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 A₁ 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) × (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.