Let g denote a Lie algebra, and let ĝ denote the tensor product of g with a ring of truncated polynomials. The Lie algebra ĝ is called a truncated current Lie algebra. The highest-weight theory of ĝ is investigated, and a reducibility criterion for the Verma modules is described. Let g be a Lie algebra over a field k of characteristic zero, and fix a positive integer N . The Lie algebra (1) ĝ =...

It is shown that the quantised enveloping algebra of sl(n) contains a quantum Lie algebra, deened by means of axioms similar to Woro-nowicz's. This gives rise to Lie algebra-like generators and relations for the locally nite part of the quantised enveloping algebra, and suggests a canonical Poincar e-Birkhoo-Witt basis.

We present an algebraic approach to string theory. An embedding of sl(2|1) in a super Lie algebra together with a grading on the Lie algebra determines a nilpotent subalgebra of the super Lie algebra. Chirally gauging this subalgebra in the corresponding Wess-Zumino-Witten model, breaks the affine symmetry of the Wess-Zumino-Witten model to some extension of the N = 2 superconformal algebra. Th...

1 Background and Motivation We start with an example of affine Kac-Moody algebras and the Virasoro algebra. In this talk, F will be a field with characteristic 0, and all the vector spaces are assumed over F. Denote by Z the ring of integers and by N the set of nonnegative integers. Let 2 ≤ n ∈ N. Set sl(n,F) = {A ∈ Mn×n(F) | tr A = 0}, (1.1) 〈A,B〉 = tr AB for A,B ∈ sl(n,F), (1.2) where Mn×n(F)...

A new N = 4 superconformal algebra (SCA) is presented. Its internal affine Lie algebra is based on the seven-dimensional Lie algebra su(2)⊕ g, where g should be identified with a fourdimensional non-reductive Lie algebra. Thus, it is the first known example of what we choose to call a non-reductive SCA. It contains a total of 16 generators and is obtained by a nontrivial Inönü-Wigner contractio...

We consider quantum dynamical systems whose degrees of freedom are described by N × N matrices, in the planar limit N → ∞. Examples are gauge theories and the M(atrix)-theory of strings. States invariant under U(N) are 'closed strings', modelled by traces of products of matrices. We have discovered that the U(N)-invariant operators acting on both open and closed string states form a remarkable ...

that g* has the same derived algebra as g itself and that every ideal in g is also an ideal in t*. Let g be any algebraic Lie algebra. Denote by b the radical of g (i.e., the largest solvable ideal in g) and by n the largest ideal of g composed only of nilpotent matrices. By Levi's theorem, g is the direct sum of t and of a semi-simple subalgebra J. It can be proved that f is the direct sum of ...

Operads are a specific tool for encoding type of algebras. For instance there are operads encoding associative algebras, commutative and associative algebras, Lie algebras, pre-Lie algebras, dendriform algebras, Poisson algebras and so on. A usual way of describing a type of algebras is by giving the generating operations and the relations among them. For instance a Lie algebra L is a vector sp...

We compute the stable homology of necklace Lie algebras associated with quivers and give a construction of stable homology classes from certain A∞-categories. Our construction is a generalization of the construction of homology classes of moduli spaces of curves due to M. Kontsevich. In the second part of the paper we produce a Moyal-type quantization of the symmetric algebra of a necklace Lie ...