Hopf algebras for ternary algebras

We construct a universal enveloping algebra associated with the ternary extension of Lie (super)algebras called Lie algebra of order three. A Poincare-Birkhoff-Witt theorem is proven is this context. It this then shown that this universal enveloping algebra can be endowed with a structure of Hopf algebra. The study of the dual of the universal enveloping algebra enables to define the parameters of the transformation of a Lie algebra of order of 3. It turns out that these variables are the variables which generate the three-exterior algebra. (C) 2009 American Institute of Physics. [DOI: 10.1063/1.3152631]