based on non-semisimple matrix Lie algebras. Applications to a kind of integrable couplings of the AKNS equations are made, along with an explicit formula for the associated Bäcklund transformation. Exact one-soliton-like solutions are computed for the integrable couplings of the second- and third-order AKNS equations, and a type of reduction is created to generate integrable couplings and their one-soliton-like solutions for the NLS and MKdV equations.

A differential operator realisation approach for constructing Casimir operators of non-semisimple Lie algebras

We introduce a search algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. To demonstrate the algorithm, we look at two classes of examples: (1) the model filiform Lie algebras and (2) the Schrödinger Lie algebras. We find that an abstract form of dimensional analysis assists us in our algorithm, and greatly reduces the complexity of the problem.

Two New Integrable Hierarchies and Their Nonlinear Integrable Couplings

By introducing an invertible linear transform, a new Lie algebra G is obtained from the Lie algebra H. Making use of the compatibility conditions of the respective isospectral problems, a generalized NLS-MKdV hierarchy and a new integrable soliton hierarchy are achieved by using the trace identity or the variational identity. Then, two special non-semisimple Lie algebras and are explicitly conducted. As an application, the nonlinear continuous integrable couplings of the obtained integrable systems as well as their bi-Hamiltonian structures are established, respectively.

Derivations of the Schrödinger algebra and their applications

The Schrodinger algebra is a non-semisimple Lie algebra and plays an important role in mathematical physics and its applications. In this paper, all derivations of the Schrodinger algebra are determined. As applications, all biderivations, linear commuting maps and commutative post-Lie algebra structures on the Schrodinger algebra are obtained.

Sextonions, Zorn matrices, and $$\mathbf {e}_{\mathbf{7} \frac{\mathbf{1}}{\mathbf{2}}}$$ e 7 1 2

By exploiting suitably constrained Zorn matrices, we present a new construction of the algebra of sextonions (over the algebraically closed field $\mathbb{C}$). This allows for an explicit construction, in terms of Jordan pairs, of the non-semisimple Lie algebra $\mathbf{e_{7 \frac12}}$, intermediate between $\mathbf{e_{7}}$ and $\mathbf{e_{8}}$, as well as of all Lie algebras occurring in the sextonionic row and column of the extended Freudenthal Magic Square.

Perturbative quantization of Yang–Mills theory with classical double as gauge algebra

Perturbative quantization of Yang-Mills theory with a gauge algebra given by the classical double of a semisimple Lie algebra is considered. The classical double of a real Lie algebra is a nonsemisimple real Lie algebra that admits a nonpositive definite invariant metric, the indefiniteness of the metric suggesting an apparent lack of unitarity. It is shown that the theory is UV divergent at one loop and that there are no radiative corrections at higher loops. One-loop UV divergences are removed through renormalization of the coupling constant, thus introducing a renormalization scale. The terms in the classical action that would spoil unitarity are proved to be cohomologically trivial with respect to the Slavnov-Taylor operator that controls gauge invariance for the quantum theory. Hence they do not contribute gauge invariant radiative corrections to the quantum effective action and the theory is unitary.

Representations of conformal Galilei algebra with integer spin and an application

Conformal Galilei algebra is a class of non-semisimple Lie algebras. Each member of the class is labelled by two parameters d (positive integer) and ℓ (positive integer or positive half-integer). We investigate the lowest weight representations of the conformal Galilei algebra for d = 1 and any integer ℓ. We start with the Verma modules and then extract all the irreducible lowest weight modules. This is done by searching and explicitly constructing singular vectors. As an application of our construction of singular vectors, we obtain the partial differential equations which are symmetric by kinematical transformation generated by the conformal Galilei algebra.

Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras

The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters $d$ and $\ell$. The aim of the present work is to investigate the lowest weight representations of CGA with $d = 1$ for any integer value of $\ell$. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if $\ell = 1$ and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when $\ell \neq 1$. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules.

Bi-integrable couplings and tri-integrable couplings of the modified Ablowitz-Kaup-Newell-Segur hierarchy with self-consistent sources

We discuss the bi-integrable couplings and tri-integrable couplings of the modified Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy with self-consistent sources from the enlarged matrix spectral problem and the enlarged zero curvature equations. By using the variational identities over non-semisimple Lie algebras consisting of block matrices, we construct Hamiltonian formulations for integrable couplings, bi-integrable couplings, and tri-integrable couplings.

Bi-integrable couplings of a new soliton hierarchy associated with a non-semisimple Lie algebra

We construct bi-integrable couplings for a new soliton hierarchy from zero curvature equations associated with a non-semisimple loop algebra. We demonstrate the Liouville integrability of the hierarchy by showing that the bi-integrable couplings generated are bi-Hamiltonian.

Representations of ℓ-conformai Galilei algebra and hierarchy of invariant equation

The ℓ-conformai Galilei algebra, denoted by gℓ(d), is a particular non-semisimple Lie algebra specified by a positive integer d and a spin value ℓ. The algebra gℓ(d) admits central extensions. We study lowest weight representations, in particular Verma modules, of gℓ(d) with the central extensions for d = 1,2. We give a classification of irreducible modules over d = 1 algebras and a condition of the Verma modules over d = 2 algebras being reducible. As an application of the representation theory, hierarchies of differential equations are derived. The Lie group generated by gℓ(d) with the central extension is a kinematical symmetry of the differential equations.

Nonlinear bi-integrable couplings with Hamiltonian structures

Bi-integrable couplings of soliton equations are presented through introducing non-semisimple matrix Lie algebras on which there exist non-degenerate, symmetric and ad-invariant bilinear forms. The corresponding variational identity yields Hamiltonian structures of the resulting bi-integrable couplings. An application to the AKNS spectral problem gives bi-integrable couplings with Hamiltonian structures for the AKNS equations.