Induced modules and central character quotients for Takiff $\mathfrak{sl}_{2}$

Abstract

We construct a large new family of simple modules over Takiff $\mathfrak{sl}_{2}$. We prove that the quotient of the universal enveloping algebra of the Takiff Lie algebra for $\mathfrak{sl}_{2}$ 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 $\mathfrak{sl}_{2}$-modules, we expect that the above ideals exhaust all primitive ideal.

Theorems & Definitions (16)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Theorem 3.3