Loop Algebra Research Articles

Trigonometric weight functions asK-theoretic stable envelope maps for the cotangent bundle of a flag variety

We consider the cotangent bundle $T^*F_\lambda$ of a $GL_n$ partial flag variety, $\lambda=(\lambda_1,...,\lambda_N)$, $|\lambda|=\sum_i\lambda_i=n$, and the torus $T=(\C^\times)^{n+1}$ equivariant K-theory algebra $K_T(T^*F_\lambda)$. We introduce K-theoretic stable envelope maps $\Stab_{\sigma}: \oplus_{|\lambda|=n} K_T((T^*F_\lambda)^T)\to\oplus_{|\lambda|=n}K_T(T^*F_\lambda)$, where $\sigma\in S_n$. Using these maps we define a quantum loop algebra action on $\oplus_{|\lambda|=n}K_T(T^*F_\lambda)$. We describe the associated Bethe algebra $B^q(K_T(T^*F_\lambda))$ by generators and relations in terms of a discrete Wronski map. We prove that the limiting Bethe algebra $B^q(K_T(T^*F_\lambda))$, called the Gelfand-Zetlin algebra, coincides with the algebra of multiplication operators of the algebra $K_T(T^*F_\lambda)$. We conjecture that the Bethe algebra $B^q(K_T(T^*F_\lambda))$ coincides with the algebra of quantum multiplication on $K_T(T^*F_\lambda)$ introduced by Givental and Lee. The stable envelope maps are defined with the help of Newton polygons of Laurent polynomials representing elements of $K_T(T^*F_\lambda)$ and with the help of the trigonometric weight functions introduced in [TV1, TV3] to construct q-hypergeometric solutions of trigonometric qKZ equations. The paper has five appendices. In particular, in Appendix 5 we describe the Bethe algebra of the XXZ model by generators and relations.

A Few Super-Integrable Hierarchies and Some Reductions, Super-Hamiltonian Structures

Super-integrable equations provide some important physical models. In the paper, we have introduced two bases of the super Lie algebra B(0, 1). One of them is used to link three super hierarchies of evolution equations under the 0-curvature equations. We have constructed its two kinds of super loop algebras. By employing the first loop algebra, an isospectral problem was introduced for which a super (1 + 1)-dimensional integrable hierarchy is obtained, whose super Hamiltonian structure was derived from the super-trace identity proposed by Tu et al. As the reduction cases, we have obtained two different explicit super Schrödinger and mKdV systems. Further reduced equations are the well-known nonlinear Schrödinger equation, the super KdV equations and the mKdV equation. By employing the second loop algebra, we have obtained a new super integrable hierarchy containing 5 even superfunctions and 4 odd ones, which possesses super bi-Hamiltonian structure. As the reduction case, we have obtained an explicit super integrable system which is further reduced to 3 super integrable coupled equations, especially we have got a super coupled constrained equation with sources with respect to the odd superfunctions. Finally, we have used the TAH scheme to obtain a super (2+1)-dimensional hierarchy of evolution equations, whose Hamiltonian structure was generated by a new trace identity developed by us in the paper, which are new results, to the best of our knowledge.

On Demazure and local Weyl modules for affine hyperalgebras

We establish the existence of Demazure flags for graded local Weyl modules for hyper current algebras in positive characteristic. If the underlying simple Lie algebra is simply laced, the flag has length one, i.e., the graded local Weyl modules are isomorphic to Demazure modules. This extends to the positive characteristic setting results of Fourier-Littelmann and Naoi for current algebras in characteristic zero. Using this result, we prove that the character of local Weyl modules for hyper loop algebras depend only on the highest weight, but not on the (algebraically closed) ground field, and deduce a tensor product factorization for them.

Degeneration of Trigonometric Dynamical Difference Equations for Quantum Loop Algebras to Trigonometric Casimir Equations for Yangians

We show that, under Drinfeld's degeneration of quantum loop algebras to Yangians, the trigonometric dynamical difference equations for the quantum affine algebra degenerate to the trigonometric Casimir differential equations for Yangians.

EXTENSIONS AND BLOCK DECOMPOSITIONS FOR FINITE-DIMENSIONAL REPRESENTATIONS OF EQUIVARIANT MAP ALGEBRAS

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. The irreducible finite-dimensional representations of these algebras were classified in [NSS12], where it was shown that they are all tensor products of evaluation representations and one-dimensional representations. In the current paper, we describe the extensions between irreducible finite-dimensional representations of an equivariant map algebra in the case that X is an affine scheme of finite type and g is reductive. This allows us to also describe explicitly the blocks of the category of finite-dimensional representations in terms of spectral characters, whose definition we extend to this general setting. Applying our results to the case of generalized current algebras (the case where the group acting is trivial), we recover known results but with very different proofs. For (twisted) loop algebras, we recover known results on block decompositions (again with very different proofs) and new explicit formulas for extensions. Finally, specializing our results to the case of (twisted) multiloop algebras and generalized Onsager algebras yields previously unknown results on both extensions and block decompositions.

Two Kinds of New Integrable Couplings of the Negative-Order Korteweg-de Vries Equation

Based on some known loop algebras with finite dimensions, two different negative-order integrable couplings of the negative-order Korteweg-de Vries (KdV) hierarchy of evolution equations are generated by making use of the Tu scheme, from which the corresponding negative-order integrable couplings of the negative-order KdV equations are followed to be obtained. The resulting Hamiltonian structure of one negative integrable coupling is derived from the variational identity.

Kinematics and Dynamics Analysis of an Innovative Face-shovel Hydraulic Excavator in Mining

A new face-shovel hydraulic excavator is proposed, which can effectively increase the thrust and stroke of the bucket cylinder. By analyzing the degrees of freedom of the working device, the result shows that the degrees of freedom are two translations and one rotation on a plane, which can meet the requirements of face-shovel excavators. Based on the graphical modularization and loop algebra theory, a universal mechanical kinematics analysis flow diagram is established, with which the kinematics of the mechanism is investigated. By numerical analysis method, five numerical examples are given to verify the correctness of kinematics analysis. Comparative analyses of the results show that the algorithm is reliable and accurate. Based on Kane's method and the equivalent torque principle, the dynamics analyses of the new-type mechanism are conducted, and the dynamic mathematical model is deduced and established. Referring to dimension parameters of the caterpillar face-shovel excavator, the structural parameters of the new hydraulic face-shovel excavator are given. Based on Matlab, simulation researches are worked according to the dynamics equations, in the case of horizontal operation, respectively for no load and variable resistance. The analysis results show that the method is reliable. The research result will lay the foundation for the design and application of the novel mechanism.

Integrable perturbations of conformal field theories and Yetter-Drinfeld modules

In this paper we relate a problem in representation theory — the study of Yetter-Drinfeld modules over certain braided Hopf algebras — to a problem in two-dimensional quantum field theory, namely, the identification of integrable perturbations of a conformal field theory. A prescription that parallels Lusztig's construction allows one to read off the quantum group governing the integrable symmetry. As an example, we illustrate how the quantum group for the loop algebra of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(2)$\end{document}sl(2) appears in the integrable structure of the perturbed uncompactified and compactified free boson.

Orbit Approach to Separation of Variables in $${\mathfrak{sl}}$$ sl (3)-Related Integrable Systems

Using the orbit method we attempt to reveal geometric and algebraic meaning of separation of variables for the integrable systems on coadjoint orbits in an $\mathfrak{sl}(3)$ loop algebra. We consider two types of generic orbits embedded into a common manifold, endowed with two nonsingular Lie-Poisson brackets. We prove that separation of variables on orbits of both types is realized by the same variables of separation. We also construct the integrable systems on these orbits: a coupled 3-component nonlinear Schr\"{o}dinger equation and an isotropic SU(3) Landau-Lifshitz equation.

Integrable couplings, bi-integrable couplings and their Hamiltonian structures of the Giachetti-Johnson soliton hierarchy

On the basis of zero curvature equations from semi-direct sums of Lie algebras, we construct integrable couplings of the Giachetti–Johnson hierarchy of soliton equations. We also establish Hamiltonian structures of the resulting integrable couplings by the variational identity. Moreover, we obtain bi-integrable couplings of the Giachetti–Johnson hierarchy and their Hamiltonian structures by applying a class of non-semisimple matrix loop algebras consisting of triangular block matrices. Copyright © 2014 John Wiley & Sons, Ltd.

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.

Hamiltonian forms of the two new integrable systems and two kinds of Darboux transformations

Based on the Lie algebra gl(2), taking a kind of corresponding loop algebra gl(2)∼, a new Lax integrable hierarchy can be obtained. Then, by means of the quadratic-form identity, the corresponding bi-Hamiltonian structure was worked out. Expanding Lie algebra gl(2), and making use of the new zero curvature equation Zhang (2008) [9], we obtain an integrable hierarchy and its Hamiltonian structure. At last, two kinds of Darboux transformations of the equation are generated.

Twisted Quantum Toroidal Algebras $${T_q^-(\mathfrak g)}$$ T q - ( g )

We construct a principally graded quantum loop algebra for the Kac-Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.

An so (3, ℝ) Counterpart of the Dirac Soliton Hierarchy and its Bi-Integrable Couplings

We derive a counterpart hierarchy of the Dirac soliton hierarchy from zero curvature equations associated with a matrix spectral problem from so (3, ℝ). Inspired by a special class of non-semisimple loop algebras, we construct a hierarchy of bi-integrable couplings for the counterpart soliton hierarchy. By applying the variational identities which cope with the enlarged Lax pairs, we generate the corresponding Hamiltonian structure for the hierarchy of the resulting bi-integrable couplings. To show Liouville integrability, infinitely many commuting symmetries and conserved densities are presented for the counterpart soliton hierarchy and its hierarchy of bi-integrable couplings.

Graded Limits of Minimal Affinizations in Type D

We study the graded limits of minimal affinizations over a quantum loop algebra of type D in the regular case. We show that the graded limits are isomorphic to multiple generalizations of Demazure modules, and also give their defining relations. As a corollary we obtain a character formula for the minimal affinizations in terms of Demazure operators, and a multiplicity formula for a special class of the minimal affinizations.

Finite-Dimensional Representations of Twisted Hyper-Loop Algebras

We investigate the category of finite-dimensional representations of twisted hyper-loop algebras, i.e., the hyperalgebras associated to twisted loop algebras over finite-dimensional simple Lie algebras. The main results are the classification of the irreducible modules, the definition of the universal highest-weight modules, called the Weyl modules, and, under a certain mild restriction on the characteristic of the ground field, a proof that the simple modules and the Weyl modules for the twisted hyper-loop algebras are isomorphic to appropriate simple and Weyl modules for the nontwisted hyper-loop algebras, respectively, via restriction of the action.

The super-bihamiltonian reduction on C∞(1, OSP(1|2))

In this article, we will show that the super-bihamiltonian structures of the Kuper-KdV equation in [3], the Kuper-CH equation in [17, 18] and the super-HS equation in [11, 16,19] can be obtained by applying a super-bihamiltonian reduction of different super-Poisson pairs defined on the loop algebra of osp(1|2).

Twisted Yangians, twisted quantum loop algebras and affine Hecke algebras of type $BC$

We study twisted Yangians of type AIII which have appeared in the literature under the name of reflection algebras. They admit q-versions which are new twisted quantum loop algebras. We explain how these can be defined equivalently either via the reflection equation or as coideal subalgebras of Yangians of gln (resp. of quantum loop algebras of gln). The connection with affine Hecke algebras of type BC comes from a functor of Schur-Weyl type between their module categories.

Multiloop algebras, iterated loop algebras and extended affine Lie algebras of nullity 2

Let \mathbb M_n be the class of all multiloop algebras of finite dimensional simple Lie algebras relative to n -tuples of commuting finite order automorphisms. It is a classical result that \mathbb M_1 is the class of all derived algebras modulo their centres of affine Kac-Moody Lie algebras. This combined with the Peterson-Kac conjugacy theorem for affine algebras results in a classification of the algebras in \mathbb M_1 . In this paper, we classify the algebras in \mathbb M_2 , and further determine the relationship between \mathbb M_2 and two other classes of Lie algebras: the class of all loop algebras of affine Lie algebras and the class of all extended affine Lie algebras of nullity 2.

Global Weyl Modules for Equivariant Map Algebras

Equivariant map algebras are Lie algebras of algebraic maps from a scheme (or algebraic variety) to a target finite-dimensional Lie algebra (in the case of the current paper, we assume the latter is a simple Lie algebra) that are equivariant with respect to the action of a finite group. In the first part of this paper, we define global Weyl modules for equivariant map algebras satisfying a mild assumption. We then identify a commutative algebra A that acts naturally on the global Weyl modules, which leads to a Weyl functor from the category of A-modules to the category of modules for the equivariant map algebra in question. These definitions extend the ones previously given for generalized current algebras (i.e. untwisted map algebras) and twisted loop algebras. In the second part of the paper, we restrict our attention to equivariant map algebras where the group involved is abelian, acts on the target Lie algebra by diagram automorphisms, and freely on (the set of rational points of) the scheme. Under these additional assumptions, we prove that A is finitely generated and the global Weyl module is a finitely generated A-module. We also define local Weyl modules via the Weyl functor and prove that these coincide with the local Weyl modules defined directly in a previous paper. Finally, we show that A is the algebra of coinvariants of the analogous algebra in the untwisted case.