Related papers: On Lie algebras associated with modules over polyn…

We study Lie bialgebroid crossed modules which are pairs of Lie algebroid crossed modules in duality that canonically give rise to Lie bialgebroids. A one-one correspondence between such Lie bialgebroid crossed modules and co-quadratic…

Let V be a finite dimensional representation of the connected complex reductive group H. Denote by G the derived subgroup of H and assume that the categorical quotient of V by G is one dimensional. In this situation there exists a…

In this paper, we study representations of the vertex operator algebra $L(k,0)$ at one-third admissible levels $k= -5/3, -4/3, -2/3$ for the affine algebra of type $G_2^{(1)}$. We first determine singular vectors and then obtain a…

For any field K and directed graph E, we completely describe the elements of the Leavitt path algebra L_K(E) which lie in the commutator subspace [L_K(E),L_K(E)]. We then use this result to classify all Leavitt path algebras L_K(E) that…

This article constructs a crossed module corresponding to the generator of the third cohomology group with trivial coefficients of a complex simple Lie algebra. This generator reads as <[,],>, constructed from the Lie bracket [,] and the…

We determine the decomposition numbers of the partition algebra when the characteristic of the ground field is zero or larger than the degree of the partition algebra. This will allow us to determine for which exact values of the parameter…

We continue the study of the vertex operator algebra $L(k,0)$ associated to a type $G_2^{(1)}$ affine Lie algebra at admissible one-third integer levels, $k = -2 + m + \tfrac{i}{3}\ (m\in \mathbb{Z}_{\ge 0}, i = 1,2)$, initiated in…

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. Examples include generalized current algebras and (twisted)…

For a natural number $m$, a Lie algebra $L$ over a field $k$ is said to be of breadth type $(0, m)$ if the co-dimension of the centralizer of every non-central element is of dimension $m$. In this article, we classify finite dimensional…

Let $\mathfrak{g}:={\rm Der}(\mathbb{C}[t_1, t_2,\cdots, t_n])$ and $\mathcal{L}:={\rm Der}(\mathbb{C}[[t_1, t_2,\cdots, t_n]])$ be the Witt Lie algebras. Clearly, $\mathfrak{g}$ is a proper subalegbra of $\mathcal{L}$. Surprisingly, we…

Let $g$ be a semisimple Lie algebra over $\mathbb C$ and $k$ be a reductive in $g$ subalgebra. We say that a simple $g$-module $M$ is a $(g; k)$-module if as a $k$-module $M$ is a direct sum of finite-dimensional $k$-modules. We say that a…

Given a grading by an abelian group G on a semisimple Lie algebra L over an algebraically closed field of characteristic 0, we classify up to isomorphism the simple objects in the category of finite-dimensional G-graded L-modules. The…

Let $W_n^+$ be the Lie algebra of the Lie algebra of vector fields on $\C^n$. In this paper, we classify all simple bounded weight $W_n^+$ modules. Any such module is isomorphic to the simple quotient of a tensor module $F(P,M)=P\otimes M$…

We study left-invariant pseudo-Riemannian metrics on Lie groups using the bracket flow of the corresponding Lie algebra. We focus on metrics where the Lie algebra is in the null cone of the $G=O(p,q)$-action; i.e., Lie algebras $\mu$ where…

In this paper, we study Whittaker modules for a Lie algebras of Block type. We define Whittaker modules and under some conditions, obtain a one to one correspondence between the set of isomorphic classes of Whittaker modules over this…

Verma modules over the $W$-algebra W(2,2) were considered by Zhang and Dong, while the Harish-Chandra modules and irreducible weight modules over the same algebra were classified by Liu and Zhu etc. In the present paper we shall investigate…

Let $f:V\times V\to F$ be a totally arbitrary bilinear form defined on a finite dimensional vector space $V$ over a a field $F$, and let $L(f)$ be the subalgebra of $\gl(V)$ of all skew-adjoint endomorphisms relative to $f$. Provided $F$ is…

For any two complex numbers $a$ and $b$, $\mathcal{V} ir(a,b)$ is a central extension of $\mathcal{W}(a,b)$ which is universal in the case $(a,b)\neq (0,1)$, where $\mathcal{W}(a,b)$ is the Lie algebra with basis $\{L_n,W_n\mid n\in\Z\}$…

In a paper by the authors, the associative and the Lie algebras of Weyl type $A[D]=A\otimes F[D]$ were introduced, where $A$ is a commutative associative algebra with an identity element over a field $F$ of any characteristic, and $F[D]$ is…

The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-${\mathcal N}$}. To facilitate this we investigate solvable Lie algebras of nilpotent length $k$,…