Special Orthogonal Lie Research Articles

Integrable reductions of a soliton hierarchy associated with so(3,R)

We construct integrable reductions of a soliton hierarchy associated with the special orthogonal Lie algebra so ( 3 , R ) . The resulting reduced integrable equations include a nonlinear Schrödinger type equation and a modified Korteweg–de Vries type equation. There are two kinds of integrable reductions in our analysis, but they lead to essentially the same scalar integrable equations. This is a particular phenomenon for soliton equations associated with so ( 3 , R ) , which is different from the one for soliton equations associated with sl ( 2 , R ) . • Scalar integrable equations associated with so(3,R). • Innovative reductions, which keep the existence of infinitely many symmetries and conservation laws.

Integrable couplings of two expanded non-isospectral soliton hierarchies and their bi-Hamiltonian structures

Starting from two matrix spectral problems associated with the real special orthogonal Lie algebra [Formula: see text], two non-isospectral hierarchies of Ablowitz–Kaup–Newell–Segur (AKNS) type and Kaup–Newell (KN) type are constructed. In addition, we enlarge the two spectral problems and then obtain non-isospectral [Formula: see text] integrable couplings of AKNS type and KN type by solving the expanded non-isospectral zero curvature equation. We find that the obtained four hierarchies have the bi-Hamiltonian structure of the combined form. It follows that all equations in the resulting soliton hierarchy are integrable in the sense of Liouville.

Integrable nonlocal nonlinear Schrödinger equations associated with 𝑠𝑜(3,ℝ)

We construct integrable PT-symmetric nonlocal reductions for an integrable hierarchy associated with the special orthogonal Lie algebra so ⁡ ( 3 , R ) \operatorname {so}(3,\mathbb {R}) . The resulting typical nonlocal integrable equations are integrable PT-symmetric nonlocal reverse-space, reverse-time and reverse-spacetime nonlinear Schrödinger equations associated with so ⁡ ( 3 , R ) \operatorname {so}(3,\mathbb {R}) .

Integrable Nonlocal PT-Symmetric Modified Korteweg-de Vries Equations Associated with so(3, \({\mathbb{R}}\))

We construct integrable PT-symmetric nonlocal reductions for an integrable hierarchy associated with the special orthogonal Lie algebra so(3,R). The resulting typical nonlocal integrable equations are integrable PT-symmetric nonlocal complex reverse-spacetime and real reverse-spacetime modified Korteweg-de Vries equations associated with so(3,R).

Orthogonal decompositions of Lie algebras over finite commutative rings

Let [Formula: see text] be a finite commutative ring with identity. In this paper, we give a necessary condition for the existence of an orthogonal decomposition of the special linear Lie algebra over [Formula: see text]. Additionally, we study orthogonal decompositions of the symplectic Lie algebra and the special orthogonal Lie algebra over [Formula: see text].

On quantum adjacency algebras of Doob graphs and their irreducible modules

For fixed integers $$n \ge 1$$ and $$m \ge 0$$ , we consider the Doob graph $$D=D(n,m)$$ formed by taking direct product of n copies of Shrikhande graph and m copies of complete graph $$K_4$$ . Fix a vertex x of D, and let $$T=T(x)$$ denote the Terwilliger algebra of D with respect to x. Let A denote the adjacency matrix of D. There exists a decomposition of A into a sum $$A = L + F + R$$ of elements in T where L, F, and R are the lowering, flat, and raising matrices, respectively. We call $$A = L + F + R$$ the quantum decomposition of A. Hora and Obata (Quantum Probability and Spectral Analysis of Graphs. Theoretical and Mathematical Physics, Springer, Berlin, 2007) introduced a semi-simple matrix algebra based on the quantum decomposition of the adjacency matrix. This algebra is generated by the quantum components of the decomposition and is called the quantum adjacency algebra of the graph. Let $$Q=Q(x)$$ denote the quantum adjacency algebra of D with respect to x. In this paper, we display an action of the special orthogonal Lie algebra $$\mathfrak {so}_4$$ on the standard module for D. We also prove Q is generated by the center and the homomorphic image of the universal enveloping algebra $$U(\mathfrak {so}_4)$$ . To do these, we exploit the work of Tanabe (JAC 6: 173–195, 1997) on irreducible T-modules of D.

Darboux transformation for a generalized Ablowitz-Kaup-Newell-Segur hierarchy equation

A generalized Ablowitz-Kaup-Newell-Segur hierarchy of integrable equation from zero curvature equation is constructed based on the special orthogonal Lie algebra so(3, R), and the second equation is solved by the Darboux transformation. Besides, the soliton solutions are presented with the help of symbolic computation. Two special cases are given to make the solution more intuitive, and the dynamical properties of the solutions are discussed through the analysis of the graphics.

EXACT SOLUTIONS FOR A DIRAC-TYPE EQUATION WITH N-FOLD DARBOUX TRANSFORMATION

Based on matrix spectral problems associated with the real special orthogonal Lie algebra so(3, $\mathbb{R}$), a Dirac-type equation is derived by virtue of the zero-curvature equation. Further, an N-fold Darboux transformation for the Dirac-type equation is constructed by means of the gauge transformation. Finally, as its application, some exact solutions and their figures are obtained via symbolic computation software (Maple).

Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)

Abstract In our paper, the theory of bi-integrable and tri-integrable couplings is generalized to the discrete case. First, based on the six-dimensional real special orthogonal Lie algebra SO(4), we construct bi-integrable and tri-integrable couplings associated with SO(4) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by the variational identities.

Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with SO(3)

Based on the three-dimensional real special orthogonal Lie algebra SO(3), by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with SO(3) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by applying the variational identities.

On matrix Lie rings over an associative commutative ring with 1 that contain the special orthogonal Lie ring

Let K be an associative and commutative ring with 1, and k a subring of K such that and the elements 2, 3 are invertible in k. Let n be an integer and an invertible diagonal n by n matrix over k. In the paper, Lie subrings of the general linear Lie ring containing the special orthogonal Lie ring are described provided that .

New soliton hierarchies associated with the real Lie algebra

We present some new matrix spectral problems, based on the real special orthogonal Lie algebra , and construct corresponding soliton hierarchies by means of zero curvature equations associated with these spectral problems. With the aid of symbolic computation by Maple, new soliton hierarchies of Kaup–Newell type, Ablowitz–Kaup–Newell–Segur type and Wadati–Konno–Ichikawa type are obtained to illustrate the use of . Copyright © 2016 John Wiley & Sons, Ltd.

A New Soliton Hierarchy Associated withso⁡3,Rand Its Conservation Laws

Based on the three-dimensional real special orthogonal Lie algebraso(3,R), we construct a new hierarchy of soliton equations by zero curvature equations and show that each equation in the resulting hierarchy has a bi-Hamiltonian structure and thus integrable in the Liouville sense. Furthermore, we present the infinitely many conservation laws for the new soliton hierarchy.

New Soliton Hierarchies Associated with the Lie Algebra so(3, [formula omitted]) and their BI-Hamiltonian Structures

We present three new matrix spectral problems, based on the real special orthogonal Lie algebra so (3, ℝ), and construct the corresponding asymmetric soliton hierarchies of AKNS type, KN type and WKI type with the aid of symbolic computation by Maple. Bi-Hamiltonian structures yielding Liouville integrability of the resulting soliton hierarchies are furnished by the trace identity, and so, all newly presented equations possess infinitely many commuting symmetries and conservation laws.

Bi-Integrable Couplings of a New Soliton Hierarchy Associated withSO(4)

Based on the six-dimensional real special orthogonal Lie algebraSO(4), a new Lax integrable hierarchy is obtained by constructing an isospectral problem. Furthermore, we construct bi-integrable couplings for this hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Hamiltonian structures of the obtained bi-integrable couplings are constructed by the variational identity.

Integrable counterparts of the D-Kaup–Newell soliton hierarchy

Two integrable counterparts of the D-Kaup–Newell soliton hierarchy are constructed from a matrix spectral problem associated with the three dimensional special orthogonal Lie algebra so(3,R). An application of the trace identity presents Hamiltonian or quasi-Hamiltonian structures of the resulting counterpart soliton hierarchies, thereby showing their Liouville integrability, i.e., the existence of infinitely many commuting symmetries and conserved densities. The involved Hamiltonian and quasi-Hamiltonian properties are shown by computer algebra systems.

Two soliton hierarchies associated with SO(4) and the applications of SU(2) ⊗ SU(2) ≅ SO(4)

Based on the six-dimensional real special orthogonal Lie algebra SO(4), we construct two different isospectral problems. And the two hierarchies of soliton equations from zero curvature equations can be generated. At last, we use the property of the isomorphism SU(2) ⊗ SU(2) ≅ SO(4), and obtain the relations between the hierarchy of SU(2) ⊗ SU(2) and the hierarchy of SO(4).

A spectral problem based on so$(3,\mathbb {R})$(3,R) and its associated commuting soliton equations

We present a spectral problem, based on the three-dimensional real special orthogonal Lie algebra $\textrm {so}(3,\mathbb {R})$ so (3,R), and construct a hierarchy of commuting bi-Hamiltonian soliton equations by zero curvature equations associated with the spectral problem. An illustrative example of soliton equations is computed, together with its associated bi-Hamiltonian structure.

Classification of embeddings of abelian extensions of [formula omitted] into [formula omitted

An abelian extension of the special orthogonal Lie algebra Dn is a nonsemisimple Lie algebra Dn⨭V, where V is a finite-dimensional representation of Dn, with the understanding that [V,V]=0. We determine all abelian extensions of Dn that may be embedded into the exceptional Lie algebra En+1, n=5,6, and 7. We then classify these embeddings, up to inner automorphism. As an application, we also consider the restrictions of irreducible representations of En+1 to Dn⨭V, and discuss which of these restrictions are or are not indecomposable.

Nilpotent Conjugacy Classes inp-adic Lie Algebras: The Odd Orthogonal Case

AbstractWe will study the following question: Are nilpotent conjugacy classes of reductive Lie algebras overp-adic fields definable? By definable, we mean definable by a formula in Pas's language. In this language, there are no field extensions and no uniformisers. Using Waldspurger's parametrization, we answer in the affirmative in the case of special orthogonal Lie algebras so(n) fornodd, overp-adic fields.