Polynomial ring representations of endomorphisms of exterior powers

Collectanea Mathematica - Tập 73 - Trang 107-133 - 2021
Ommolbanin Behzad1, André Contiero2, Letterio Gatto3, Renato Vidal Martins2
1Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
2Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
3Dipartimento di Scienze Matematiche, Politecnico di Torino, Turin, Italy

Tóm tắt

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.

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