**Urmie Ray**

University of Reims, France

March 3, 2011

**Weyl Character and super-character formulae for finite dimensional and affine Lie superalgebras wit**:
For finite dimensional simple and affine Lie algebras, the roots
of positive norm are all real, i.e. act locally finitely on
integrable modules, whereas the roots of non-positive norm do
not, and the non-diagonal entries of the Cartan matrices are
non-positive. For most finite dimensional and affine Lie superalgebras there are roots both positive, negative and isotropic (norm 0) that are real and there are both positive and negative non-diagonal entries in the Cartan matrices. Hence the usual proofs of the character formula do not work and computing these characters in
all cases has remained an open problem for the last twenty years. The greater the dimension of the maximal isotropic subspace of the set of roots, the greater the complication. In this talk, I will explain my recent proof of these formulae.