Noether bound for invariants in relatively free algebras

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• 作者：Má ; tyá ; s Domokos^a ; ^{domokos.matyas@renyi.mta.hu" class="auth_mail" title="E-mail the corresponding author} ; Vesselin Drensky^b ; ^{drensky@math.bas.bg" class="auth_mail" title="E-mail the corresponding author}
• 关键词：16R10 ; 13A50 ; 15A72 ; 16W22
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 October 2016
• 年：2016
• 卷：463
• 期：Complete
• 页码：152-167
• 全文大小：395 K

文摘

Let R$\mathfrak{R}$ be a weakly noetherian variety of unitary associative algebras (over a field K   of characteristic 0), i.e., every finitely generated algebra from R$\mathfrak{R}$ satisfies the ascending chain condition for two-sided ideals. For a finite group G and a d-dimensional G-module V   denote by F(R,V)$F(\mathfrak{R},V)$ the relatively free algebra in R$\mathfrak{R}$ of rank d freely generated by the vector space V  . It is proved that the subalgebra F(R,V)^G$F{(\mathfrak{R},V)}^G$ of G  -invariants is generated by elements of degree at most b(R,G)$b(\mathfrak{R},G)$ for some explicitly given number b(R,G)$b(\mathfrak{R},G)$ depending only on the variety R$\mathfrak{R}$ and the group G (but not on V  ). This generalizes the classical result of Emmy Noether stating that the algebra of commutative polynomial invariants K[V]^G$K{[V]}^G$ is generated by invariants of degree at most |G|$|G|$.