Finite-dimensional Lie Algebra Research Articles

New Simple Lie Algebras in Characteristic 2

Several improvements of the Kostrikin-Shafarevich method conjecturally producing all simple finite-dimensional Lie algebras over algebraically closed fields of any positive characteristic were rece ...

5-dimensional indecomposable contact Lie algebras as double extensions

In this work we shall show that a suitable double extension of a finite dimensional indecomposable contact Lie algebra is a contact Lie algebra again. In particular, with exception of the family of 5-dimensional indecomposable contact solvable Lie algebras A 5 , 35 , any 5-dimensional indecomposable contact solvable Lie algebra can be obtained as a double extension of a 3-dimensional Lie algebra. The family A 5 , 35 can be generalized to a family of ( 4 n + 1 ) -dimensional indecomposable contact solvable Lie algebras that cannot be obtained neither as a suspension of a symplectic Lie algebra of codimension 1 or as a double extension of a contact Lie subalgebra of codimension 2.

Lie symmetries for Lie systems: Applications to systems of ODEs and PDEs

A Lie system is a nonautonomous system of first-order differential equations admitting a superposition rule, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants. Using that a Lie system can be considered as a curve in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot–Guldberg Lie algebra, we associate every Lie system with a Lie algebra of Lie point symmetries induced by the Vessiot–Guldberg Lie algebra. This enables us to derive Lie symmetries of relevant physical systems described by first- and higher-order systems of differential equations by means of Lie systems in an easier way than by standard methods. A generalization of our results to partial differential equations is introduced. Among other applications, Lie symmetries for several new and known generalizations of the real Riccati equation are studied.

Grušin operators, Riesz transforms and nilpotent Lie groups

We establish that the Riesz transforms of all orders corresponding to the Grusin operator \(H_N=-\nabla _{x}^2-|x|^{2N}\,\nabla _{y}^2\), and the first-order operators \((\nabla _{x},x^\nu \,\nabla _{y})\) where \(x\in \mathbf{R}^n\), \(y\in \mathbf{R}^m\), \(N\in \mathbf{N}_+\), and \(\nu \in \{1,\ldots ,n\}^N\), are bounded on \(L_p(\mathbf{R}^{n+m})\) for all \(p\in \langle 1,\infty \rangle \) and are also weak-type (1, 1). Moreover, the transforms of order less than or equal to \(N+1\) corresponding to \(H_N\) and the operators \((\nabla _{x}, |x|^N\nabla _{y})\) are bounded on \(L_p(\mathbf{R}^{n+m})\) for all \(p\in \langle 1,\infty \rangle \). But if N is odd all transforms of order \(N+2\) are bounded if and only if \(p\in \langle 1,n\rangle \). The proofs are based on the observation that the \((\nabla _{x},x^\nu \,\nabla _{y})\) generate a finite-dimensional nilpotent Lie algebra, the corresponding connected, simply connected, nilpotent Lie group is isometrically represented on the spaces \(L_p(\mathbf{R}^{n+m})\) and \(H_N\) is the corresponding sublaplacian.

Classification of linked indecomposable modules of a family of solvable Lie algebras over an arbitrary field of characteristic 0

Let 𝔤 be a finite-dimensional Lie algebra over a field of characteristic 0, with solvable radical 𝔯 and nilpotent radical 𝔫 = [𝔤, 𝔯]. Given a finite-dimensional 𝔤-module U, its nilpotency series 0 ⊂ U(1) ⊂ ⋯ ⊂ U(m) = U is defined so that U(1) is the 0-weight space of 𝔫 in U, U(2)/U(1) is the 0-weight space of 𝔫 in U/U(1), and so on. We say that U is linked if each factor of its nilpotency series is a uniserial 𝔤/𝔫-module, i.e. its 𝔤/𝔫-submodules form a chain. Every uniserial 𝔤-module is linked, every linked 𝔤-module is indecomposable with irreducible socle, and both converses fail. In this paper, we classify all linked 𝔤-modules when 𝔤 = 〈x〉 ⋉ 𝔞 and ad x acts diagonalizably on the abelian Lie algebra 𝔞. Moreover, we identify and classify all uniserial 𝔤-modules amongst them.

Cohomologically rigid solvable Lie algebras with a nilradical of arbitrary characteristic sequence

It is shown that for a finite-dimensional solvable rigid Lie algebra r, its rank is upper bounded by the length of the characteristic sequence c(n) of its nilradical n. For any characteristic sequence c=(n1,⋯,nk,1), it is proved that there exists at least a solvable Lie algebra rc the nilradical of which has this characteristic sequence and that satisfies the conditions Hp(rc,rc)=0 for p≤3.

2-Local automorphisms of finite-dimensional simple Lie algebras

Let F be an algebraically closed field of characteristic 0, L be a finite-dimensional simple Lie algebra of type Al (l≥1), Dl (l≥4), Ek (k=6,7,8) over F. A not necessarily linear map φ:L→L is called a 2-local automorphism if for every x,y∈L there is an automorphism φx,y:L→L, depending on x and y, such that φ(x)=φx,y(x), φ(y)=φx,y(y). In this paper, we prove that any 2-local automorphism of L is an automorphism.

On nilpotent Lie algebras of small breadth

A Lie algebra L is said to be of breadth k if the maximal dimension of the images of left multiplication by elements of the algebra is k. In this paper we give characterization of finite dimensional nilpotent Lie algebras of breadth less than or equal to two. Furthermore, using these characterizations we determined the isomorphism classes of these algebras.

Spin norm, pencils, and the u-small convex hull

Let g \mathfrak {g} be any finite-dimensional complex simple Lie algebra. In this paper, we show that the spin norm increases strictly along any pencil once it goes beyond the u-small convex hull of g \mathfrak {g} .

Lie–Hamilton systems on the plane: applications and superposition rules

A Lie–Hamilton (LH) system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. We provide new algebraic/geometric techniques to easily determine the properties of such Lie algebras on the plane, e.g., their associated Poisson bivectors. We study new and known LH systems on with physical, biological and mathematical applications. New results cover Cayley–Klein Riccati equations, the here defined planar diffusion Riccati systems, complex Bernoulli differential equations and projective Schrödinger equations. Constants of motion for planar LH systems are explicitly obtained which, in turn, allow us to derive superposition rules through a coalgebra approach.

Superalgebras, constraints and partition functions

We consider Borcherds superalgebras obtained from semisimple finite-dimensional Lie algebras by adding an odd null root to the simple roots. The additional Serre relations can be expressed in a covariant way. The spectrum of generators at positive levels are associated to partition functions for a certain set of constrained bosonic variables, the constraints on which are complementary to the Serre relations in the symmetric product. We give some examples, focusing on superalgebras related to pure spinors, exceptional geometry and tensor hierarchies, of how construction of the content of the algebra at arbitrary levels is simplified.

Some homological properties of category [formula omitted]. III

We prove that thick category O associated to a semi-simple complex finite dimensional Lie algebra is extension full in the category of all modules. We also prove the weak Alexandru conjecture both for regular blocks of thick category O and the associated categories of Harish–Chandra bimodules, but disprove it for singular blocks.

The action of a compact Lie group on nilpotent Lie algebras of type {{n,2}}

Abstract We classify finite-dimensional real nilpotent Lie algebras with 2-dimensional central commutator ideals admitting a Lie group of automorphisms isomorphic to SO 2 ⁢ ( ℝ ) ${{\mathrm{SO}}_{2}(\mathbb{R})}$ . This is the first step to extend the class of nilpotent Lie algebras 𝔥 ${{\mathfrak{h}}}$ of type { n , 2 } ${\{n,2\}}$ to solvable Lie algebras in which 𝔥 ${{\mathfrak{h}}}$ has codimension one.

Comparison of categorical characteristic classes of a transitive Lie algebroid with the Chern-Weil homomorphism

This paper is devoted to analyzing two approaches to characteristic classes of transitive Lie algebroids. The first approach is due to Kubarski [5] and is a version of the Chern-Weil homomorphism. The second approach is related to the so-called categorical characteristic classes (see, e.g., [6]). The construction of transitive Lie algebroids due to Mackenzie [1] can be considered as a homotopy functor T LAg from the category of smooth manifolds to the transitive Lie algebroids. The functor T LAg assigns to every smooth manifold M the set T LAg(M) of all transitive algebroids with a chosen structural finite-dimensional Lie algebra g. Hence, one can construct [2, 3] a classifying space Bg such that the family of all transitive Lie algebroids with the chosen Lie algebra g over the manifold M is in one-to-one correspondence with the family of homotopy classes of continuous maps [M, Bg]: T LAg(M) ≈ [M, Bg]. This enables us to describe characteristic classes of transitive Lie algebroids from the point of view of a natural transformation of functors similar to the classical abstract characteristic classes for vector bundles and to compare them with those derived from the Chern-Weil type homomorphism by Kubarski [5]. As a matter of fact, we show that the Chern-Weil type homomorphism by Kubarski does not cover all characteristic classes from the categorical point of view.

Weight modules for current algebras

For any finite-dimensional simple Lie algebra g and finitely generated commutative associative algebra S, we give a complete classification of the simple weight modules of g⊗S with bounded weight multiplicities.

Geometry of Lie integrability by quadratures

In this paper, we extend the Lie theory of integration by quadratures of systems of ordinary differential equations in two different ways. First, we consider a finite-dimensional Lie algebra of vector fields and discuss the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a prescribed way. It turns out that the conditions can be expressed in a purely algebraic way. In the second step, we generalize the construction to the case in which we substitute the Lie algebra of vector fields by a module (generalized distribution). We obtain a much larger class of explicitly integrable systems, replacing standard concepts of solvable (or nilpotent) Lie algebra with distributional solvability (nilpotency).

K-step nilpotent Lie algebras

Abstract The classification of complex or real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example, the nilpotent Lie algebras are classified only up to dimension 7. Moreover, to recognize a given Lie algebra in the classification list is not so easy. In this work, we propose a different approach to this problem. We determine families for some fixed invariants and the classification follows by a deformation process or a contraction process. We focus on the case of 2- and 3-step nilpotent Lie algebras. We describe in both cases a deformation cohomology for this type of algebras and the algebras which are rigid with respect to this cohomology. Other p-step nilpotent Lie algebras are obtained by contraction of the rigid ones.

The Ideal Index of Subalgebras of a Finite Dimensional Lie Algebra

D. A. Towers introduced the notion of ideal index of a maximal subalgebra of a Lie algebra, and used it to analyze the influence of maximal subalgebras on the structure of a finite dimensional Lie algebras. In this article, we generalize the ideal index from maximal subalgebras to all subalgebras, and obtain some new characterizations of solvable and supersolvable Lie algebras by the ideal indices of some certain subalgebras.

The p.i.m.s for the restricted Zassenhaus algebras in characteristic 2

It is shown that for the restricted Zassenhaus algebra \({\mathfrak{W} = \mathfrak{W}(1; n)}\), n > 1, defined over an algebraically closed field \({\mathbb{F}}\) of characteristic 2, any projective indecomposable restricted \({\mathfrak{W}}\) -module has maximal possible dimension \({2^{2^n-1}}\), and thus is isomorphic to some induced module \({{\rm idn}^{\mathfrak{W}}_{\mathfrak{t}}(\mathbb{F}(\mu))}\) for some torus of maximal dimension \({\mathfrak{t}}\). This phenomenon is in contrast to the behavior of finite-dimensional non-solvable restricted Lie algebras in characteristic p > 3 (cf. Feldvoss et al. Restricted Lie algebras with maximal 0-pim, 2014, Theorem 6.3).

Subalgebras that cover or avoid chief factors of Lie algebras

We call a subalgebra $U$ of a Lie algebra $L$ a $CAP$-subalgebra of $L$ if for any chief factor $H/K$ of $L$, we have $H \cap U = K \cap U$ or $H+U = K+U$. In this paper we investigate some properties of such subalgebras and obtain some characterizations for a finite-dimensional Lie algebra $L$ to be solvable under the assumption that some of its maximal subalgebras or $2$-maximal subalgebras be $CAP$-subalgebras.