Monograf́ıas de la Real Academia de Ciencias de Zaragoza. 29: 197–205, (2006). Cartan An

We present a Drinfeld double structure for the Cartan series An of semisimple Lie algebras (that can be extended to the other three series). This algebraic structure is obtained from two disjoint solvable subalgebras s ± related by a Weyl transformation and containing the positive and negative roots, respectively. The new Lie algebra g ̄ = s+ + s − is a central extension of the corresponding semisimple Lie algebra An by an Abelian kernel, whose dimension is the rank of An. In order to construct such Drinfeld double algebra we need a particular basis: all its generators are explicitly described, the generators of the extended Cartan subalgebra are orthonormal and the length of all the root vectors is fixed.