Finite-dimensional Lie Algebra Research Articles

Group-graded twisted Calabi–Yau algebras

Historically, the study of graded (twisted or otherwise) Calabi–Yau algebras has meant the study of such algebras under an N-grading. In this paper, we propose a suitable definition for a twisted G-graded Calabi–Yau algebra, for G an arbitrary abelian group. Building on the work of Reyes and Rogalski, we show that a G-graded algebra is twisted Calabi–Yau if and only if it is G-graded twisted Calabi–Yau. In the second half of the paper, we prove that localizations of twisted Calabi–Yau algebras at elements which form both left and right denominator sets remain twisted Calabi–Yau. As such, we obtain a large class of Z-graded twisted Calabi–Yau algebras arising as localizations of Artin–Schelter regular algebras. Throughout the paper, we survey a number of concrete examples of G-graded twisted Calabi–Yau algebras, including the Weyl algebras, families of generalized Weyl algebras, and universal enveloping algebras of finite dimensional Lie algebras.

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.

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].

$\mathfrak{sl}_{2}$ Triples Whose Nilpositive Elements Are in a Space Which Is Spanned by the Real Root Vectors in Rank 2 Symmetric Hyperbolic Kac–Moody Lie Algebras

In analogy with the theory of nilpotent orbits in finite-dimensional semisimple Lie algebras, it is known that the principal \mathfrak{sl}_{2} subalgebras can be constructed in hyperbolic Kac–Moody Lie algebras. We obtained a series of \mathfrak{sl}_{2} subalgebras in rank 2 symmetric hyperbolic Kac–Moody Lie algebras by extending the aforementioned construction. We present this result and also discuss \mathfrak{sl}_{2} modules obtained by the action of the \mathfrak{sl}_{2} subalgebras on the original Lie algebras.

New cluster algebras from old: integrability beyond Zamolodchikov periodicity

Abstract We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodchikov periodicity: they are affine birational maps for which every orbit is periodic with the same period. Following on from preliminary work by one of us with Kouloukas, here we present integrable maps obtained from deformations of cluster mutations related to the simple root systems 
$A_3$, $B_2$, $B_3$ and $D_4$. We further show how new cluster algebras arise, by considering Laurentification, that is, a lifting to a higher-dimensional map expressed in a set of new variables (tau functions), for which the dynamics exhibits the Laurent property. For the integrable map obtained by deformation of type $A_3$, which already appeared in our previous work, we show that there is a commuting map of Quispel-Roberts-Thompson (QRT) type which is built from a composition of mutations and a permutation applied to 
the same cluster algebra of rank 6, with an additional 2 frozen variables. Furthermore, both the deformed $A_3$ map and the QRT map correspond to translation by a generator in the Mordell-Weil group of a rational elliptic surface of rank two, and the underlying cluster algebra comes from a quiver that is mutation equivalent to the $q$-Painlev\'e III quiver found by Okubo. 
The deformed integrable maps of types $B_2$, $B_3$ and $D_4$ are also related to elliptic surfaces. 

From a dynamical systems viewpoint, the message of the paper is that special families of birational maps with completely periodic dynamics under iteration admit natural deformations that are aperiodic yet completely integrable.

Optimal systems of Lie subalgebras: A computational approach

Lie groups of symmetries of differential equations constitute a fundamental tool for constructing group-invariant solutions. The number of subgroups is potentially infinite and so the number of invariant solutions; thus, it is crucial to obtain a classification of subgroups in order to have an optimal system of inequivalent solutions from which all other solutions can be derived by action of the group itself. Since Lie groups are intimately connected to Lie algebras, a classification of inequivalent subgroups induces a classification of inequivalent Lie subalgebras, and vice versa. A general method for classifying the Lie subalgebras of a finite–dimensional Lie algebra uses inner automorphisms that are obtained by exponentiating the adjoint groups. In this paper, we present an effective algorithm able to automatically determine optimal systems of Lie subalgebras of a generic finite–dimensional Lie algebra abstractly assigned by means of its structure constants, or realized in terms of matrices or vector fields, or defined by a basis and the set of non-zero Lie brackets. The algorithm is implemented in the computer algebra system Wolfram Mathematica™; some meaningful and non-trivial examples are considered.

Lie semisimple algebras of derivations and varieties of PI-algebras with almost polynomial growth

We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras U T 2 ( W λ ) UT_2(W_\lambda ) or E n d ( W μ ) End(W_\mu ) for some integral dominant weight λ , μ \lambda ,\mu with μ ≠ 0 \mu \neq 0 . In the special case L = s l 2 L=\mathfrak {sl}_2 we prove that this is a sufficient condition too.

Local and 2-Local $$\frac{1}{2}$$-Derivations on Finite-Dimensional Lie Algebras

Local and 2-Local $$\frac{1}{2}$$-Derivations on Finite-Dimensional Lie Algebras

Simple Harish-Chandra modules over the superconformal current algebra

In this paper, we classify the simple Harish-Chandra modules over the superconformal current algebra gˆ, which is the semi-direct sum of the N=1 superconformal algebra with the affine Lie superalgebra g˙⊗A⊕CC1, where g˙ is a finite-dimensional simple Lie algebra, and A is the tensor product of the Laurent polynomial algebra and the Grassmann algebra. As an application, we can directly get the classification of the simple Harish-Chandra modules over the N=1 Heisenberg-Virasoro algebra.

THE LEVI DECOMPOSITION OF THE LIE ALGEBRA M_2 (R)⋊gl_2 (R)

The idea of the Lie algebra is studied in this research. The decomposition between Levi sub-algebra and the radical can be used to define the finite dimensional Lie algebra. The Levi decomposition is the name for this type of decomposition. The goal of this study is to obtain a Levi decomposition of the Lie algebra of dimension 8. We compute its Levi sub-algebra and the radical of Lie algebra with respect to its basis to achieve this goal. We use literature studies on the Levi decomposition and Lie algebra in Dagli result to produce the radical and Levi sub-algebra. It has been shown that can be decomposed in the terms of the Levi sub-algebra and its radical. In this resulst, it has been given by direct computations and we obtained that the explicit formula of Levi decomposition of the affine Lie algebra whose basis is is written by with is is the Levi sub-algebra of .

Cohomological varieties associated to vertex operator algebras

Given a vertex operator algebra V, one can attach a graded Poisson algebra called the C2-algebra R(V). The associate Poisson scheme provides an important invariant for V and has been studied by Arakawa as the associated variety. In this article, we define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point. We study its basic properties and then construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational C2-cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. In this paper, we study R(V) as a commutative algebra only and do not use the property of its Poisson structure, which is expected to provide more refined invariants. The goal of this work is to study the cohomological supports of modules for vertex algebras as the cohomological support varieties for finite groups and restricted Lie algebras.

Investigation of ID-derivations of finite-dimensional nilpotent Lie algebras

Let [Formula: see text] be a Lie algebra and let [Formula: see text] be the set of all derivations of [Formula: see text]. Then the derivation [Formula: see text] of [Formula: see text] is called an ID-derivation if [Formula: see text] for all [Formula: see text]. The set of all ID-derivations is denoted by [Formula: see text]. Let [Formula: see text] be the set of all central derivations and let [Formula: see text] and [Formula: see text] be, respectively, the set of all center derivations and ID-derivations that map [Formula: see text] to zero. In this paper, we verify relations between [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text]. Also, we prove that if [Formula: see text] is a nilpotent Lie algebra of class [Formula: see text], then [Formula: see text] is a nilpotent Lie algebra of class [Formula: see text].

A computational approach to almost-inner derivations

We present a computational approach to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table. We also present an example of a Lie algebra for which the quotient algebra of the almost-inner derivations modulo the inner derivations is non-abelian. This answers a question of Kunyavskii and Ostapenko.

A study of Bianchi type I spacetime according to their Ricci collineations

Rici collineations (RCs) have been used in this research to study the locally rotationally symmetric (LRS) Bianchi type I spacetimes. To accomplish our objectives, the RC equations are typically integrated for both situations of the Ricci tensor, degenerate and non-degenerate. Throughout this work, a number of situations occur that provide various finite and infinite dimensional Lie algebras of RCs.

Classification of simple smooth modules over the Heisenberg–Virasoro algebra

In this paper, we classify simple smooth modules over the mirror Heisenberg–Virasoro algebra ${\mathfrak {D}}$ , and simple smooth modules over the twisted Heisenberg–Virasoro algebra $\bar {\mathfrak {D}}$ with non-zero level. To this end we generalize Sugawara operators to smooth modules over the Heisenberg algebra, and develop new techniques. As applications, we characterize simple Whittaker modules and simple highest weight modules over ${\mathfrak {D}}$ . A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg–Virasoro vertex algebras. We also present a few examples of simple smooth ${\mathfrak {D}}$ -modules and $\bar {\mathfrak {D}}$ -modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over $\mathfrak {D}$ and $\bar {\mathfrak {D}}$ which are always tensor products of simple Virasoro modules and simple Heisenberg modules.

Transposed Poisson structures on solvable and perfect Lie algebras

We described all transposed Poisson algebra structures on oscillator Lie algebras, i.e. on one-dimensional solvable extensions of the -dimensional Heisenberg algebra; on solvable Lie algebras with naturally graded filiform nilpotent radical; on -dimensional solvable extensions of the -dimensional Heisenberg algebra; and on n-dimensional solvable extensions of the n-dimensional algebra with trivial multiplication. We also answered one question on transposed Poisson algebras early posted in a paper by Beites, Ferreira and Kaygorodov. Namely, we found that the semidirect product of and irreducible module gives a finite-dimensional Lie algebra with non-trivial -derivations, but without non-trivial transposed Poisson structures.

Bounds for the dimension of the Schur multiplier of finite dimensional nilpotent Lie algebras

Let L be an n-dimensional nilpotent Lie algebra of nilpotency class c≥2 over an arbitrary field F with dim⁡γ2(L)=m and dim(Z(L)Z(L)∩γ2(L))=t, where Z(L) and γ2(L) denote the center ideal and the derived subalgebra of L, respectively. The purpose of this work is to obtain a new bound on the dimension of the Schur multiplier for L, which is given in terms of n,m,c, and t. In the virtue of an old bound due to Niroomand and Russo, namely 12(n−m−1)(n+m−2)+1, we find a finite dimensional nilpotent Lie algebra L of nilpotency class 3 over a field F of characteristic 3 that attains this bound. We then present a complete list of all finite dimensional nilpotent Lie algebras of nilpotency class at least 2 that reach Niroomand–Russo's Formula. Finally, we give the structure of all finite dimensional nilpotent Lie algebras that attain some recent generalizations of Niroomand–Russo's Formula.

Realization of arbitrary Lie algebras by automorphisms of $\mathrm{CR}$ manifolds and symmetries of differential equations

For any finite-dimensional real Lie algebra $\mathfrak{h}$, we construct a germ of a real analytic hypersurface in complex space such that its Lie algebra of infinitesimal holomorphic automorphisms is isomorphic to $\mathfrak{h}$. For any $\mathfrak{h}$, we also construct a system of partial differential equations whose Lie algebra of symmetries is isomorphic to the complexification of the algebra $\mathfrak{h}$.

U0-free quantum group representations

Let g be a finite dimensional complex simple Lie algebra, and let U=Uq(g) be its quantized enveloping algebra with a triangular decomposition U=U−U0U+. We classify all U-module structures on U0 with the regular action of U0 on itself, which are shown to have direct connections with the q-Weyl algebra and the bounded weight representations. We obtain that the necessary and sufficient condition for the existence of such modules is that U has to be of type An(n≥1), Bn(n≥2) or Cn(n≥3). Moreover, we study their module structures and associated weight modules for each type.

Geometry-Preserving Numerical Methods for Physical Systems with Finite-Dimensional Lie Algebras

Geometry-Preserving Numerical Methods for Physical Systems with Finite-Dimensional Lie Algebras