Associating geometry to the Lie superalgebra $\mathfrak {sl}(1|1)$ and to the color Lie algebra $\mathfrak {sl}^c_2(\Bbbk )$

Abstract:

In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra. In particular, in the case of the Lie algebra $\mathfrak {sl}_2(\mathbb {C})$, there is a correspondence between Verma modules and certain line modules that associates a pair $(\mathfrak {h}, \phi )$, where $\mathfrak {h}$ is a $2$-dimensional Lie subalgebra of $\mathfrak {sl}_2(\mathbb {C})$ and $\phi \in \mathfrak {h}^*$ satisfies $\phi ([\mathfrak {h}, \mathfrak {h}]) = 0$, to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra $\mathfrak {sl}(1|1)$ and for a color Lie algebra associated to the Lie algebra $\mathfrak {sl}_2$.