COMPATIBLE ALGEBRA STRUCTURES OF LIEALGEBRAS

摘要

A compatible algebra products of a finite-dimensional semisimple Lie algebrag with a Lie bracket [-,-] are studied. If g is simple of type A1 or not oftype A_nof n2, then the compatible algebra products must be the scalarmultiples of the Lie bracket [-, In case that g is simple of type An of n 2,such a product is a sum of a scalar multiple of [-, -] and a deformed one ofthe 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 associativealgebra structures of a semisimple Lie algebra g.