REMARKS ON THE EXTENSION RELATION AND APP LICATIONS TO THE STUDY OF THE NOTION OF BIALGEBRAS

Let A and B two algebras; a large class of linear operators from A to B is formed by the set of linear operators which possess the splitting property: x is-an-element-of L(s) (A; B) if and only if there exists [GRAPHICS] such that [GRAPHICS] for-all u, v is-an-element-of A; this defines the extension relation DELTA(r), which presents the interest to be constructive, in particular through the notion of elementary fam ily which is studied in the case of associative algebras. When this re lation is considered for operators from A to A, some subalgebras k sub set-of L(s) (A; A) are of interest: a subalgebra K of L(s) (A; A) is c alled splitted by A, if there exists an homomorphism DELTA : K --> K x K such that p o DELTA (x) = x o p where p is the product from A x A to A. We show how these notions naturally occured in the study of th e notion of bialgebras and we illustrate them by the construction of t he left action of U(q) (Sl(2)) on Sl(q) (2).