Conformal embeddings in basic Lie superalgebras

In this thesis we deal with some aspects concerning embeddings of regular subalgebras in basic Lie superalgebras. A first problem is to find a criterion to determine whether a regular subalgebra is maximal or not, similarly to what has been done for semisimple Lie algebras. We discovered that the the maximal equal rank subalgebras are obtained by taking all possible affine diagrams and deleting a node corresponding to a simple root with prime coefficient. The second and main purpose of the thesis is, for every maximal equal rank regular subalgebra g_0 in g, to compute numbers k such that we have a conformal embedding of the vertex algebra generated by g_0 in the affine vertex algebra of level k associated with g. We used the Adamovic-Perse criterion to compute all these levels and we noted that they rarely depend on the choice of g_0 in g. We also used the "fusion rule argument" to find some decompositions, as a g_0-module, of the simple vertex algebra associated with g of conformal level k.