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# Mathematical Research Letters

## Volume 7 (2000)

### Number 6

### Proof of the GGS Conjecture

Pages: 801 – 826

DOI: http://dx.doi.org/10.4310/MRL.2000.v7.n6.a12

#### Author

#### Abstract

We prove the GGS conjecture \cite{GGS} (1993), which gives a particularly simple explicit quantization of classical $r$-matrices for Lie algebras $\mathfrak{gl}(n)$, in terms of a matrix $R \in Mat_n({\mathbb C}) \o Mat_n({\mathbb C})$ which satisfies the quantum Yang-Baxter equation (QYBE) and the Hecke condition, whose quasiclassical limit is $r$. The $r$-matrices were classified by Belavin and Drinfeld in the 1980’s in terms of combinatorial objects known as Belavin-Drinfeld triples. We prove this conjecture by showing that the GGS matrix coincides with another quantization from \cite{ESS}, which is a more general construction. We do this by explicitly expanding the product from \cite{ESS} using detailed combinatorial analysis in terms of Belavin-Drinfeld triples.