Smith Algebra from $\mathfrak{sl}_2$ Tensor Product Branching via the Mickelsson-Zhelobenko Algebra
Let $L_n$ be the finite-dimensional irreducible $\mathfrak{sl}_2$-module of highest weight $n$. We study the space of highest-weight vectors in $L_n\otimes L_m$ for the diagonal $\mathfrak{sl}_2$-action. The extremal projector constructs these singular vectors and gives operators connecting the highest-weight lines in the Clebsch-Gordan decomposition. We interpret these operators in the Mickelsson-Zhelobenko algebra of the diagonal pair $(\mathfrak{sl}_2\oplus\mathfrak{sl}_2, \mathfrak{sl}_2^{\mathrm{diag}})$. A rescaling of their quadratic relation yields the cubic Smith relation $[E,F]=2(H^3-cH)$. This cubic algebra is also known as a Higgs algebra in the physics literature. We express both the central parameter $c$ and the Smith Casimir in terms of the quadratic $\mathfrak{sl}_2$ Casimir elements of the two tensor factors. The singular subspace becomes an irreducible Smith module of dimension $\min\{n,m\}+1$, with explicit formulas for the actions of the raising and lowering operators $E$ and $F$. Finally, comparison with finite-dimensional simple Smith modules gives a necessary and sufficient condition for a module to arise from this tensor-product construction.