We construct a birational invariant for certain algebraic group actions. We use this invariant to classify linear representations of finite abelian groups up to birational equivalence, thus answering, in a special case, a question of E. B. Vinberg and giving a family of counterexamples to a related conjecture of P. I. Katsylo. We also give a new proof of a theorem of M. Lorenz on birational equivalence of quantum tori (in a slightly expanded form) by applying our invariant in the setting of PGL(n)-varieties. [References: 20]
展开▼
机译：我们为某些代数群动作构造一个双线性不变式。我们使用这个不变量对有限的阿贝尔群的线性表示进行分类，直到双边对等，从而在特殊情况下回答了E. B. Vinberg的问题，并为P. I. Katsylo的一个相关猜想提供了一系列反例。通过将不变式应用于PGL（n）变量的设置中，我们还给出了M. Lorenz关于量子托里双边对等价定理的新证明（略微扩展）。 [参考：20]
展开▼
