Special Lie Algebra Research Articles

Полная группа изометрий специальной трёхмерной группы Ли

We consider a special three-dimensional Lie algebra 3 of Bianchi type V. A matrix representation of this Lie algebra and the corresponding connected simply connected Lie group 𝑆3 is found and natural coordinates are introduced, which are determined by the matrix representation. Formulas for exponential mapping with respect to natural coordinates are found. Complete groups of autoisometries of the special Lie algebra with respect to the Euclidean or Lorentz scalar product are written out. The leftinvariant Riemannian metric of the Lie group 𝑆3 and the complete group of isometries of the resulting homogeneous manifold are found. This manifold turned out to be a space of constant Ricci curvature.

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.

The Weakly Solvable Radical and Locally Strongly Algebraic Derivations of Locally Generalized Special Lie Algebras

In this paper, the classical theorem on the image of the solvable radical of a finite-dimensional Lie algebra over a field of characteristic zero under the action of its derivation is generalized to locally generalized special Lie algebras.

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.

On a locally nilpotent radical Jacobson for special Lie algebras

In the paper investigates the possibility of homological description of Jacobson radical and locally nilpotent radical for Lie algebras, and their relation with a $$PI$$ - irreducibly represented radical, and some properties of primitive Lie algebras are studied. We prove an analog of The F. Kubo theorem for almost locally solvable Lie algebras with a zero Jacobson radical. It is shown that the Jacobson radical of a special almost locally solvable Lie algebra $$L$$ over a field $$F$$ of characteristic zero is zero if and only if the Lie algebra $$L$$ has a Levi decomposition $$L=S\oplus Z(L)$$, where $$Z(L)$$ is the center of the algebra $$L$$, $$S$$ is a finite-dimensional subalgebra $$L$$ such that $$J(L)=0$$. For an arbitrary special Lie algebra $$L$$, the inclusion of $$IrrPI(L)\subset J(L)$$ is shown, which is generally strict. An example of a Lie algebra $$L$$ with strict inclusion $$J(L)\subset IrrPI(L)$$ is given. It is shown that for an arbitrary special Lie algebra $$L$$ over the field $$F$$ of characteristic zero, the inclusion of $$N (L)\subset IrrPI(L)$$, which is generally strict. It is shown that most Lie algebras over a field are primitive. An example of an Abelian Lie algebra over an algebraically closed field that is not primitive is given. Examples are given showing that infinite-dimensional commutative Lie algebras are primitive over any fields; a finite-dimensional Abelian algebra of dimension greater than 1 over an algebraically closed field is not primitive; an example of a non-Cartesian noncommutative Lie algebra is primitive. It is shown that for special Lie algebras over a field of characteristic zero $$PI$$-an irreducibly represented radical coincides with a locally nilpotent one. An example of a Lie algebra whose locally nilpotent radical is neither locally nilpotent nor locally solvable is given. Sufficient conditions for the primitiveness of a Lie algebra are given, and examples of primitive Lie algebras and non-primitive Lie algebras are given.

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.

Modular Vector Fields Attached to Dwork Family: s l 2 ( C ) Lie algebra

This paper aims to show that a certain moduli space $\textsf{T}$, which arises from the so-called Dwork family of Calabi-Yau $n$-folds, carries a special complex Lie algebra containing a copy of $\mathfrak{sl}_2(\mathbb{C})$. In order to achieve this goal, we introduce an algebraic group $\sf G$ acting from the right on $\textsf{T}$ and describe its Lie algebra ${\rm Lie}({\sf G})$. We observe that ${\rm Lie}({\sf G})$ is isomorphic to a Lie subalgebra of the space of the vector fields on $\textsf{T}$. In this way, it turns out that ${\rm Lie}({\sf G})$ and the modular vector field ${\sf R}$ generate another Lie algebra $\mathfrak{G}$, called AMSY-Lie algebra, satisfying $\dim (\mathfrak{G})=\dim (\textsf{T})$. We find a copy of $\mathfrak{sl}_2(\mathbb{C})$ containing ${\sf R}$ as a Lie subalgebra of $\mathfrak{G}$. The proofs are based on an algebraic method calling "Gauss-Manin connection in disguise". Some explicit examples for $n=1,2,3,4$ are stated as well.

О почти локально разрешимых алгебрах Ли с нулевым радикалом Джекобсона и локально нильпотентном радикале для алгебр Ли

В статье доказывается аналог теоремы Ф. Кубо [1] для почти локально разрешимых алгебр Ли с нулевым радикалом Джекобсона. Первый раздел направлен на выяснение некоторых аспектов гомологического описания радикала Джекобсона. Доказана теорема, обобщающая теорему Е. Маршалла на случай почти локально разрешимых алгебр Ли, следствием которой и является аналог теоремы Кубо. Во втором разделе исследуются некоторые свойства локально нильпотентного радикала алгебры Ли. Рассматриваются примитивные алгебры Ли. Приведены примеры, показывающие, что бесконечномерные коммутативные алгебры Ли являются примитивными над любыми полями; конечномерная абелева алгебра, размерности больше 1, над алгебраически замкнутым полем не является примитивной; пример неартиновой некоммутативной алгебры Ли являющейся примитивной. Показано, что для специальных алгебр Ли над полем характеристики нуль PI-неприводимо представленный радикал совпадает с локально нильпотентным. Приведен пример алгебры Ли, локально нильпотентный радикал которой не является ни локально нильпотентным, ни локально разрешимым. Даются достаточные условия примитивности алгебры Ли, приводятся примеры примитивных алгебр Ли и алгебры Ли не являющейся примитивной.

Models of Lie algebra sl(2,C) and special matrix functions by means of a matrix integral transformation

In this paper, we present four models of irreducible representations of special complex Lie algebra s l ( 2 , C ) from the special matrix functions point of view. These models, which involve differential operators, are transformed into matrix difference–differential operators using an integral transformation motivated by the integral representation of beta matrix function. We also obtain the matrix identities involving one or two variable special matrix functions.

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).

Some novel soliton solution, breather solution and Darboux transformation for a generalized coupled Toda soliton hierarchy

A few of discrete integrable coupling systems(DICSs) of previous papers are linear discrete integrable couplings(LDICS). We take a special matrix Lie algebra system(non-semisimple) to construct the Lax pairs, and establish a method for deriving the nonlinear discrete integrable coupling systems(NDICS). From the Lax pairs of the generalized Toda(G-Toda) spectral problem, we can derive a novel NDICS, which is a real NDICS. For the obtained lattice integrable coupling equation, we establish a Darboux transformation (DT) with 4 × 4 Lax pairs, and apply the gauge transformation to a specific equation, then the explicit solutions of the lattice integrable coupling equation are given, which contains discrete soliton solution, breather solution and rogue wave solution. Furthermore, we can derive the discrete explicit solutions with free parameters to depict their dynamic behaviors.

Развитие понятия «артиновость» для алгебр Ли

В статье рассматривается развитие понятия "артиновость"для алгебр Ли. Понятие артиновости было введено для ассоциативных колец с условием минимальности. Одновременно с этим оно распространилось на модули и подалгебры. Чуть позже стали рассматривать артиновы йордановы алгебры. Для таких алгебр роль одностороннего идеала играет квадратичный идеал или, как назвал его Н.Джекобсон, вутренний идеал. Артиновость для алгебр Ли через идеалы определяли Ю.А. Бахтурин, С.А. Пихтильков и В.М. Поляков. Они рассматривали специальные артиновы алгебры Ли. С.А. Пихтильков применял артиновы алгебры Ли для построения структурной теории специальных алгебр Ли. Джорджия Бенкарт определила артиновость для алгебр Ли через внутренние идеалы. Ф. Лопес, Е. Гарсия, Г. Лозано исследовали понятие внутреннего идеала применительно к артиновости с помощью йордановых пар. Определение артиновости для алгебр Ли в данной статье представлено в трёх смыслах: через подалгебры, идеалы и внутренние идеалы. Представлена установленная авторами ранее связь между данными определениями. Рассмотрены примеры артиновых алгебр Ли. Описано применение артиновых алгебр Ли к решению проблемы Михалева: первичный радикал артиновой алгебры Ли является разрешимым.

О ГОМОЛОГИЧЕСКОМ ОПИСАНИИ РАДИКАЛА ДЖЕКОБСОНА ДЛЯ АЛГЕБР ЛИ И ЛОКАЛЬНО НИЛЬПОТЕНТНОГО РАДИКАЛА ДЛЯ СПЕЦИАЛЬНЫХ АЛГЕБР ЛИ

One way to study the properties of rings, algebras, Lie algebras and their ideals presupposes their description via the properties of modules over these rings, algebras, Lie algebras. This article deals with the study of radicals of Lie algebras. We discuss the possibility of homological descriptions of the Jacobson radical of Lie algebras and nilpotent radical of the special Lie algebra. The first section introduces the concepts of radicals of Lie algebras. The second section is devoted to the Jacobson radical of Lie algebras. It is proved that the intersection of all annihilators of irreducible modules over an arbitrary Lie algebra L coincides with the intersection of the Lie algebras L and the Jacobson radical of the universal enveloping algebra. This section contains examples that prove this fact. This examples allows to prove the equality of the nilpotent radical of PI -irreducible represented radical of finite-dimensional Lie algebra over a field of characteristic zero. We find the correlation between the locally nilpotent radical and others radicals of Lie algebras such that the irreducible represented radical, the PI -irreducible represented radical and the finitely irreducible represented radical. In the third section it is shown that the locally nilpotent radical is included in the PI -irreducible represented radical for an arbitrary special Lie algebra L over a field F of characteristics zero. We have proved that the prime radical is not included in the PI -irreducible represented radical. The reverse inclusion for these radicals does not hold. The PI -irreducible represented radical is not locally solvable in the general case. Shows an example of a special Lie algebra L over a field F with the locally nilpotent radical, which has is equal to zero.

Leibniz Algebras Constructed by Representations of General Diamond Lie Algebras

In this paper we construct a minimal faithful representation of the (2 m+ 2) -dimensional complex general Diamond Lie algebra, Dm(C) , which is isomorphic to a subalgebra of the special linear Lie algebra sl(m+ 2 , C). We also construct a faithful representation of the real general Diamond Lie algebra Dm which is isomorphic to a subalgebra of the special symplectic Lie algebra sp(2 m+ 2 , R). Furthermore, we describe Leibniz algebras with corresponding (2 m+ 2) -dimensional real general Diamond Lie algebra Dm and such that the ideal generated by the squares of elements provides a faithful representation of Dm.

On M. V. Zaicev problem for Noetherian special Lie algebras

We solve the M. V. Zaicev problem in the following sense: Any Noetherian semiprime special Lie algebra is embedded into algebra of matrices over commutative ring which is the direct sum of fields.

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.

On special Lie algebras having a faithful module with Krull dimension

For special Lie algebras we prove an analogue of Markov's theorem on -algebras having a faithful module with Krull dimension: the solubility of the prime radical. We give an example of a semiprime Lie algebra that has a faithful module with Krull dimension but cannot be represented as a subdirect product of finitely many prime Lie algebras. We prove a criterion for a semiprime Lie algebra to be representable as such a subdirect product.