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

Abstract

Given an arbitrary field $\mathbb{F}$ of characteristic 0, we study Lie bialgebra structures on $sl(n,\mathbb{F})$, based on the description of the corresponding classical double. For any Lie bialgebra structure $\delta$, the classical double $D(sl(n,\mathbb{F}),\delta )$ is isomorphic to $sl(n,\mathbb{F}){\otimes }_{\mathbb{F}}A$, where $A$ is either $\mathbb{F}[\epsilon ]$, with ${\epsilon }^2=0$, 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--Drinfeld cohomology can be introduced which enables one to classify Lie bialgebras on $sl(n,\mathbb{F})$, up to gauge equivalence. The Belavin--Drinfeld untwisted and twisted cohomology sets associated to an $r$-matrix are computed. For the Cremmer--Gervais $r$-matrix in $sl(3)$, we also construct a natural map of sets between the total Belavin--Drinfeld twisted cohomology set and the Brauer group of the field $\mathbb{F}$.

arXiv admin note: text overlap with arXiv:1303.4046, arXiv:1309.7133

Cited by in corpus (2)

• Quantum groups: from Kulish-Reshetikhin discovery to classification
• Belavin-Drinfeld quantum groups and Lie bialgebras: Galois cohomology considerations