An explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

中文翻译：

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外部幂同态的多项式环表示

给出了r中有理多项式环的明确描述，作为可数无限维向量空间的k次外幂的内同态的Lie代数的表示形式。我们的描述基于Laksov和Throup关于多项式环的外部幂的对称结构的结果。我们的结果基于在著名的玻色子顶点表示中出现的顶点算子的近似版本，这归因于Date，Jimbo，Kashiwara和Miwa，这是无限大小的所有矩阵的Lie代数，其所有项均为零，但数量有限。