Finite-dimensional Lie Research Articles

Nilpotent Lie algebras of class 3 with 1-dimensional center and 3-dimensional derived subalgebra

In this paper, we obtain the structure of all non-isomorphic finite-dimensional nilpotent Lie algebras L of class 3 with derived subalgebra of dimension 3 when

On the universal central extension of superelliptic affine Lie algebras

We describe explicitly in terms of generators and relations the universal central extension of infinite-dimensional superelliptic affine Lie algebras with a finite-dimensional simple Lie algebra and the coordinate ring where and p(t) is a polynomial with distinct roots.

Classification of Nilpotent Lie Superalgebras of Multiplier-Rank Less than or Equal to 6

In this paper, we classify all the finite-dimensional nilpotent Lie superalgebras of multiplier-rank less than or equal to 6 over an algebraically closed field of characteristic zero. We also determine the covers of all the nilpotent Lie superalgebras mentioned above.

Partial actions on reductive Lie algebras

In this paper, we study partial group actions on Lie algebras. We describe the structure of the inverse semigroup of all partial automorphisms (isomorphisms between ideals) of a finite-dimensional reductive Lie algebra. Also, we show that every partial group action on a finite-dimensional semisimple Lie algebra admits a globalization, unique up to isomorphism. As a consequence, we obtain that the globalization problem for partial group actions on reductive Lie algebra is equivalent to the globalization problem on its center.

Cohomology of Lie Conformal Algebra Vir⋉Cur g

In this article, we compute cohomology groups of the semisimple Lie conformal algebra [Formula: see text] with coefficients in its irreducible modules for a finite-dimensional simple Lie algebra [Formula: see text].

Analysing ‘Simple’ Image Registrations

Processes such as growth and atrophy cause changes through time that can be visible in a series of medical images, following the hypothesis that form follows function. As was hypothesised by D’Arcy Thompson more than 100 years ago, models of the changes inherent in these actions can aid understanding of the processes at work. We consider how image registration using finite-dimensional planar Lie groups (in contrast to general diffeomorphisms) can be used in this process. The deformations identified can be described as points in the Lie algebra, thus enabling processes such as evolutionary change, growth, and deformation from disease, to be described in a linear space. The choice of appropriate Lie group becomes a modelling choice and can be selected using model selection; Occam’s razor suggests that groups with the smallest number of parameters (which Thompson referred to as ‘simple transformations’) are to be preferred. We demonstrate our method on an example from Thompson of the cannon-bones of three hoofed mammals and a set of outline curves of the development of the human skull, with promising results.

Symmetry-Based Algorithms for Invertible Mappings of Polynomially Nonlinear PDE to Linear PDE

This paper is a sequel to our previous work where we introduced the MapDE algorithm to determine the existence of analytic invertible mappings of an input (source) differential polynomial system (DPS) to a specific target DPS, and sometimes by heuristic integration an explicit form of the mapping. A particular feature was to exploit the Lie symmetry invariance algebra of the source without integrating its equations, to facilitate MapDE, making algorithmic an approach initiated by Bluman and Kumei. In applications, however, the explicit form of a target DPS is not available, and a more important question is, can the source be mapped to a more tractable class? We extend MapDE to determine if a source nonlinear DPS can be mapped to a linear differential system. MapDE applies differential-elimination completion algorithms to the various over-determined DPS by applying a finite number of differentiations and eliminations to complete them to a form for which an existence-uniqueness theorem is available, enabling the existence of the linearization to be determined among other applications. The methods combine aspects of the Bluman–Kumei mapping approach with techniques introduced by Lyakhov, Gerdt and Michels for the determination of exact linearizations of ODE. The Bluman–Kumei approach for PDE focuses on the fact that such linearizable systems must admit a usually infinite Lie sub-pseudogroup corresponding to the linear superposition of solutions in the target. In contrast, Lyakhov et al. focus on ODE and properties of the so-called derived sub-algebra of the (finite) dimensional Lie algebra of symmetries of the ODE. Examples are given to illustrate the approach, and a heuristic integration method sometimes gives explicit forms of the maps. We also illustrate the powerful maximal symmetry groups facility as a natural tool to be used in conjunction with MapDE.

Regular Hom-Lie structures on Borel subalgebras of finite-dimensional simple Lie algebras

A Hom-structure on a Lie algebra is a linear map which satisfies the Hom-Jacobi identity for all A Hom-structure is called regular if is also a Lie algebra isomorphism. Let be a Borel subalgebra of a finite-dimensional simple Lie algebra L over an algebraically closed field of characteristic zero. In this paper, we prove that any regular Hom-structure on is an extremal inner automorphism induced by a highest root vector of Communicated by Sudarshan Kumar Sehgal

Endotrivial modules for nilpotent restricted Lie algebras

Let $$\mathfrak {g}$$ be a finite dimensional nilpotent p-restricted Lie algebra over a field k of characteristic p. For $$p\geqslant 5$$, we show that every endotrivial $$\mathfrak {g}$$-module is a direct sum of a syzygy of the trivial module and a projective module. The proof includes a theorem that the intersection of the maximal linear subspaces of the null cone of a nilpotent restricted p-Lie algebra for $$p \geqslant 5$$ has dimension at least two. We give an example to show that the statement about endotrivial modules is false in characteristic two. In characteristic three, another example shows that our proof fails, and we do not know a characterization of the endotrivial modules in this case.

Local derivations on Borel subalgebras of finite-dimensional simple Lie algebras

Let be an algebraically closed field of characteristic 0, let be a finite-dimensional simple Lie algebra over , and let be the standard Borel subalgebra of . In this paper we prove that every local derivation of is a derivation.

C-capability of Lie algebras with the derived subalgebra of dimension two

The current article is devoted to classify the c-capability of finite dimensional nilpotent Lie algebras with the derived subalgebra of dimension two.

Extended affine Lie superalgebras and their affinizations

Extended affine Lie superalgebras are super versions of the defining axioms of extended affine Lie algebras or more generally invariant affine reflection algebras. This class includes finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras. In this paper, an affinization process is introduced for the class of extended affine Lie superalgebras, and the necessary conditions for an extended affine Lie superalgebra to be invariant under this process are presented. Moreover, new extended affine Lie superalgebras are constructed by means of the affinization process.

On the dimension of the Schur multiplier of nilpotent lie algebras

We give a bound on the dimension of the Schur multiplier of a finite dimensional nilpotent Lie algebra which sharpens the earlier known bounds.

Algorithm to compute minimal matrix representation of nilpotent lie algebras

ABSTRACTAs it is well-known there exist matrix representations of any given finite-dimensional complex Lie algebra. More concretely, such representations can be obtained by means of an isomorphic matrix Lie algebra consisting of upper-triangular square matrices. However, there is no general information about the minimal order for the matrices involved in such representations. In this way, our main goal is to revisit, debug and implement an algorithm which provides the minimal order for matrix representations of any finite-dimensional nilpotent Lie algebra from its law, as well as returning a matrix representative of such an algebra by using the minimal order previously computed. In order to show the applicability of this procedure, we have computed minimal representative for each nilpotent Lie algebra of dimensions 6 and 7 and we have also obtained the representation of some families with an arbitrary dimension.

On truncated Weyl modules

We study structural properties of truncated Weyl modules. A truncated Weyl module is a local Weyl module for , where is a finite-dimensional simple Lie algebra. It has been conjectured that, if N is sufficiently small with respect to λ, the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when λ is a multiple of certain fundamental weights, including all minuscule ones for simply laced . We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov–Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari–Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that related to Demazure flags and chains of inclusions of truncated Weyl modules.

Separability of Topological Groups: A Survey with Open Problems

Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and ℓ p , for p ≥ 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey.

Cosets of affine vertex algebras inside larger structures

Given a finite-dimensional reductive Lie algebra g equipped with a nondegenerate, invariant, symmetric bilinear form B, let Vk(g,B) denote the universal affine vertex algebra associated to g and B at level k. Let Ak be a vertex (super)algebra admitting a homomorphism Vk(g,B)→Ak. Under some technical conditions on Ak, we characterize the coset Com(Vk(g,B),Ak) for generic values of k. We establish the strong finite generation of this coset in full generality in the following cases: Ak=Vk(g′,B′), Ak=Vk−l(g′,B′)⊗F, and Ak=Vk−l(g′,B′)⊗Vl(g″,B″). Here g′ and g″ are finite-dimensional Lie (super)algebras containing g, equipped with nondegenerate, invariant, (super)symmetric bilinear forms B′ and B″ which extend B, l∈C is fixed, and F is a free field algebra admitting a homomorphism Vl(g,B)→F. Our approach is essentially constructive and leads to minimal strong finite generating sets for many interesting examples. As an application, we give a new proof of the rationality of the simple N=2 superconformal algebra with c=3kk+2 for all positive integers k.

Extended geometries

We present a unified and completely general formulation of extended geometry, characterised by a Kac-Moody algebra and a highest weight coordinate module. Generalised diffeomorphisms are constructed, as well as solutions to the section constraint. Generically, additional (“ancillary”) gauge transformations are present, and we give a concrete criterion determining when they appear. A universal form of the (pseudo-)action determines the dynamics in all cases without ancillary transformations, and also for a restricted set of cases based on the adjoint representation of a finite-dimensional simple Lie group. Our construction reproduces (the internal sector of) all previously considered cases of double and exceptional field theories.

On the centers of quantum groups of An-type

Let g be the finite dimensional simple Lie algebra of type An, and let U = U q (g,Λ) and U = U q (g,Q) be the quantum groups defined over the weight lattice and over the root lattice, respectively. In this paper, we find two algebraically independent central elements in U for all n ≥ 2 and give an explicit formula of the Casimir elements for the quantum group U = U q (g,Λ), which corresponds to the Casimir element of the enveloping algebra U(g). Moreover, for n = 2 we give explicitly generators of the center subalgebras of the quantum groups U = U q (g,Λ) and U = U q (g,Q).

Orbifolds and Cosets of Minimal $${\mathcal{W}}$$-Algebras

Let $\mathfrak{g}$ be a simple, finite-dimensional Lie (super)algebra equipped with an embedding of $\mathfrak{s} \mathfrak{l}_2$ inducing the minimal gradation on $\mathfrak{g}$. The corresponding minimal $\mathcal{W}$-algebra $\mathcal{W}^k(\mathfrak{g}, e_{-\theta})$ introduced by Kac and Wakimoto has strong generators in weights $1,2,3/2$, and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra $V^{k'}(\mathfrak{g}^{\natural})$ where $\mathfrak{g}^{\natural} \subset \mathfrak{g}$ denotes the centralizer of $\mathfrak{s} \mathfrak{l}_2$. Therefore $\mathcal{W}^k(\mathfrak{g}, e_{-\theta})$ has an action of a connected Lie group $G^{\natural}_0$ with Lie algebra $\mathfrak{g}^{\natural}_0$, where $\mathfrak{g}^{\natural}_0$ denotes the even part of $\mathfrak{g}^{\natural}$. We show that for any reductive subgroup $G \subset G^{\natural}_0$, and for any reductive Lie algebra $\mathfrak{g}' \subset \mathfrak{g}^{\natural}$, the orbifold $\mathcal{O}^k = \mathcal{W}^k(\mathfrak{g}, e_{-\theta})^{G}$ and the coset $\mathcal{C}^k = \text{Com}(V(\mathfrak{g}'),\mathcal{W}^k(\mathfrak{g}, e_{-\theta}))$ are strongly finitely generated for generic values of $k$. Here $V(\mathfrak{g}')$ denotes the affine vertex algebra associated to $\mathfrak{g}'$. We find explicit minimal strong generating sets for $\mathcal{C}^k$ when $\mathfrak{g}' = \mathfrak{g}^{\natural}$ and $\mathfrak{g}$ is either $\mathfrak{s} \mathfrak{l}_n$, $\mathfrak{s}\mathfrak{p}_{2n}$, $\mathfrak{s}\mathfrak{l}(2|n)$ for $n\neq 2$, $\mathfrak{p}\mathfrak{s}\mathfrak{l}(2|2)$, or $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|4)$. Finally, we conjecture some surprising coincidences among families of cosets $\mathcal{C}_k$ which are the simple quotients of $\mathcal{C}^k$, and we prove several cases of our conjecture.