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Defining relations associated with the principal sl(2)-subalgebras of simple Lie algebras

The notion of defining relations is well-defined for any nilpotent Lie algebra. Therefore a conventional way to present a simple Lie algebra G is by splitting it into the direct sum of a commutative Cartan subalgebra and two maximal nilpotent subalgebras (positive and negative) and together the generators of both these nilpotent subalgebras together generate G. Though there are many relations between these generators, they are neat (Serre relations). It is possible to determine the relations for generators of different type, e.g, with the principal embeddings of sl(2) into G one can associate only TWO elements that generate G. We explicitly describe the corresponding presentations of simple Lie algebras, for all finite dimensional and certain infinite dimensional ones; namely, for the Lie algebra "of matrices of a complex size" realized as a subalgebra of the Lie algebra of differential operators in 1 indeterminate. The relations obtained are rather simple. Our results might be of interest in applications to integrable systems (like vector-valued Liouville (or Leznov-Saveliev, or 2-dimensional Toda) equations and KdV-type equations). They also indicate how to q-quantize the Lie algebra of matrices of complex size.