Loop Algebra Research Articles

A Vector-Product Lie Algebra of a Reductive Homogeneous Space and Its Applications

A new vector-product Lie algebra is constructed for a reductive homogeneous space, which can lead to the presentation of two corresponding loop algebras. As a result, two integrable hierarchies of evolution equations are derived from a new form of zero-curvature equation. These hierarchies can be reduced to the heat equation, a special diffusion equation, a general linear Schrödinger equation, and a nonlinear Schrödinger-type equation. Notably, one of them exhibits a pseudo-Hamiltonian structure, which is derived from a new vector-product identity proposed in this paper.

Carrollian $\mathscr Lw_{1+\infty}$ representation from twistor space

We construct an explicit realization of the action of the \mathscr Lw_{1+∞}ℒw1+∞ loop algebra on fields at null infinity. This action is directly derived by Penrose transform of the geometrical action of \mathscr Lw_{1+∞}ℒw1+∞ symmetries in twistor space, ensuring that it forms a representation of the algebra. Finally, we show that this action coincides with the canonical action of \mathscr Lw_{1+∞}ℒw1+∞ Noether charges on the asymptotic phase space.

Theta Series for Quantum Loop Algebras and Yangians

Theta Series for Quantum Loop Algebras and Yangians

Generation of higher-dimensional isospectral-nonisospectral integrable hierarchies associated with a new class of higher-dimensional column-vector loop algebra

Построен новый класс петлевых алгебр многомерных вектор-столбцов. На их основе предложен метод получения изоспектрально-неизоспектральных многомерных интегрируемых иерархий. В качестве приложения выведена обобщенная неизоспектральная интегрируемая иерархия Шредингера, которую можно свести к знаменитому нелинейному уравнению Шредингера с производной, а также получена расширенная изоспектрально-неизоспектральная интегрируемая иерархия Шредингера, которая в частных случаях сводится к разнообразным классическим и новым уравнениям, таким как расширенная неизоспектральная система уравнений Шредингера с производной, уравнение теплопроводности и уравнение Фоккера-Планка, имеющее широкий спектр приложений в стохастических динамических системах. Также выведена неизоспектральная интегрируемая $Z_N^\varepsilon$-иерархия Шредингера, и этот результат показывает, что представленный подход можно распространить на произвольную размерность системы. Кроме того, для рассмотренных иерархий обсуждаются гамильтоновы структуры, которые получаются из квадратичного следового тождества.

Generation of higher-dimensional isospectral–nonisospectral integrable hierarchies associated with a new class of higher-dimensional column-vector loop algebras

Generation of higher-dimensional isospectral–nonisospectral integrable hierarchies associated with a new class of higher-dimensional column-vector loop algebras

From quantum loop superalgebras to super Yangians

The goal of this paper is to generalize a statement by Drinfeld, asserting that Yangians can be constructed as limit forms of the quantum loop algebras, to the super case. We establish a connection between quantum loop superalgebra and super Yangian of the general linear Lie superalgebra glM|N in RTT type presentation. In particular, we derive the Poincaré-Birkhoff-Witt(PBW) theorem for the quantum loop superalgebra Uq(LglM|N). Additionally, we investigate the application of the same argument to twisted super Yangian of the ortho-symplectic Lie superalgebra. For this purpose, we introduce the twisted quantum loop superalgebra as a one-sided coideal of Uq(LglM|2n) with respect to the comultiplication.

Representations of the affine ageing algebra agê(1)

In this paper, we investigate the affine ageing algebra agê(1), which is a central extension of the loop algebra of the one-spatial ageing algebra age(1). Certain Verma-type modules including Verma modules and imaginary Verma modules of agê(1) are studied. Particularly, the simplicity of these modules are characterized and their irreducible quotient modules are determined. We also study the restricted modules of agê(1) which are also the modules of the affine vertex algebra arising from the one-spatial ageing algebra age(1). We present certain constructions of simple restricted agê(1)-modules and an explicit such example of simple restricted module via the Whittaker module of agê(1) is given.

A discrete basis for celestial holography

Celestial holography provides a reformulation of scattering amplitudes in four dimensional asymptotically flat spacetimes in terms of conformal correlators of operators on the two dimensional celestial sphere in a basis of boost eigenstates. A basis of massless particle states has been previously identified in terms of conformal primary wavefunctions labeled by a boost weight ∆ = 1+iλ with λ ∈ ℝ. Here we show that a discrete orthogonal and complete basis exists for ∆ ∈ ℤ. This new basis consists of a tower of discrete memory and Goldstone observables, which are conjugate to each other and allow to reconstruct gravitational signals belonging to the Schwartz space. We show how generalized dressed states involving the whole tower of Goldstone operators can be constructed and evaluate the higher spin Goldstone 2-point functions. Finally, we recast the tower of higher spin charges providing a representation of the w1+∞ loop algebra (in the same helicity sector) in terms of the new discrete basis.

On infinite symmetry algebras in Yang-Mills theory

Similar to gravity, an infinite tower of symmetries generated by higher-spin charges has been identified in Yang-Mills theory by studying collinear limits or celestial operator products of gluons. This work aims to recover this loop symmetry in terms of charge aspects constructed on the gluonic Fock space. We propose an explicit construction for these higher spin charge aspects as operators which are polynomial in the gluonic annihilation and creation operators. The core of the paper consists of a proof that the charges we propose form a closed loop algebra to quadratic order. This closure involves using the commutator of the cubic order expansion of the charges with the linear (soft) charge. Quite remarkably, this shows that this infinite-dimensional symmetry constrains the non-linear structure of Yang-Mills theory. We provide a similar all spin proof in gravity for the so-called global quadratic (hard) charges which form the loop wedge subalgebra of w1+∞.

Prime representations in the Hernandez–Leclerc category: classical decompositions

Abstract We use the dual functional realization of loop algebras to study the prime irreducible objects in the Hernandez–Leclerc (HL) category for the quantum affine algebra associated with $\mathfrak {sl}_{n+1}$ . When the HL category is realized as a monoidal categorification of a cluster algebra (Hernandez and Leclerc (2010, Duke Mathematical Journal 154, 265–341); Hernandez and Leclerc (2013, Symmetries, integrable systems and representations, 175–193)), these representations correspond precisely to the cluster variables and the frozen variables are minimal affinizations. For any height function, we determine the classical decomposition of these representations with respect to the Hopf subalgebra $\mathbf {U}_q(\mathfrak {sl}_{n+1})$ and describe the graded multiplicities of their graded limits in terms of lattice points of convex polytopes. Combined with Brito, Chari, and Moura (2018, Journal of the Institute of Mathematics of Jussieu 17, 75–105), we obtain the graded decomposition of stable prime Demazure modules in level two integrable highest weight representations of the corresponding affine Lie algebra.

The [formula omitted]-symmetric down-up algebra

In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra A. The algebra A is associative, noncommutative, and infinite-dimensional. It is defined by two generators A,B and two relations called the down-up relations. In the present paper, we introduce the Z3-symmetric down-up algebra A. We define A by generators and relations. There are three generators A,B,C and any two of these satisfy the down-up relations. We describe how A is related to some familiar algebras in the literature, such as the Weyl algebra, the Lie algebras sl2 and sl3, the sl3 loop algebra, the Kac-Moody Lie algebra A2(1), the q-Weyl algebra, the quantized enveloping algebra Uq(sl2), and the quantized enveloping algebra Uq(A2(1)). We give some open problems and conjectures.

A functor for constructing R$R$‐matrices in the category O$\mathcal {O}$ of Borel quantum loop algebras

AbstractWe tackle the problem of constructing ‐matrices for the category associated to the Borel subalgebra of an arbitrary untwisted quantum loop algebra . For this, we define an invertible exact functor from the category linked to to the one linked to . This functor is compatible with tensor products, preserves irreducibility, and interchanges the subcategories and of Hernandez and Leclerc (Algebra Number Theory 10 (2016) 2015–2052). We construct ‐matrices for by applying on the braidings already found for by Hernandez (Represent. Theory 26 (2022) 179–210). We also use the factorization of the latter intertwiners in terms of stable maps to deduce an analogous factorization for our new braidings. We finally obtain as byproducts new relations for the Grothendieck ring as well as a functorial interpretation of a remarkable ring isomorphism of Hernandez–Leclerc.

Derivation of Painlev\'e type system with $D_4^{(1)}$ affine Weyl group symmetry in a self-similarity limit

We show how the zero-curvature equations based on a loop algebra of $D_4$ with a principal gradation reduce via self-similarity limit to a polynomial Hamiltonian system of coupled Painlev\'e III models with four canonical variables and $D_4^{(1)}$ affine Weyl group symmetry.

The celestial chiral algebra of self-dual gravity on Eguchi-Hanson space

We consider the twistor description of classical self-dual Einstein gravity in the presence of a defect operator wrapping a certain ℂℙ1. The backreaction of this defect deforms the flat twistor space to that of Eguchi-Hanson space. We show that the celestial chiral algebra of self-dual gravity on the Eguchi-Hanson background is likewise deformed to become the loop algebra of a certain scaling limit of the family of W(μ)-algebras, where the scaling limit is controlled by the radius of the Eguchi-Hanson core. We construct this algebra by computing the Poisson algebra of holomorphic functions on the deformed twistor space, and check this result with a space-time calculation of the leading contribution to the gravitational splitting function. The loop algebra of a general W(μ)-algebra (away from the scaling limit) similarly arises as the celestial chiral algebra of Moyal-deformed self-dual gravity on Eguchi-Hanson space. We also obtain corresponding results for self-dual Yang-Mills.

Using Vector-Product Loop Algebra to Generate Integrable Systems

A new three-dimensional Lie algebra and its loop algebra are proposed by us, whose commutator is a vector product. Based on this, a positive flow and a negative flow are obtained by introducing a new kind of spectral problem expressed by the vector product, which reduces to a generalized KdV equation, a generalized Schrödinger equation, a sine-Gordon equation, and a sinh-Gordon equation. Next, the well-known Tu scheme is generalized for generating isospectral integrable hierarchies and non-isospectral integrable hierarchies. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy presented in the paper is worded out. Finally, we further enlarge the three-dimensional loop algebra into a six-dimensional loop algebra so that a new isospectral integrable hierarchy which is a type of extended integrable model is produced whose bi-Hamiltonian structure is also derived from the vector-product trace identity. This new approach presented in the paper possesses extensive applications in the aspect of generating integrable hierarchies of evolution equations.

Hamiltonian structure of rational isomonodromic deformation systems

The Hamiltonian approach to isomonodromic deformation systems is extended to include generic rational covariant derivative operators on the Riemann sphere with irregular singularities of arbitrary Poincaré rank. The space of rational connections with given pole degrees carries a natural Poisson structure corresponding to the standard classical rational R-matrix structure on the dual space L*gl(r) of the loop algebra Lgl(r). Nonautonomous isomonodromic counterparts of isospectral systems generated by spectral invariants are obtained by identifying deformation parameters as Casimir elements on the phase space. These are shown to coincide with higher Birkhoff invariants determining local asymptotics near to irregular singular points, together with the pole loci. Pairs consisting of Birkhoff invariants, together with the corresponding dual spectral invariant Hamiltonians, appear as “mirror images” matching, at each pole, the negative power coefficients in the principal part of the Laurent expansion of the fundamental meromorphic differential on the associated spectral curve with the corresponding positive power terms in the analytic part. Infinitesimal isomonodromic deformations are shown to be generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation. The Casimir elements serve as coordinates complementing those along the symplectic leaves, defining a local symplectomorphism between them. The explicit derivative vector fields preserve the Poisson structure and define a flat transversal connection, spanning an integrable distribution whose leaves may be identified as the orbits of a free Abelian local group action. The projection of infinitesimal isomonodromic deformation vector fields to the quotient manifold under this action gives commuting Hamiltonian vector fields corresponding to the spectral invariants dual to the Birkhoff invariants and the pole loci.

On the finite loop algebra F[M(Cm p x C2, 2)

Let G = Cm p o C2 be a generalized dihedral group for an odd prime p and a natural number m, L = M(G; 2) be the RA2 loop obtained from G and F be a finite field of characteristic 2. For the loop algebra F[L], we determine the Jacobson radical J(F[L]) of F[L] and the Wedderburn decomposition of F[L]=J(F[L]). The structure of 1 + J(F[L]) is also determined.

From Moyal deformations to chiral higher-spin theories and to celestial algebras

We study the connection of Moyal deformations of self-dual gravity and self-dual Yang-Mills theory to chiral higher-spin theories, and also to deformations of operator algebras in celestial holography. The relation to Moyal deformations illuminates various aspects of the structure of chiral higher-spin theories. For instance, the appearance of the self-dual kinematic algebra in all the theories considered here leads via the double copy to vanishing tree-level scattering amplitudes. Regarding celestial holography, the Moyal deformation of self-dual gravity was recently shown to lead to the loop algebra of W∧, and we obtain here the analogous deformation of a Kac-Moody algebra corresponding to Moyal-deformed self-dual Yang-Mills theory. We also introduce the celestial algebras for various chiral higher-spin theories.

Detecting topological order from modular transformations of ground states on the torus

The ground states encode the information of the topological phases of a 2-dimensional system, which makes them crucial in determining the associated topological quantum field theory (TQFT). Most numerical methods for detecting the TQFT relied on the use of minimum entanglement states (MESs), extracting the anyon mutual statistics and self statistics via overlaps and/or the entanglement spectra. The MESs are the eigenstates of the Wilson loop operators, and are labeled by the anyons corresponding to their eigenvalues. Here we revisit the definition of the Wilson loop operators and MESs. We derive the modular transformation of the ground states purely from the Wilson loop algebra, and as a result, the modular $S$ and $T$ matrices naturally show up in the overlap of MESs. Importantly, we show that due to the phase degree of freedom of the Wilson loop operators, the MES-anyon assignment is not unique. This ambiguity obstructs our attempt to detect the topological order, that is, there exist different TQFTs that cannot be distinguished solely by the overlap of MESs. In this paper, we provide the upper limit of the information one may obtain from the overlap of MESs without other additional structure. Finally, we show that if the phase is enriched by rotational symmetry, there may be additional TQFT information that can be extracted from overlap of MESs.

Moyal deformations, W1+∞ and celestial holography

We consider the Moyal deformation of self-dual gravity. In the conformal primary basis, holomorphic collinear limits of the amplitudes of this theory show that it enjoys a perturbatively exact symmetry algebra LW∧ that generalises Lw∧, the loop algebra of the wedge algebra of w1+∞, which appears in self-dual gravity.