Abstract
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.
Highlights
1.1 Statement of the main resultThe purpose of this paper is to supply an explicit description of the polynomial ringBr ∶= Q[e1, ... , er] as a module⨁over the Lie algebra of endomorphisms of k-th exterior powers of a vector space V ∶= i≥0 Q ⋅ bi of infinite countable dimension
If r is big with respect to the length of the partition they can be explicitly written as Polynomial ring representations of endomorphisms of exterior
The case r = k = 1 recovers the well known general fact that every vector space is a module over the Lie algebra of its own endomorphisms
Summary
The purpose of this paper is to supply an explicit description of the polynomial ring. Er] as a module⨁over the Lie algebra of endomorphisms of k-th exterior powers of a vector space V ∶= i≥0 Q ⋅ bi of infinite countable dimension. Vgl→(⋀⋀k Vr−)-kaVctiios nth(e1s)taonndBarrdthcroonutrgahctaiocnoomppearacttofro(rSmeuclt.a,2.a2)s.tandard philosophy suggests to use generating functions To this purpose, let us introduce some notation. If r is big with respect to the length of the partition they can be explicitly written as Polynomial ring representations of endomorphisms of exterior. The case r = k = 1 recovers the well known general fact that every vector space is a module over the Lie algebra of its own endomorphisms. The linear extension of the set map ei1 ↦ bi is a vector space isomorphism B1 → V , making B1 into a gl(V)module, by pulling back that structure from V. Our Main Theorem takes into account the general case
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