We use a fermionic extension of the bosonic module to obtain a class of B(0, N)-graded Lie superalgebras with nontrivial central extensions. 0 Introduction B(M − 1, N)-graded Lie superalgebras were first investigated and classified up to central extension by Benkart-Elduque (see also Garcia-Neher’s work in [GN]). Those root graded Lie superalgebras are a super-analog of root graded Lie algebras...

J. Lepowsky and R. L. Wilson initiated the approach to combinatorial Rogers-Ramanujan type identities via the vertex operator constructions of representations of affine Lie algebras. In a joint work with Arne Meurman this approach is developed further in the framework of vertex operator algebras. The main ingredients of that construction are defining relations for standard modules and relations...

a unital $c^*$ -- algebra $mathcal a,$ endowed withthe lie product $[x,y]=xy- yx$ on $mathcal a,$ is called a lie$c^*$ -- algebra. let $mathcal a$ be a lie $c^*$ -- algebra and$g,h:mathcal a to mathcal a$ be $bbb c$ -- linear mappings. a$bbb c$ -- linear mapping $f:mathcal a to mathcal a$ is calleda lie $(g,h)$ -- double derivation if$f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all ...

For any finite graph Γ and any field K of characteristic unequal to 2 we construct an algebraic variety X over K whose K-points parameterise K-Lie algebras generated by extremal elements, corresponding to the vertices of the graph, with prescribed commutation relations, corresponding to the non-edges. After that, we study the case where Γ is a connected, simply laced Dynkin diagram of finite or...

An important theorem in the theory of infinite dimensional Lie algebras states that any affine Kac-Moody algebra can be realized (that is to say constructed explicitly) using loop algebras. In this paper, we consider the corresponding problem for a class of Lie algebras called extended affine Lie algebras (EALAs) that generalize affine algebras. EALAs occur in families that are constructed from...

We obtain Drinfeld second realization of the quantum affine superalgebras associated with the affine Lie superalgebra D(1)(2, 1;x). Our results are analogous to those obtained by Beck for the quantum affine algebras. Beck’s analysis uses heavily the (extended) affine Weyl groups of the affine Lie algebras. In our approach the structures are based on a Weyl groupoid. Preprint numbers: MIT-CTP 38...

We investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac-Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras have been under intensive investigation in recent years. They have also been call...

Iterated loop algebras are by definition obtained by repeatedly applying the loop construction, familiar from the theory of affine Kac-Moody Lie algebras, to a given base algebra. Our interest in this iterated construction is motivated by its use in the realization of extended affine Lie algebras, but the construction also appears naturally in the study of other classes of algebras. This paper ...

We study a class of representations called “calibrated representations” of the degenerate double affine Hecke algebra and those of the rational Cherednik algebra of type GLn. We give a realization of calibrated irreducible modules as spaces of coinvariants constructed from integrable modules over the affine Lie algebra ĝl m . Moreover, we give a character formula of these irreducible modules in...

We continue a previous study on Γ-vertex algebras and their quasimodules. In this paper we refine certain known results and we prove that for any Z-graded vertex algebra V and a positive integer N , the category of V -modules is naturally isomorphic to the category of quasimodules of a certain type for V ⊗N . We also study certain generalizations of twisted affine Lie algebras and we relate suc...