#### Z. Reichstein, B. Youssin, * A birational
invariant for algebraic
group actions*, Pacific Journal of Math., 204, no. 1 (2002),
223--246.

**Abstract:**
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.

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