On the structure tensor of $\mathfrak{sl}_n$

The structure tensor of $\mathfrak{sl}_n$, denoted $T_{\mathfrak{sl}_n}$, is the tensor arising from the Lie bracket bilinear operation on the set of traceless $n \times n$ matrices over $\mathbb{C}$. This tensor is intimately related to the well studied matrix multiplication tensor. Studying the structure tensor of $\mathfrak{sl}_n$ may provide further insight into the complexity of matrix multiplication and the "hay in a haystack" problem of finding explicit sequences tensors with high rank or border rank. We aim to find new bounds on the rank and border rank of this structure tensor in the case of $\mathfrak{sl}_3$ and $\mathfrak{sl}_4$. We additionally provide bounds in the case of the lie algebras $\mathfrak{so}_4$ and $\mathfrak{so}_5$. The lower bounds on the border ranks were obtained via various recent techniques, namely Koszul flattenings, border substitution, and border apolarity. Upper bounds on the rank of $T_{\mathfrak{sl}_3}$ are obtained via numerical methods that allowed us to find an explicit rank decomposition.