COMPATIBLE ALGEBRA STRUCTURES OF LIE ALGEBRAS

A compatible algebra products of a finite-dimensional semisimple Lie algebra g with a Lie bracket [-, -] are studied. If g is simple of type A(1) or not of type A(n) of n >= 2, then the compatible algebra products must be the scalar multiples of the Lie bracket [-, -]. In case that g is simple of type A(n) of n >= 2, such a product is a sum of a scalar multiple of [-, -] and a deformed one of the ordinal associative products on the full (n + 1) x (n + 1) matix algebra. Then we give a alternative proof to the triviality of the compatible associative algebra structures of a semisimple Lie algebra g.