This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module L(k, 0) of positive integer level k for any affine Kac-Moody Lie algebra ĝ, where g is a finite dimensional simple Lie algebra. In particular, the generators and the C 2-cofiniteness of the parafermion vertex operator algebras are discussed. A proof of the well-known fact that the parafermion vertex operator algebra can be realized as the commutant of a lattice vertex operator algebra in L(k, 0) is also given.
In this paper, we prove the isomorphic criterion theorem for (n+2)-dimensional n-Lie algebras, and give a complete classification of (n + 2)-dimensional n-Lie algebras over an algebraically closed field of characteristic zero.
In this paper, we study the fermionic and bosonic representations for a class of BC-graded Lie algebras coordinatized by skew Laurent polynomial rings. This generalizes the fermionic and bosonic constructions for the affine Kac-Moody algebras of type A N (2).
In this paper, we investigate Lie bialgebra structures on a twisted Schrödinger-Virasoro type algebra
The notion of generating index of Lie algebras is introduced. We characterize some Lie algebras with full generating index and classify the Lie algebras with generating index 2 and 3. As a corollary, we give a characterization of 2-step nilpotent Lie algebras.
Let U = ℂ2, Γ = ℤ2, and let ℂ[x 1 ±1, x 2 ±1] be the ring of Laurent polynomials. The Witt algebra L is the Lie algebra of derivations over ℂ[x 1 ±1, x 2 ±1], which is spanned by elements of the form D(u, r) = x r(u 1 d 1 + u 2 d 2), u = (u 1, u 2) ∈ U, r ∈ Γ, where d 1 and d 2 are the degree derivations of ℂ[x 1 ±1, x 2 ±1]. The image of gl 2-module V under Larsson functor F α, denoted by W = F α(V), gives a class of L-modules, often called the Larsson-modules of L. In this paper, we study the derivations from the Witt algebra L to its Larsson-modules W, and we determine the first cohomology group H 1(L,W).
We associate quantum vertex algebras and their ϕ-coordinated quasi modules to certain deformed Heisenberg algebras.
In this paper, we study Whittaker modules for a Lie algebra of Block type. We define Whittaker modules and under some conditions, obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules over this algebra and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra.
Let
Singular vectors of a representation of a finite-dimensional simple Lie algebra are weight vectors in the underlying module that are nullified by positive root vectors. In this article, we use partial differential equations to explicitly find all the singular vectors of the polynomial representation of the simple Lie algebra of type F 4 over its 26-dimensional basic irreducible module, which also supplements a proof of the completeness of Brion’s abstractly described generators. Moreover, we show that the number of irreducible submodules contained in the space of homogeneous harmonic polynomials with degree k ⩾ 2 is greater than or equal to 〚k/3〛 + 〚(k − 2)/3〛 + 2.
In this paper, we study the support varieties for Lie algebras L of Cartan type. We give some description for the support varieties of any finitedimensional L-module with character χ, whenever the height of the character χ is not too large. And a more concrete computation can be made for a class of modules with semisimple characters.
Double graded ideals and simplicity of elementary unitary Lie algebra eu n(R,−, γ) and Steinberg unitary Lie algebra stu n(R,−, γ) are characterized, where R is a unital involutory associative algebra over a field F of characteristic zero, n ⩾ 5.