Specht type problems and Gelfand Conjecture

Belov, A.J.

Автор: Belov A.J.
Сборник: Международная конференция МАЛЬЦЕВСКИЕ ЧТЕНИЯ 13–17 ноября 2023 г
Тезисы
Год издания: 2023
Издательство: Изд-во СО РАН
Местоположение издательства: Новосибирск
Первая страница: 198
Последняя страница: 198
Аннотация: This is a joint work with Anton Khoroshkin, Ivan Vorobiev and Lucio Centrone.Let Wn be the Lie algebra of formal vector fields on the n-dimensional plane and byWpolnthe Lie algebra of polynomial vector fields on the plane. The Lie subalgebra of linearvector fields {xi∂∂xj} is denoted by gln. We have the following decomposition of gln modules:Wn ∼Y∞k=0SkV ⊗ V∗; Wpoln ∼ ⊕∞k=0SkV ⊗ V∗The subalgebras of finite codimension ⊕∞k=dSkV ⊗V∗ are called the basic subalgebras and aredenoted by Ld(n). With each irreducible gln-representation Vλ one associates the irreducibleL0(n)-module Vλ with the trivial action of subalgebra L1(n) ⊂ L0(n). The induced andcoinduced modulesTλ := IndWnL0(n)Vλ = U(Wpoln) ⊗U(L0(n)) Vλ ≃ ⊕∞k=0SkV ⊗ Vλ,T∗λ:= CoIndWnL0(n)Vλ = HomU(L0(n))(U(Wn), Vλ)of the Lie algebras Wpoln and Wn correspondingly are called the modules of tensor fields.Conjecture. (I.M.Gelfand) For all k and n the continious cohomology Hk(Wn; T∗λ) aswell as the Lie algebra homology Hk(Wpoln, Tλ) are finite-dimensional for all k and λ.Note that thanks to the Chevalley–Eilenberg complex C•(g; M) = Hom(Λkg; M) thatcomputes the Lie algebra cohomology, it is enough to prove the followingTheorem 1. For any collection of weights λ1, . . . , λr the tensor product of Wn-modulesTλ1 ⊗ . . . Tλris a Noetherian L0(n)-module.This theorem can be deduced from the followingTheorem 2. With polynomials in two variables over field of zero characteristics, onecan do the following operations Suppose there are or have already been derived polynomialsP1, P2. Then the following polynomials are output:λP1, P1 + P2; P1(R(x1), . . . , R(xn)),where R is an arbitrary polynomial in one variable, independent on i.Then any system of polynomials is deduced from a finite subsystem.Remark. Same question for positive characteristics fails if n > 1.Moscow State University, Moscow (Russia)
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