Catenarity in quantum algebras
In a celebrated but never-published work [Equidimensionalité de la variété caractéristique, Exposé de O. Gabber rédigé par T. Levasseur, Université de Paris VI (1982)] \textit{O. Gabber} proved that the enveloping algebra \(U\) of a finite dimensional complex solvable Lie algebra is catenary, meaning that all saturated chains of prime ideals between any two fixed primes \(P<Q\) of \(U\) have the same length. (Gabber's proof is included in [\textit{T. Levasseur} and \textit{J. T. Stafford}, Rings of differential operators on classical rings of invariants (Mem. Am. Math. Soc. 412, 1989; Zbl 0691.16019)].) The idea behind Gabber's theorem is that catenarity is a consequence of good homological and growth properties of the algebra. In the present paper the authors first state and prove an ``abstract'' version of Gabber's result, making precise the rough statement of the previous sentence; simultaneously, they derive an ``abstract'' version of Tauvel's height formula [\textit{P. Tauvel}, Bull. Soc. Math. Fr. 106, 177-205 (1978; Zbl 0399.17003)]. In the second part of the paper they show that the hypotheses required for these ``abstract'' results are satisfied by various classes of algebras associated with quantum groups.
finite dimensional complex solvable Lie algebras
scientific article; zbMATH DE number 920025
scientific article