文摘
Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras \({V_{\mathbf{k}}({\mathfrak{g}}^0)\subset V_{k}({\mathfrak{g}})}\), corresponding to an embedding of a maximal equal rank reductive subalgebra \({{\mathfrak{g}}^0}\) into a simple Lie algebra \({{\mathfrak{g}}}\), is conformal if and only if the corresponding central charges are equal. We classify the equal rank conformal embeddings. Furthermore we describe, in almost all cases, when \({V_{k}({\mathfrak{g}})}\) decomposes finitely as a \({V_{\mathbf{k}}({\mathfrak{g}}^0)}\)-module.