The Dynkin index and sl(2)-subalgebras of simple Lie algebras

Abstract: This is a continuation of arXiv:0903.0398 [math.RT]. Let g be a simple Lie algebra. In this note, we provide simple formulae for the index of sl(2)-subalgebras in the classical Lie algebras and a new formula for the index of the principal sl(2). We also compute the difference, D, of the indices of principal and subregular sl(2)-subalgebras. Our formula for D involves some data related to the McKay correspondence for g. Using the index of sl(2)-subalgebras of classical Lie algebras, we also obtain three series of interesting combinatorial identities parameterised by partitions.