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In this paper we classify all irreducible linear representations of the simple Lie algebra An which admit a one-dimensional weight space with respect to some Cartan subalgebra H of A". We first show that the problem is equivalent to determining all algebra homomorphisms from the centralizer of the Cartan subalgebra H in the universal enveloping algebra of An to the base field. We construct all such algebra homomorphisms and provide conditions under which two such algebra homomorphisms provide inequivalent irreducible representations of An.

Irreducible representations of $A\sb{n}$ with a $1$-dimensional weight space