Irreducible representations of 𝐴_{𝑛} with a 1-dimensional weight space

Britten, D. J.; Lemire, F. W.

doi:10.1090/S0002-9947-1982-0667158-4

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

Authors: D. J. Britten and F. W. Lemire
Journal: Trans. Amer. Math. Soc. 273 (1982), 509-540
MSC: Primary 17B10
DOI: https://doi.org/10.1090/S0002-9947-1982-0667158-4
MathSciNet review: 667158
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Abstract: In this paper we classify all irreducible linear representations of the simple Lie algebra ${A_n}$ which admit a one-dimensional weight space with respect to some Cartan subalgebra of ${A_n}$ . We first show that the problem is equivalent to determining all algebra homomorphisms from the centralizer of the Cartan subalgebra in the universal enveloping algebra of ${A_n}$ to the base field. We construct all such algebra homomorphisms and provide conditions under which two such algebra homomorphisms provide inequivalent irreducible representations of ${A_n}$ .

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