Lie bialgebra structures over the Lie algebra of truncated polynomials

Let K be a field with characteristic zero, the algebra of polynomials g=C[t can be endowed with a Lie algebra structure with the bracket [¿,¿:g¿g¿g givenby t^i¿t^j=(j-i)t^(i+j). (1.1) An immediate fact about this Lie algebra is that the set ¿t^k¿, the set of multiples of t^k, is an ideal. Then the quotient C_k [t=(C[t)/(¿t^(k+1)¿) inherit a Lie algebra structure. The main purpose of this work is the classification of the Lie bialgebra structure over the Lie algebra of truncated polynomials C_k [t=(C[t)/(¿t^(k+1)¿), at degree k, for any positive integer k. Although, in its full generality, the classification of Lie bialgebras is a wild problem, we will concentrate in this Lie algebra as a fist step towards the study of the Lie bialgebra structures over current Lie algebras in general [11, with non semisimple Lie algebras as background. This choice was motivated by the relevance of current Lie algebras in the study of conformal field theories and vertex operator algebras.