Seminarium KMMF "Teoria Dwoistości"

A \textit{Lie bialgebra} is a pair $(\mathfrak{g}, \delta)$, where $\mathfrak{g}$ is a Lie algebra and $\delta: \mathfrak{g} \to \mathfrak{g} \wedge \mathfrak{g}$ is a map, a so-called \textit{cocommutator}, that is closed relative to the Chevalley--Eilenberg cohomology of $(\mathfrak{g}\wedge\mathfrak{g})$-valued forms and whose transpose induces a Lie algebra structure on $\mathfrak{g}^*$. If $\delta(\cdot) = [\cdot, r]_{S}$ for a bivector $r \in \mathfrak{g} \wedge \mathfrak{g}$ and $[\cdot, \cdot]_{S}$ is the Schouten-Nijenhuis bracket, the Lie bialgebra $(\mathfrak{g},\delta)$ is called \textit{coboundary}. The classification of coboundary Lie bialgebras is carried out generally through {\it ad-hoc} methods to solve the modified Yang--Baxter equations determining all possible $r$. In this talk I will present several much more unifying approaches to classifying three-dimensional real coboundary Lie bialgebras by extending Lie algebra theory techniques to Grassmann algebras. To illustrate our techniques, several examples will be discussed.