Solvable Lie Algebras Research Articles

Spectral invariants for finite dimensional Lie algebras

For a Lie algebra L with basis {x1,x2,…,xn}, its associated characteristic polynomial QL(z) is the determinant of the linear pencil z0I+z1adx1+⋯+znadxn. This paper shows that QL is invariant under the automorphism group Aut(L). The zero variety and factorization of QL reflect the structure of L. In the case L is solvable QL is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincaré polynomial for solvable Lie algebras. Application is given to 1-dimensional extensions of nilpotent Lie algebras.

Poisson algebras and symmetric Leibniz bialgebra structures on oscillator Lie algebras

Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures whose underlying Lie algebra is an oscillator Lie algebra. We give also all the symmetric Leibniz bialgebra structures whose underlying Lie bialgebra structure is a Lie bialgebra structure on an oscillator Lie algebra. We derive some geometric consequences on oscillator Lie groups.

On solvability of the first Hochschild cohomology of a finite-dimensional algebra

For an arbitrary finite-dimensional algebra A A , we introduce a general approach to determining when its first Hochschild cohomology H H 1 ( A ) \mathrm {HH}^1(A) , considered as a Lie algebra, is solvable. If A A is, moreover, of tame or finite representation type, we are able to describe H H 1 ( A ) \mathrm {HH}^1(A) as the direct sum of a solvable Lie algebra and a sum of copies of s l 2 \mathfrak {sl}_2 . We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of A A . As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll, and Solotar.

Solvable Lie Algebras of Vector Fields and a Lie's Conjecture

We present a local and constructive differential geometric description of finite-dimensional solvable and transitive Lie algebras of vector fields. We show that it implies a Lie's conjecture for such Lie algebras. Also infinite-dimensional analytical solvable and transitive Lie algebras of vector fields whose derivative ideal is nilpotent can be adapted to this scheme.

Variation of complex structures and variation of Lie algebras II: new Lie algebras arising from singularities

Finite dimensional Lie algebras are semi-direct product of the semi-simple Lie algebras and solvable Lie algebras. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. It is extremely important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this article, a new natural connection between the set of complex analytic isolated hypersurface singularities and the set of finite dimensional solvable (nilpotent) Lie algebras has been constructed. We construct finite dimensional solvable (nilpotent) Lie algebras naturally from isolated hypersurface singularities. These constructions help us to understand the solvable (nilpotent) Lie algebras from the geometric point of view. Moreover, it is known that the classification of nilpotent Lie algebras in higher dimensions ($\gt 7$) remains to be a vast open area. There are one-parameter families of non-isomorphic nilpotent Lie algebras (but no two-parameter families) in dimension seven. Dimension seven is the watershed of the existence of such families. It is well-known that no such family exists in dimension less than seven, while it is hard to construct one-parameter family in dimension greater than seven. In this article, we construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $11$ (resp. $10$) from $\tilde{E}_7$ singularities and show that the weak Torelli-type theorem holds. We shall also construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $12$ (resp. $11$) from $\tilde{E}_8$ singularities and show that the Torelli-type theorem holds. Moreover, we investigate the numerical relation between the dimensions of the new Lie algebras and Yau algebras. Finally, the new Lie algebras arising from fewnomial isolated singularities are also computed.

Global stability of a class of difference equations on solvable Lie algebras

Motivated by the ubiquitous sampled-data setup in applied control, we examine the stability of a class of difference equations that arises by sampling a right- or left-invariant flow on a solvable matrix Lie group. The map defining such a difference equation has three key properties that facilitate our analysis: (1) its Lie series expansion enjoys a type of strong convergence; (2) the origin is an equilibrium; (3) the algebraic ideals enumerated in the lower central series of the Lie algebra are dynamically invariant. We show that certain global stability properties are implied by stability of the Jacobian linearization of the dynamics at the origin, in particular, global asymptotic stability. If the Lie algebra is nilpotent, then the origin enjoys semiglobal exponential stability.

A new symmetry-based method for constructing nonlocally related PDE systems from admitted multi-parameter groups

Nonlocally related partial differential equation (PDE) systems can play an important role in the analysis of a given PDE system. In this paper, a new systematic method for obtaining nonlocally related PDE systems is developed. In particular, it is shown that if a PDE system admits q point symmetries whose infinitesimal generators form a q-dimensional solvable Lie algebra, then, for each resulting q-dimensional solvable algebra chain, one can obtain systematically q nonlocally related PDE systems. Such nonlocally related systems are obtained for a general class of nonlinear reaction–diffusion equations admitting two- to four-dimensional solvable algebras.

On the classification of rational K-matrices

This paper presents a derivation of the possible residual symmetries of rational K-matrices which are invertible in the ‘classical limit’ (the spectral parameter goes to infinity). This derivation uses only the boundary Yang–Baxter equation and the asymptotic expansions of the R-matrices. The result proves the previous assumption of the literature: if the original and the residual symmetry algebras are and then there exists a Lie-algebra involution of for which the invariant sub-algebra is . In addition, we study a K-matrix which is not invertible in the ‘classical limit’. It is shown that its symmetry algebra is not reductive but a semi-direct sum of reductive and solvable Lie-algebras.

Algorithmic construction of solvable rigid Lie algebras determined by generating functions

ABSTRACT We introduce the notion of saturation of a nilradical with respect to the eigenvalue spectrum of a maximal torus of derivations and show the existence of a generating function and a factor sequence that completely determine the nilradical. As an application, a cohomologically rigid series of solvable Lie algebras of rank one with saturated nilradical is obtained for any and dimensions , where the eigenvalues of the torus are given by the sequence . An algorithmic procedure to search cohomologically rigid Lie algebras is proposed, based on the imposition of certain constraints on the generating function.

A Classification of Line Bundles over a Scheme

In this paper, we attempt to indicate the links between representations over a Lie algebra and line bundles over a scheme. Our hypothesis is that we can make a correspondence between representations of a solvable Lie algebra and line bundles. Finally, we study the decomposition of these bundles by using the representation theory over solvable Lie algebras.

On the Lie algebra structure of 𝐻𝐻¹(𝐴) of a finite-dimensional algebra 𝐴

Let A A be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext \operatorname {Ext} -quiver of A A is a simple directed graph, then H H 1 ( A ) H\!H^1(A) is a solvable Lie algebra. The second main result shows that if the Ext \operatorname {Ext} -quiver of A A has no loops and at most two parallel arrows in any direction, and if H H 1 ( A ) H\!H^1(A) is a simple Lie algebra, then char ⁡ ( k ) ≠ \operatorname {char}(k)\neq 2 2 and H H 1 ( A ) ≅ H\!H^1(A)\cong s l 2 ⁡ ( k ) \operatorname {\mathfrak {sl}}_2(k) . The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.

Nilpotent decomposition of solvable Lie algebras

Semisimple Lie algebras have been completely classified by Cartan and Killing. The Levi theorem states that every finite dimensional Lie algebra is isomorphic to a semidirect sum of its largest solvable ideal and a semisimple Lie algebra. These focus the classification of solvable Lie algebras as one of the main challenges of Lie algebra research. One approach towards this task is to take a class of nilpotent Lie algebras and construct all extensions of these algebras to solvable ones. In this paper, we propose another approach, i.e., to decompose a solvable nonnilpotent Lie algebra to two nilpotent Lie algebras which are called the left and right nilpotent algebras of the solvable algebra. The right nilpotent algebra is the smallest ideal of the lower central series of the solvable algebra, while the left nilpotent algebra is the factor algebra of the solvable algebra and its right nilpotent algebra. We show that the solvable algebras are decomposable if its left nilpotent algebra is an Abelian algebra of dimension higher than one and its right algebra is an Abelian algebra of dimension one. We further show that all the solvable algebras are isomorphic if their left nilpotent algebras are Heisenberg algebras of fixed dimension and their right algebras are Abelian algebras of dimension one.

Vaisman solvmanifolds and relations with other geometric structures

Fil: Andrada, Adrian Marcelo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Centro de Investigacion y Estudios de Matematica. Universidad Nacional de Cordoba. Centro de Investigacion y Estudios de Matematica; Argentina. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica; Argentina

Solvable Lie Algebras of Derivations of Rank One

Let K be a field of characteristic zero, A = K[x1, . . . , xn] the polynomial ring and R = K(x1, . . . , xn) the field of rational functions in n variables over K. The Lie algebra Wn(K) of all K-derivations on A is of great interest since its elements may be considered as vector fields on Kn with polynomial coefficients. If L is a subalgebra of Wn(K), then one can define the rank rkAL of L over A as the dimension of the vector space RL over the field R. Finite dimensional (over K) subalgebras of Wn(K) of rank 1 over A were studied by the first author jointly with I. Arzhantsev and E. Makedonskiy. We study solvable subalgebras L of Wn(K) with rkAL = 1, without restrictions on dimension over K. Such Lie algebras are described in terms of Darboux polynomials.

Applying matrix theory to classify real solvable Lie algebras having 2-dimensional derived ideals

We present a new approach to the problem of classifying real solvable Lie algebras having 2-dimensional derived ideals. Partial results on this problem were obtained by Schöbel in 1993 and by Janisse in 2010. In association with the results of Eberlein in 2003, we achieve a full list of real solvable Lie algebras having 2-dimensional derived ideals.

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.

Locally conformally Kähler structures on four-dimensional solvable Lie algebras

AbstractWe classify and investigate locally conformally Kähler structures on four-dimensional solvable Lie algebras up to linear equivalence. As an application we can produce many examples in higher dimension, here including lcK structures on Oeljeklaus-Toma manifolds, and we also give a geometric interpretation of some of the 4-dimensional structures in our classification.

Classification of Symmetry Lie Algebras of the Canonical Geodesic Equations of Five-Dimensional Solvable Lie Algebras

In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras A 5 , 7 a b c to A 18 a . For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized.

Heun algebras of Lie type

We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For s u ( 2 ) \mathfrak {su}(2) , this leads to the Heun–Krawtchouk algebra. The corresponding Heun–Krawtchouk operator is identified as the Hamiltonian of the quantum analogue of the Zhukovsky–Voltera gyrostat. For s u ( 1 , 1 ) \mathfrak {su}(1,1) , one obtains the Heun algebras attached to the Meixner, Meixner–Pollaczek, and Laguerre polynomials. These Heun algebras are shown to be isomorphic to the the Hahn algebra. Focusing on the harmonic oscillator algebra h o \mathfrak {ho} leads to the Heun–Charlier algebra. The connections to orthogonal polynomials are achieved through realizations of the underlying Lie algebras in terms of difference and differential operators. In the s u ( 1 , 1 ) \mathfrak {su}(1,1) cases, it is observed that the Heun operator can be transformed into the Hahn, Continuous Hahn, and Confluent Heun operators, respectively.

A Role for Symmetry in the Bayesian Solution of Differential Equations

The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical paradigms can be considered and Bayesian probabilistic numerical methods (PNMs) are obtained when Bayesian statistical principles are deployed. Bayesian PNM have the appealing property of being closed under composition, such that uncertainty due to different sources of discretisation in a numerical method can be jointly modelled and rigorously propagated. Despite recent attention, no exact Bayesian PNM for the numerical solution of ordinary differential equations (ODEs) has been proposed. This raises the fundamental question of whether exact Bayesian methods for (in general nonlinear) ODEs even exist. The purpose of this paper is to provide a positive answer for a limited class of ODE. To this end, we work at a foundational level, where a novel Bayesian PNM is proposed as a proof-of-concept. Our proposal is a synthesis of classical Lie group methods, to exploit underlying symmetries in the gradient field, and non-parametric regression in a transformed solution space for the ODE. The procedure is presented in detail for first and second order ODEs and relies on a certain strong technical condition – existence of a solvable Lie algebra – being satisfied. Numerical illustrations are provided.