The Gelfand-Kirillov conjecture for Lie algebras of dimension at most eight
In their seminal paper \textit{I. M. Gelfand} and \textit{A. A. Kirillov} [Publ. Math., Inst. Hautes Étud. Sci. 31, 509-523 (1966; Zbl 0144.02104)] made the following conjecture (GK-conjecture): Let \(L\) be a finite dimensional algebraic Lie algebra over an algebraically closed field \(k\) of characteristic zero, let \(U(L)\) be its enveloping algebra, and let \(D(L)\) be its field of fractions; then \(D(L)\) is isomorphic to the skew field of fractions \(D_n(F)\) of the Weyl algebra over a purely transcendental extension \(F\) of \(k\). In that paper the conjecture was proved for \(gl(n)\), \(sl(n)\), and nilpotent Lie algebras. Now the GK-conjecture is proved for the solvable case, semisimple Lie algebras and some other cases. Recently, a family of counterexamples to the GK-conjecture was constructed [\textit{J. Alev, A. Ooms} and \textit{M. Van den Bergh}, Trans. Am. Math. Soc. 348, 1709-1716 (1996; Zbl 0853.17006)]. However, in these cases one recovers the GK-conjecture after an appropriate extension of the center. The smallest obtained counterexample to the GK-conjecture is a 9-dimensional semi-direct product of \(sl(2)\) with two copies of the adjoint representation. The main result of the paper is as follows: Theorem: Let \(L\) be an algebraic Lie algebra over an algebraically closed field \(k\) of characteristic zero with \(\dim L\leq 8\). Then \(L\) satisfies the Gelfand-Kirillov conjecture.
scientific article; zbMATH DE number 1471032
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