Finite-dimensional Lie subalgebras of the universal enveloping algebra of $\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$

In this work we provide the complete classification of all non-zero finite-dimensional complex Lie subalgebras of the universal enveloping algebra of the special linear Lie algebra $\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$. That is, we show that every such nilpotent Lie subalgebra is abelian and every non-solvable Lie subalgebra is reductive, with semisimple component isomorphic to $\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$ itself. Every non-nilpotent solvable Lie subalgebra is the direct sum of an abelian center and a member of the family of solvable Lie algebras $\mathfrak{r}(\mathit{d},\mathit{m})$, parameterized by a vector $\mathit{d}$ whose entries are strictly increasing positive integers that have greatest common divisor one together with a positive integer vector $\mathit{m}$ of the same length recording their multiplicities. Moreover, we discuss the corresponding classification in a real skew-hermitian setting and its relation to the realizations in the single- and two-mode Weyl algebras.