Dirac operators and the Very Strange Formula for Lie superalgebras

Using a super-affine version of Kostant's cubic Dirac operator, we prove a very strange formula for quadratic finite-dimensional Lie superalgebras with a reductive even subalgebra.Comment: Latex file, 25 pages. A few misprints corrected. To appear in the forthcoming volume "Advances in Lie Superalgebras", Springer INdAM Serie

Irreducible modules over finite simple Lie conformal superalgebras of type K

We construct all finite irreducible modules over Lie conformal superalgebras of type KComment: Accepted for publication in J. Math. Phys

Parafermionic Representation of the Affine sl(2/1)sl(2/1) Algebra at Fractional Level

The four fermionic currents of the affine superalgebra $sl(2/1)$ at fractional level $k=1/u-1$, u positive integer, are shown to be realised in terms of a free scalar field, an $sl(2)$ doublet field and a primary field of the parafermionic algebra $Z_{u-1}$.Comment: 5 pages, Latex 2

Admissible sl(2/1) Characters and Parafermions

The branching functions of the affine superalgebra $sl(2/1)$ characters into characters of the affine subalgebra $sl(2)$ are calculated for fractional levels $k=1/u-1$, u positive integer. They involve rational torus $A_{u(u-1)}$ and $Z_{u-1}$ parafermion characters.Comment: 14 pages, Latex 2

W_{1+\infty} and W(gl_N) with central charge N

We study representations of the central extension of the Lie algebra of differential operators on the circle, the W-infinity algebra. We obtain complete and specialized character formulas for a large class of representations, which we call primitive; these include all quasi-finite irreducible unitary representations. We show that any primitive representation with central charge N has a canonical structure of an irreducible representation of the W-algebra W(gl_N) with the same central charge and that all irreducible representations of W(gl_N) with central charge N arise in this way. We also establish a duality between "integral" modules of W(gl_N) and finite-dimensional irreducible modules of gl_N, and conjecture their fusion rules.Comment: 29 pages, Latex, uses file amssym.def (a few remarks added, typos corrected

Representation theory of the stabilizer subgroup of the point at infinity in Diff(S^1)

The group Diff(S^1) of the orientation preserving diffeomorphisms of the circle S^1 plays an important role in conformal field theory. We consider a subgroup B_0 of Diff(S^1) whose elements stabilize "the point of infinity". This subgroup is of interest for the actual physical theory living on the punctured circle, or the real line. We investigate the unique central extension K of the Lie algebra of that group. We determine the first and second cohomologies, its ideal structure and the automorphism group. We define a generalization of Verma modules and determine when these representations are irreducible. Its endomorphism semigroup is investigated and some unitary representations of the group which do not extend to Diff(S^1) are constructed.Comment: 34 pages, no figur

On the Calculation of Group Characters

It is known that characters of irreducible representations of finite Lie algebras can be obtained using theWeyl character formula including Weyl group summations which make actual calculations almost impossible except for a few Lie algebras of lower rank. By starting from the Weyl character formula, we show that these characters can be re-expressed without referring to Weyl group summations. Some useful technical points are given in detail for the instructive example of G2 Lie algebra.Comment: 6 pages, no figure, Plain Te