Lie Algebra Research Articles

On Lie Homomorphisms of Complex IntuitionisticFuzzy Lie Algebras

In this paper, we investigate the properties of complex intuitionistic fuzzy Lie subalgebras and ideals under Lie algebra homomorphisms, focusing on both images and inverse images. We provide detailed proofs for several new results concerning the preservation of complex intuitionistic fuzzy structures through homomorphisms, and we introduce additional homomorphism-related properties for these structures. This work extends known results to the context of complex intuitionistic fuzzy Lie algebras, contributing new insights into their behavior under algebraic mappings.

The algebraic classification of low-dimensional nilpotent n-Lie algebras

In this paper, we study the low-dimensional nilpotent [Formula: see text]-Lie algebras and classify nilpotent [Formula: see text]-Lie algebras of dimension [Formula: see text] and nilpotent [Formula: see text]-Lie algebras of class two with dimension [Formula: see text].

The solenoidal Virasoro algebra and its simple weight modules

The aim of this paper is to construct a one dimensional universal central extension of the solenoidal Lie algebra W ( n ) μ introduced by Y. Billig and V. Futorny. The obtained Lie algebra is higher rank Virasoro algebra denoted by Vir ( n ) μ . Moreover, we study Vir ( n ) μ -Harich-Chandra modules. We obtain two kinds of Harich-Chandra modules, the intermediate series modules and generalized weight modules. Furthermore, we use the triangular decomposition of Vir ( n ) μ given by lexicographic order on Z n to construct new weight modules having infinite dimensional weight spaces.

Lie Rota–Baxter operators on the Sweedler algebra H4

If [Formula: see text] is an associative algebra, then we can define the adjoint Lie algebra [Formula: see text] and Jordan algebra [Formula: see text]. It is easy to see that any associative Rota–Baxter (RB) operator on [Formula: see text] induces a Lie and Jordan RB operator on [Formula: see text] and [Formula: see text], respectively. Are there Lie (Jordan) RB operators, which are not associative RB operators? In this paper, we explore these questions for the Sweedler algebra [Formula: see text], which is a 4-dimensional non-commutative Hopf algebra. More precisely, we describe the RB operators on the adjoint Lie algebra [Formula: see text].

An isomorphism between unitals and between related classical groups

Abstract An isomorphism between two hermitian unitals is provided, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).

A Vector-Product Lie Algebra of a Reductive Homogeneous Space and Its Applications

A new vector-product Lie algebra is constructed for a reductive homogeneous space, which can lead to the presentation of two corresponding loop algebras. As a result, two integrable hierarchies of evolution equations are derived from a new form of zero-curvature equation. These hierarchies can be reduced to the heat equation, a special diffusion equation, a general linear Schrödinger equation, and a nonlinear Schrödinger-type equation. Notably, one of them exhibits a pseudo-Hamiltonian structure, which is derived from a new vector-product identity proposed in this paper.

Horosymmetric limits of Kähler–Ricci flow on Fano $\mathbf{G}$-manifolds

We prove that on a Fano \mathbf{G} -manifold (M,J) , the Gromov–Hausdorff limit of the Kähler–Ricci flow with initial metric in 2\pi c_{1}(M) must be a \mathbb{Q} -Fano horosymmetric variety M_{\infty} which admits a singular Kähler–Ricci soliton. Moreover, M_{\infty} is the limit of a \mathbb{C}^{*} -degeneration of M induced by an element in the Lie algebra of the Cartan torus of \mathbf{G} . A similar result can be proved for Kähler–Ricci flows on any Fano horosymmetric manifolds. As an application, we generalize our previous result about the existence of type II singularities of Kähler–Ricci flows on Fano \mathbf{G} -manifolds to Fano horosymmetric manifolds.

The Lubin–Tate theory of configuration spaces: I

AbstractWe construct a spectral sequence converging to the Lubin–Tate theory, that is, Morava ‐theory, of unordered configuration spaces and identify its ‐page as the homology of a Chevalley–Eilenberg‐like complex for Hecke Lie algebras. Based on this, we compute the ‐theory of the weight summands of iterated loop spaces of spheres (parameterizing the weight operations on ‐algebras), as well as the ‐theory of the configuration spaces of points on a punctured surface. We read off the corresponding Morava ‐theory groups, which appear in a conjecture by Ravenel. Finally, we compute the ‐homology of the space of unordered configurations of particles on a punctured surface.

Canonical structure constants for simple Lie algebras

AbstractLet $$\mathfrak {g}$$ g be a finite-dimensional simple Lie algebra over $$\mathbb {C}$$ C . In the 1950s Chevalley showed that $$\mathfrak {g}$$ g admits particular bases, now called “Chevalley bases”, for which the corresponding structure constants are integers. Such bases are not unique but, using Lusztig’s theory of canonical bases, one can single out a “canonical” Chevalley basis which is unique up to a global sign. In this paper, we give explicit formulae for the structure constants with respect to such a basis.

Local derivations of the modular Lie algebra $$\mathfrak {sl}(2)$$ to its all simple modules

Local derivations of the modular Lie algebra $$\mathfrak {sl}(2)$$ to its all simple modules

Strong Kähler with torsion solvable lie algebras with codimension 2 nilradical

AbstractIn this paper, we study strong Kähler with torsion (SKT) and generalized Kähler structures on solvable Lie algebras with (not necessarily abelian) codimension 2 nilradical. We treat separately the case of ‐invariant nilradical and non‐‐invariant nilradical. A classification of such SKT Lie algebras in dimension 6 is provided. In particular, we give a general construction to extend SKT nilpotent Lie algebras to SKT solvable Lie algebras of higher dimension, and we construct new examples of SKT and generalized Kähler compact solvmanifolds.

Spin relaxation dynamics with a continuous spin environment: The dissipaton equation of motion approach.

We investigate the quantum dynamics of a spin coupling to a bath of independent spins via the dissipaton equation of motion (DEOM) approach. The bath, characterized by a continuous spectral density function, is composed of spins that are independent level systems described by the su(2) Lie algebra, representing an environment with a large magnitude of anharmonicity. Based on the previous work by Suarez and Silbey [J. Chem. Phys. 95, 9115 (1991)] and by Makri [J. Chem. Phys. 111, 6164 (1999)] that the spin bath can be mapped to a Gaussian environment under its linear response limit, we use the time-domain Prony fitting decomposition scheme to the bare-bath time correlation function (TCF) given by the bosonic fluctuation-dissipation theorem to generate the exponential decay basis (or pseudo modes) for DEOM construction. The accuracy and efficiency of this strategy have been explored by a variety of numerical results. We envision that this work provides new insights into extending the hierarchical equations of motion and DEOM approach to certain types of anharmonic environments with arbitrary TCF or spectral density.

Finiteness of 3D higher spin gravity Landscape

Abstract We give Swampland constraints on the three dimensional Landscape of Anti-de Sitter higher spin gravity in the Chern-Simons formulation with connection valued in various split real forms of Lie algebras. We derive the finiteness conjecture by computing the upper bound on the rank of possible gauge groups then we refine it using the AdS distance conjecture. We discuss the implications of this Swampland constraint on the spectrum of higher spin gravity theories and we compare it with the gravitational exclusion principle required from BTZ black hole consideration to excerpt a constraint on the Chern-Simons level k.

Graded Lie algebras, compactified Jacobians and arithmetic statistics

A simply laced Dynkin diagram gives rise to a family of curves over \mathbb{Q} and a coregular representation, using deformations of simple singularities and Vinberg theory, respectively. Thorne conjectured and partially proved a strong link between the arithmetic of these curves and the rational orbits of these representations. In this paper, we complete Thorne’s picture and show that 2-Selmer elements of the Jacobians of the smooth curves in each family can be parametrised by integral orbits of the corresponding representation. Using geometry-of-numbers techniques, we deduce statistical results on the arithmetic of these curves. We prove these results in a uniform manner. This recovers and generalises results of Bhargava, Gross, Ho, Shankar, Shankar and Wang. The main innovations are an analysis of torsors on affine spaces using results of Colliot-Thélène and the Grothendieck–Serre conjecture, a study of geometric properties of compactified Jacobians using the Białynicki-Birula decomposition, and a general construction of integral orbit representatives.

Cyclic wide subalgebras of semisimple Lie algebras

Let s ∈ + r be a Levi decomposable Lie algebra, with Levi factor s , and radical r . A module V of s ∈ + r is cyclic indecomposable if it is indecomposable and the quotient module V / r · V is a simple s -module. A Levi decomposable subalgebra of a semisimple Lie algebra is cyclic wide if the restriction of every simple module of the semisimple Lie algebra to the subalgebra is cyclic indecomposable. We establish a condition for a regular Levi decomposable subalgebra of a semisimple Lie algebra to be cyclic wide. Then, in the case of a regular Levi decomposable subalgebra whose radical is an ad-nilpotent subalgebra, we show that the condition is necessary and sufficient for the subalgebra to be cyclic wide. All Lie algebras, and modules in this article are finite-dimensional, and over the complex numbers.

The common solution space of general relativity

We review the solution space for the field equations of Einstein's General Relativity for various static, spherically symmetric spacetimes. We consider the vacuum case, represented by the Schwarzschild black hole; the de Sitter-Schwarzschild geometry, which includes a cosmological constant; the Reissner-Nordström geometry, which accounts for the presence of charge. Additionally we consider the homogenenous and anisotropic locally rotational Bianchi II spacetime in the vacuum. Our analysis reveals that the field equations for these scenarios share a common three-dimensional group of point transformations, with the generators being the elements of the D⊗sT2 Lie algebra, known as the semidirect product of dilations and translations in the plane. Due to this algebraic property the field equations for the aforementioned gravitational models can be expressed in the equivalent form of the null geodesic equations for conformally flat geometries. Consequently, the solution space for the field equations is common, and it is the solution space for the free particle in a flat space. This approach open new directions on the construction of analytic solutions in gravitational physics and cosmology.

The classification of nilpotent Bol algebras

In this paper, we generalize the classical method used to classify nilpotent low-dimensional Lie algebras to serve for nilpotent Bol algebras. The algebraic classification of the nilpotent Bol algebras up to dimension four is given.

Lie nilpotency and Lie solvability of tensor product of multiplicative Lie algebras

In this article, we discuss Lie nilpotency and Lie solvability of non-abelian tensor product of multiplicative Lie algebras. As our main results, we provide upper bounds on the Lie nilpotency class and Lie solvability length of quotient of tensor product of multiplicative Lie algebras.

Smooth representations of the extended Schrödinger–Virasoro algebra

The Schrödinger–Virasoro algebra was originally introduced during the process of investigating the free Schrödinger equations. In order to investigate vertex representations of the Schrödinger–Virasoro algebra, Unterberger introduced a new infinite-dimensional Lie algebra [Formula: see text] called the extended Schrödinger–Virasoro algebra in the context of two-dimensional conformal field theory and statistical physics. In this paper, we study simple smooth modules over [Formula: see text], which can be viewed as an extension of the Schrödinger–Virasoro algebra by a conformal current with conformal weight 1. More specifically, these modules are determined by simple modules over finite-dimensional solvable Lie algebras. Moreover, we present several equivalent descriptions for simple smooth modules over [Formula: see text].

Counting gradings on Lie algebras of block-triangular matrices

We study the number of isomorphism classes of gradings on Lie algebras of block-triangular matrices. Let G be a finite abelian group, for m∈Ns we determine the number E(−)(G,m) of isomorphism classes of elementary G-gradings on the Lie algebra UT(m)(−) of block-triangular matrices over an algebraically closed field of characteristic zero. We study the asymptotic growth of E(−)(G,m) and as a consequence prove that the E(−)(G,⋅) determines G up to isomorphism. We also study the asymptotic growth of the number N(−)(G,m) of isomorphism classes of G-gradings on UT(m)(−) and prove that N(−)(G,m))∼|G|E(−)(G,m).