Abstract
Let L denote a simple Lie algebra over the complex number field C with H a fixed Cartan subalgebra and C(L) the centralizer of H in the universal enveloping algebra U of L. It is known [cf. 2, 5] that one can construct from each algebra homomorphism ϕ:C(L) → C a unique algebraically irreducible representation of L which admits a weight space decomposition relative to H in which the weight space corresponding to ϕ ↓ H ∈ H* is one-dimensional. Conversely, if (ρ, V) is an algebraically irreducible representation of L admitting a one-dimensional weight space Vλ for some λ ∈ H*, then there exists a unique algebra homomorphism ϕ:C(L) → C which extends λ such that (ρ, V) is equivalent to the representation constructed from ϕ. Any such representation will be said to be pointed.
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