A basis for representations of symplectic Lie algebras

Research Report MRR98-012

A basis for representations of symplectic Lie algebras

Alexander Molev

Abstract: A basis for each finite-dimensional irreducible representation of the symplectic Lie algebra $\frak{sp}(2n)$ is constructed. The basis vectors are expressed in terms of the Mickelsson lowering operators. Explicit formulas for the matrix elements of generators of $\frak{sp}(2n)$ in this basis are given. The basis is natural from the viewpoint of the representation theory of the Yangians. The key role in the construction is played by the fact that the subspace of $\frak{sp}(2n-2)$ -highest vectors in any finite-dimensional irreducible representation of $\frak{sp}(2n)$ admits a natural structure of a representation of the Yangian $\text{Y}(\frak{gl}(2))$ .

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