Homogenized $\mathfrak {sl}(2)$

Abstract:

This note studies a special case of Artin’s projective geometry (Geometry of quantum planes, MIT, preprint, 1990) for noncommutative graded algebras. It is shown that (most of) the line modules over the homogenization of the enveloping algebra $U(\mathfrak {s}\mathfrak {l}(2,\mathbb {C}))$ are in bijection with the lines lying on the quadrics that are the (closures of the) conjugacy classes in $\mathfrak {s}\mathfrak {l}(2,\mathbb {C})$. Furthermore, these line modules are the homogenization of the Verma modules for $\mathfrak {s}\mathfrak {l}(2,\mathbb {C})$.