G-分次代数的可表示性与Specht问题
Representability and Specht problem for G-graded algebras
摘要：
设W是特征为零的域F上的结合PI代数，被有限群G分次.设id_{G}（W）表示W的G-分次恒等式的T-理想。我们证明：1. {[G-分次PI等价]}存在F的域扩张K和K上的有限维Z/2 Z × G-分次代数A使得id_{G}（W）= id_{G}（A^{*}），其中A^{*}是A的Grassmann包络. 2. {[G-分次Specht问题]} T-理想id_{G}（W）是T-理想的逆生成。3. {[仿射代数的G-分次PI-等价]}设W是F上的G-分次仿射代数。则存在F的域扩张K和K上的有限维代数A使得id_{G}（W）= id_{G}（A）.
英文摘要：
Let W be an associative PI algebra over a field F of characteristic zero, graded by a finite group G. Let id_{G}(W) denote the T-ideal of G-graded identities of W. We prove: 1. {[G-graded PI equivalence]} There exists a field extension K of F and a finite dimensional Z/2ZxG-graded algebra A over K such that id_{G}(W)=id_{G}(A^{*}) where A^{*} is the Grassmann envelope of A. 2. {[G-graded Specht problem]} The T-ideal id_{G}(W) is finitely generated as a T-ideal. 3. {[G-graded PI-equivalence for affine algebras]} Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite dimensional algebra A over K such that id_{G}(W)=id_{G}(A).
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