Lie algebras with quadratic dimension equal to 2

The quadratic dimension of a Lie algebra is defined as the dimension of the linear space spanned by all its invariant non-degenerate symmetric bilinear forms. We prove that a quadratic Lie algebra with quadratic dimension equal to 2 is a local Lie algebra, this is to say, it admits a unique maximal ideal. We describe local quadratic Lie algebras using the notion of double extension and characterize those with quadratic dimension equal to 2 by the study of the centroid of such Lie algebras. We also give some necessary or sufficient conditions for a Lie algebra to have quadratic dimension equal to 2. Examples of local Lie algebras with quadratic dimension larger than 2 are given.     

On abelian subalgebras and ideals of maximal dimension in supersolvable Lie algebras

In this paper, the main objective is to compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable Lie algebras. We characterise the maximal abelian subalgebras of solvable Lie algebras and study solvable Lie algebras containing an abelian subalgebra of codimension 2. Finally, we prove that nilpotent Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not 2. Throughout the paper, we also give several examples to clarify some results.     

Classification of quasifinite modules over the Lie algebras of Weyl type

For a nondegenerate additive subgroup Γ of the n-dimensional vector space over an algebraically closed field of characteristic zero, there is an associative algebra and a Lie algebra of Weyl type spanned by all differential operators uD₁^m₁?D_n^m_n for (the group algebra), and m₁,…,m_n?0, where D₁,…,D_n are degree operators. In this paper, it is proved that an irreducible quasifinite -module is either a highest or lowest weight module or else a module of the intermediate series; furthermore, a classification of uniformly bounded -modules is completely given. It is also proved that an irreducible quasifinite -module is a module of the intermediate series and a complete classification of quasifinite -modules is also given, if Γ is not isomorphic to .     

Nonzero level Harish-Chandra modules over the Virasoro-like algebra

In the present paper, we study the nonzero level Harish-Chandra modules for the Virasoro-like algebra. We prove that a nonzero level Harish-Chandra module of the Virasoro-like algebra is a generalized highest weight (GHW for short) module. Then we prove that a GHW module of the Virasoro-like algebra is induced from an irreducible module of a Heisenberg subalgebra.     

Modified regular representations of affine and Virasoro algebras, VOA structure and semi-infinite cohomology

We find a counterpart of the classical fact that the regular representation R(G) of a simple complex group G is spanned by the matrix elements of all irreducible representations of G. Namely, the algebra of functions on the big cell G₀⊂G of the Bruhat decomposition is spanned by matrix elements of big projective modules from the category O of representations of the Lie algebra g of G, and has the structure of a g⊕g-module.The standard regular representation of the affine group has two commuting actions of the Lie algebra with total central charge 0, and carries the structure of a conformal field theory. The modified versions and , originating from the loop version of the Bruhat decomposition, have two commuting -actions with central charges shifted by the dual Coxeter number, and acquire vertex operator algebra structures derived from their Fock space realizations, given explicitly for the case G=SL(2,C).The quantum Drinfeld-Sokolov reduction transforms the representations of the affine Lie algebras into their W-algebra counterparts, and can be used to produce analogues of the modified regular representations. When g=sl(2,C) the corresponding W-algebra is the Virasoro algebra. We describe the Virasoro analogues of the modified regular representations, which are vertex operator algebras with the total central charge equal to 26.The special values of the total central charges in the affine and Virasoro cases lead to the non-trivial semi-infinite cohomology with coefficients in the modified regular representations. The inherited vertex algebra structure on this cohomology degenerates into a supercommutative associative superalgebra. We describe these superalgebras in the case when the central charge is generic, and identify the 0th cohomology with the Grothendieck ring of finite-dimensional G-modules. We conjecture that for the integral values of the central charge the 0th semi-infinite cohomology coincides with the Verlinde algebra and its counterpart associated with the big projective modules.     

Quasifinite modules of a Lie algebra related to Block type

Let B be the universal central extension of a graded Lie algebra of Block type. In this paper, it is proved that any quasifinite irreducible B-module is either highest weight, lowest weight or uniformly bounded. Furthermore, the quasifinite irreducible highest weight B-modules are classified, and the intermediate series B-modules are classified and constructed.     

gl(λ) and differential operators preserving polynomials

We discuss a certain generalization of gl _n (), and show how it is connected to polynomial differential operators that leave the polynomial space invariant.     

Classification of irreducible weight modules over higher rank Virasoro algebras

Let G be a rank n additive subgroup of C and Vir[G] the corresponding Virasoro algebra of rank n. In the present paper, irreducible weight modules with finite dimensional weight spaces over Vir[G] are completely determined. There are two different classes of them. One class consists of simple modules of intermediate series whose weight spaces are all 1-dimensional. The other is constructed by using intermediate series modules over a Virasoro subalgebra of rank n−1. The classification of such modules over the classical Virasoro algebra was obtained by O. Mathieu in 1992 using a completely different approach.     

Invariant polynomials on truncated multicurrent algebras

We construct invariant polynomials on truncated multicurrent algebras, which are Lie algebras of the form $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$, where $\mathfrak{g}$ is a finite-dimensional Lie algebra over a field $\mathbb{F}$ of characteristic zero, and I is a finite-codimensional ideal of $\mathbb{F}[t_1,\dots ,t_{?}]$ generated by monomials. In particular, when $\mathfrak{g}$ is semisimple and $\mathbb{F}$ is algebraically closed, we construct a set of algebraically independent generators for the algebra of invariant polynomials. In addition, we describe a transversal slice to the space of regular orbits in $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$. As an application of our main result, we show that the center of the universal enveloping algebra of $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$ acts trivially on all irreducible finite-dimensional representations provided I has codimension at least two.     

On seaweed subalgebras and meander graphs in type D

In 2000, Dergachev and Kirillov introduced subalgebras of “seaweed type” in ${\mathfrak{gl}}_n$ and computed their index using certain graphs, which we call type-A meander graphs. Then the subalgebras of seaweed type, or just “seaweeds”, have been defined by Panyushev (2001) [9] for arbitrary reductive Lie algebras. Recently, a meander graph approach to computing the index in types B and C has been developed by the authors. In this article, we consider the most difficult and interesting case of type $\mathsf{D}$. Some new phenomena occurring here are related to the fact that the Dynkin diagram has a branching node.     

Abelian subalgebras in some particular types of Lie algebras

It is well-known that there exists a close link between Lie Theory and Relativity Theory. Indeed, the set of all symmetries of the metric in our four-dimensional spacetime is a Lie group. In this paper we try to study this link in depth, by dealing with three particular types of Lie algebras: h_n algebras, g_n algebras and Heisenberg algebras. Our main goal is to compute the maximal abelian dimensions of each of them, which will allow us to move a step forward in the advancement of this subject.     

Bosonic and fermionic representations of Lie algebra central extensions

Given any representation of an arbitrary Lie algebra g over a field K of characteristic 0, we construct representations of a central extension of g on bosonic and fermionic Fock space. The method gives an explicit formula for a (sometimes trivial) 2-cocycle in H²(g;K). We illustrate these techniques with several concrete examples.     

On the derived algebra of a centraliser

Let g be a classical Lie algebra, e∈g a nilpotent element and ge⊂g the centraliser of e. We prove that ge=[ge,ge] if and only if e is rigid. It is also shown that if e∈[ge,ge], then the nilpotent radical of ge coincides with [ge(1),ge], where ge(1)⊂ge is an eigenspace of a characteristic of e corresponding to the eigenvalue 1.     

On the Leibniz (co)homology of the Lie algebra of the Euclidean group

The Lie algebra of the Euclidean group is an abelian extension of the orthogonal Lie algebra. We compute its Leibniz (co)homology. It is computed via the identification of certain orthogonal invariants and shown to be an algebra generated by a n−1-fold tensor and an n-fold tensor.