This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module ? (k, 0) of positive integer level k for any affine Kac-Moody Lie algebra g^, where g is a finite dimensional simple Lie algebra. In particular, the generators and the C₂-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 ? (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 AN(2).

The aim of this paper is to give an alternative proof of Kac’s theorem for weighted projective lines over the complex field. The geometric realization of complex Lie algebras arising from derived categories is essentially used.

In this paper, we investigate Lie bialgebra structures on a twisted Schr?dinger-Virasoro type algebra L. All Lie bialgebra structures on L are triangular coboundary, which is different from the relative result on the original Schr?dinger-Virasoro type Lie algebra. In particular, we find for this Lie algebra that there are more hidden inner derivations from itself to L?L and we develop one method to search them.

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=?2, Γ=?2, and let ?[x1±1,x2±1] be the ring of Laurent polynomials. The Witt algebra ? is the Lie algebra of derivations over ?[x1±1,x2±1], which is spanned by elements of the form D(u, r) = x^r(u₁d₁ + u₂d₂), u=(u1,u2)∈U, r∈Γ, where d₁ and d₂ are the degree derivations of ?[x1±1,x2±1]. The image of gl₂-module V under Larsson functor Fα, denoted by W=Fα(V), gives a class of ?-modules, often called the Larsson-modules of ?. In this paper, we study the derivations from the Witt algebra ? to its Larsson-modules W, and we determine the first cohomology group H¹(?, W).

A class of piecewise linear paths, as a generalization of Littelmann’s paths, are introduced, and some operators, acting on the above paths with fixed parametrization, are defined. These operators induce the ordinary Littelmann’s root operators’ action on the equivalence classes of paths. With these induced operators, an explicit realization of B(∞) is given in terms of equivalence classes of paths, where B(∞) is the crystal base of the negative part of a quantum group Uq(g). Furthermore, we conjecture that there is a complete set of representatives for the above model by fixing a parametrization, and we prove the case when g is of finite type.

We associate quantum vertex algebras and their ?-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 F be a field of characteristic 0, and let G be an additive subgroup of F. We define a class of infinite-dimensional Lie algebras W with an F-basis {Lμ,Vμ,Wμ|μ∈G}, which are very closely related to W-algebras. In this paper, the second cohomology group of W is determined.

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 F₄ 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 eun(R,-,γ) and Steinberg unitary Lie algebra stun(R,-,γ) are characterized, where R is a unital involutory associative algebra over a field F of characteristic zero, n≥5.