Induced modules and central character quotients for Takiff ${\mathfrak{s}\mathfrak{l}}_2$

We construct a large new family of simple modules over Takiff ${\mathfrak{s}\mathfrak{l}}_2$. We prove that the quotient of the universal enveloping algebra of the Takiff Lie algebra for ${\mathfrak{s}\mathfrak{l}}_2$ by the ideal generated by a non-trivial central character is a simple algebra. In the case of the trivial central character, we show that the corresponding ideal is primitive by explicitly constructing a simple module whose annihilator coincides with that ideal. Together with the annihilators of simple ${\mathfrak{s}\mathfrak{l}}_2$-modules, we expect that the above ideals exhaust all primitive ideal.