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Given an arbitrary field $\mathbb{F}$ of characteristic 0, we study Liebialgebra structures on $sl(n,\mathbb{F})$, based on the description of thecorresponding classical double. For any Lie bialgebra structure $\delta$, theclassical double $D(sl(n,\mathbb{F}),\delta)$ is isomorphic to$sl(n,\mathbb{F})\otimes_{\mathbb{F}} A$, where $A$ is either$\mathbb{F}[\varepsilon]$, with $\varepsilon^{2}=0$, or $\mathbb{F}\oplus\mathbb{F}$ or a quadratic field extension of $\mathbb{F}$. In the first case,the classification leads to quasi-Frobenius Lie subalgebras of$sl(n,\mathbb{F})$. In the second and third cases, a Belavin--Drinfeldcohomology can be introduced which enables one to classify Lie bialgebras on$sl(n,\mathbb{F})$, up to gauge equivalence. The Belavin--Drinfeld untwistedand twisted cohomology sets associated to an $r$-matrix are computed. For theCremmer--Gervais $r$-matrix in $sl(3)$, we also construct a natural map of setsbetween the total Belavin--Drinfeld twisted cohomology set and the Brauer groupof the field $\mathbb{F}$.

Classification of quantum groups and Lie bialgebra structures on $sl(n,\mathbb{F})$. Relations with Brauer group