Proof of the GGS Conjecture

Details Proof of the GGS Conjecture Proof of the GGS Conjecture

2000-09-19

arxiv.org/abs/math/0009173v2

Mathematics

Quantum Algebra

AMSLaTeX; uses mrlart2e.cls (included-- MRL's document class, based on amsart)

Scientific paper

We prove the GGS conjecture (1993), due to Gerstenhaber, Giaquinto, and Schack, which gives a particularly simple explicit quantization of classical r-matrices for Lie algebras gl(n) in terms of an element R satisfying the quantum Yang-Baxter equation and the Hecke condition. The r-matrices were classified by Belavin and Drinfeld in the 1980s in terms of combinatorial objects known as Belavin-Drinfeld triples. We prove this conjecture by showing that the GGS matrix coincides with another quantization due to Etingof, Schiffmann, and the author, which is a more general construction. We do this by explicitly expanding the product from the aforementioned paper using detailed combinatorial analysis in terms of Belavin-Drinfeld triples.

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