Loop Algebra Research Articles

Hidden symmetries and killing tensors on curved spaces

Higher-order symmetries corresponding to Killing tensors are investigated. The intimate relation between Killing-Yano tensors and nonstandard supersymmetries is pointed out. In the Dirac theory on curved spaces, Killing-Yano tensors generate Dirac-type operators involved in interesting algebraic structures as dynamical algebras or even infinite dimensional algebras or superalgebras. The general results are applied to space-times which appear in modern studies. One presents the infinite dimensional superalgebra of Dirac type operators on the 4-dimensional Euclidean Taub-NUT space that can be seen as a twisted loop algebra. The existence of the conformal Killing-Yano tensors is investigated for some spaces with mixed 3-Sasakian structures.

Some classes of semisimple group (and loop) algebras over finite fields

We determine the structure of the semisimple group algebra of certain groups over the rationals and over those finite fields where the Wedderburn decompositions have the least number of simple components. We apply our work to obtain similar information about the loop algebras of indecomposable RA loops and to produce negative answers to the isomorphism problem over various fields.

Two new integrable couplings of the soliton hierarchies with self-consistent sources

A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with s̃l(4) has been presented (Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6613). Based on this method, we construct two integrable couplings of the soliton hierarchy with self-consistent sources by using the loop algebra s̃l(4). In this paper, we also point out that there are some errors in these references and we have corrected these errors and set up new formula. The method can be generalized to other soliton hierarchy with self-consistent sources.

An integrable couplings of G-WKI equations hierarchy with self-consistent sources

We present a kind of integrable couplings of soliton equations hierarchy with self-consistent sources by using the loop algebra s l ̃ ( 4 ) . As an application, a new integrable couplings of the generalized Wadati–Konono–Ichikawa (G-WKI) equations hierarchy with self-consistent sources is derived through the enlarged loop algebra s l ̃ ( 4 ) .

Constructing nonlinear discrete integrable Hamiltonian couplings

Beginning with Lax pairs from special non-semisimple matrix Lie algebras, we establish a scheme for constructing nonlinear discrete integrable couplings. Discrete variational identities over the associated loop algebras are used to build Hamiltonian structures for the resulting integrable couplings. We illustrate the application of the scheme by means of an enlarged Volterra spectral problem and present an example of nonlinear discrete integrable Hamiltonian couplings for the Volterra lattice equations.

A New Lie Algebra and Its Related Liouville Integrable Hierarchies

A new Lie algebra G and its two types of loop algebras G̃1 and G̃2 are constructed. Basing on G̃1 and G̃2, two different isospectral problems are designed, furthermore, two Liouville integrable soliton hierarchies are obtained respectively under the framework of zero curvature equation, which is derived from the compatibility of the isospectral problems expressed by Hirota operators. At the same time, we obtain the Hamiltonian structure of the first hierarchy and the bi-Hamiltonian structure of the second one with the help of the quadratic-form identity.

Kac's Theorem for weighted projective lines

We prove an analogue of Kac’s Theorem, describing the dimension types of indecomposable coherent sheaves (or parabolic bundles) over weighted projective lines in terms of root systems for loop algebras of Kac–Moody Lie algebras. We use a theorem of Peng and Xiao to associate a Lie algebra to the category of coherent sheaves for a weighted projective line over a finite field, and find elements of this Lie algebra which satisfy the relations defining the loop algebra. We use these elements in the proof of our analogue of Kac’s Theorem.

Four Lie algebras associated with R6 and their applications

The first part in the paper reads that a three-dimensional Lie algebra is first introduced, whose corresponding loop algebra is constructed, for which isospectral problems are established. By employing zero curvature equations, a modified Kaup–Newell (mKN) soliton hierarchy of evolution equations is obtained. The corresponding hereditary operator and Hamiltonian structure are worked out, respectively. Then two types of enlarging semisimple Lie algebras isomorphic to the linear space R6 are followed to construct, one of them is a complex Lie algebra. Their corresponding loop algebras are also given so that two types of new isospectral problems are introduced to generate two kinds of integrable couplings of the above mKN hierarchy. The hereditary operators, Hamiltonian structures of the hierarchies are produced again, respectively. The exact computing formulas of the constant γ appearing in the trace identity and the variational identity are derived under the semisimple algebras. The second part of this paper is devoted to constructing two kinds of Lie algebras by using product of complex vectors, which are also isomorphic to the linear space R6. Then we make use of the corresponding loop algebras to produce two integrable hierarchies along with bi-Hamiltonian structures. From various aspects, we give some ways for constructing Lie algebras which have extensive applications in generating integrable Hamiltonian systems.

THE STRUCTURE OF THIN LIE ALGEBRAS WITH CHARACTERISTIC TWO

Thin Lie algebras are graded Lie algebras [Formula: see text] with dim Li ≤ 2 for all i, and satisfying a more stringent but natural narrowness condition modeled on an analogous condition for pro-p-groups. The two-dimensional homogeneous components of L, which include L1, are named diamonds. Infinite-dimensional thin Lie algebras with various diamond patterns have been produced, over fields of positive characteristic, as loop algebras of suitable finite-dimensional simple Lie algebras, of classical or of Cartan type depending on the location of the second diamond. The goal of this paper is a description of the initial structure of a thin Lie algebra, up to the second diamond. Specifically, if Lk is the second diamond of L, then the quotient L/Lk is a graded Lie algebras of maximal class. In odd characteristic p, the quotient L/Lk is known to be metabelian, and hence uniquely determined up to isomorphism by its dimension k, which ranges in an explicitly known set of possible values: 3, 5, a power of p, or one less than twice a power of p. However, the quotient L/Lk need not be metabelian in characteristic two. We describe here all the possibilities for L/Lk up to isomorphism. In particular, we prove that k + 1 equals a power of two.

Coupling integrable couplings and bi-Hamiltonian structure associated with the Boiti–Pempinelli–Tu hierarchy

A kind of new Lie algebra is introduced whose corresponding loop algebra is followed to define. From which two isospectral problems are established whose compatibility condition gives rise to a coupling integrable coupling of the Boiti–Pempinelli–Tu (BPT) hierarchy. By making use of the quadratic-form identity, its bi-Hamiltonian structure is obtained. Finally, we reduce the coupling integrable coupling to obtain two integrable couplings of the BPT hierarchy. Furthermore, the well-known mKdV equation and a new coupled equation with two dependent variables are produced, respectively.

A New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent Sources

A kind of integrable couplings of soliton equations hierarchy with self-consistent sources associated with is presented by Yu. Based on this method, we construct a new integrable couplings of the classical-Boussinesq hierarchy with self-consistent sources by using of loop algebra . In this paper, we also point out that there exist some errors in Yu's paper and have corrected these errors and set up new formula. The method can be generalized other soliton hierarchy with self-consistent sources.

Expanding integrable models of the nonlinear Schrödinger equation

By using a few Lie algebras and the corresponding loop algebras, we establish some isospectral problems whose compatibility conditions give rise to a few various expanding integrable models (including integrable couplings) of the well-known nonlinear Schrödinger equation. The Hamiltonian forms of two of them are generated by making use of the variational identity. Finally, we propose an efficient method for generating a nonlinear integrable coupling of the nonlinear Schrödinger equation.

The trigonometric Casimir connection of a simple Lie algebra

Let g be a complex, semisimple Lie algebra, G the corresponding simply-connected Lie group and H ⊂ G a maximal torus. We construct a flat connection on H with logarithmic singularities on the root hypertori and values in the Yangian Y ( g ) of g . By analogy with the rational Casimir connection of g , we conjecture that the monodromy of this trigonometric connection is described by the quantum Weyl group operators of the quantum loop algebra U ℏ ( L g ) .

INTEGRABLE HAMILTONIAN HIERARCHIES ASSOCIATED WITH THE EQUATION OF HEAT CONDUCTION

Using a 4-dimensional Lie algebra g, an isospectral Lax pair is introduced, whose compatibility condition is equivalent to a soliton hierarchy of evolution equations with three components of potential functions. Its Hamiltonian structure is obtained by employing the quadratic-form identity proposed by Guo and Zhang. In order to obtain explicit Hamiltonian functions, a detailed computing formula for the constant appearing in the quadratic-form identity is obtained. One type of reduction equations of the hierarchy is also produced, which is further reduced to the standard equation of heat conduction. By introducing a loop algebra of the Lie algebra g, we obtain a soliton hierarchy with an arbitrary parameter which can be reduced to the previous equation hierarchy obtained, whose quasi-Hamiltonian structure is also worked out by the quadratic-form identity. Finally, we extend the Lie algebra g into a higher-dimensional Lie algebra so that a new integrable Hamiltonian hierarchy, which comprise integrable couplings, is produced; its reduced equations in particular contain two arbitrary parameters.

Finite dimensional modules for the q-tetrahedron algebra

In [7] the $q$ tetrahedron algebra $\boxtimes_{q}$ was introduced as a $q$ analogue of the universal enveloping algebra of the three point loop algebra $\mathit{sl}_{2} \otimes \mathbf{C}[t,t^{-1},(t-1)^{-1}]$. In this paper the relation between finite dimensional $\boxtimes_{q}$ modules and finite dimensional modules for $U_{q}(L(\mathit{sl}_{2}))$, a $q$ analogue of the loop algebra $L(\mathit{sl}_{2})$, is studied. A connection between the $\boxtimes_{q}$ module structure and $L$-operators for $U_{q}(L(\mathit{sl}_{2}))$ is also discussed.

A new non-isospectral integrable couplings for generalized Volterra lattice hierarchy

In this paper, we apply the new algebraic method to construct a integrable couplings of non-isospectral soliton equations by a loop algebra sl ˜ ( 4 ) . Furthermore, the non-isospectral integrable coupling system of the generalized Volterra lattice hierarchy is obtained.

On the classification of compact quantum groups U θ(2)

In this paper, we show that the corepresentation of compact group Uθ(2) on the vector space is determined by its infinitesimal generators B0, B2, A0, A1 and A2, where θ is an irrational number. We also show that B0 and B2 commute with Aj, j = 0, 1, 2, so B0, A0, A1 and A2 are sl(2, ℂ) loop algebra. Then we exhibit all irreducible representations of Uθ(2), which are different from those of group U(2), and use above results to give the classification of quantum groups Uθ(2), which is analogous to that of irrational algebra Aθ. At the same time, we also give all the forms of automorphisms on quantum group Uθ(2).

Integrable couplings of the (2 + 1)-dimensional dispersive long wave hierarchy and its Hamiltonian structure

A higher loop algebra is constructed, from which the integrable couplings associated with the (2 + 1)-dimensional dispersive long wave hierarchy is obtained with the help of the (2 + 1) zero curvature equation generated from one of reduced equations of the self-dual Yang–Mills equations. Furthermore, the Hamiltonian structure of the integrable couplings is worked out by taking use of the variational identity.

Three kinds of coupling integrable couplings of the Korteweg–de Vries hierarchy of evolution equations

We introduce three kinds of column-vector Lie algebras Ls(s=1,2,3). By making invertible linear transformations we get the corresponding three induced Lie algebras. According to the defined loop algebras L̃s of the Lie algebras Ls(s=1,2,3), we establish three various isospectral problems. Then by applying Tu scheme, we obtain three different coupling integrable couplings of the Korteweg–de Vries (KdV) hierarchy and further reduce them to three kinds of explicit coupling integrable couplings of the KdV equation. One of the coupling integrable couplings of the KdV hierarchy of evolution equations possesses Hamiltonian structure obtained by using the quadratic-form identity and it is Liouville integrable.

A LIOUVILLE INTEGRABLE MULTI-COMPONENT INTEGRABLE SYSTEM AND ITS INTEGRABLE COUPLINGS

A Liouville integrable multi-component integrable system is obtained by the vector loop algebra. Then, the integrable couplings of the above system are presented by using the expanding vector loop algebra [Formula: see text] of the [Formula: see text]. Finally, the bi -Hamiltonian structure of the obtained system is given, respectively, by the variational identity.