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

Abstract

The structure tensor of sl n , denoted T sln , is the tensor arising from the Lie bracket bilinear operation on the set of traceless n × n matrices over C. This tensor is intimately related to the well studied matrix multiplication tensor. Studying the structure tensor of 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 sl 3 and sl 4 . We additionally provide bounds in the case of the lie algebras so 4 and 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 sl3 are obtained via numerical methods that allowed us to find an explicit rank decomposition.

Two classes of minimal generic fundamental invariants for tensors

“…In particular, let T sl n denote the structure tensor of sl n . From [5], we know that T sl n ∈ C n 2 −1 ⊗ C n 2 −1 ⊗ C n 2 −1 whose minimal generic fundamental invariant is given in Section 3. 7.2.…”

“…Moreover, we get that δ(n 2 −1) = √ n 2 − 1 = n, which partially answered Problem 1.2 above. Just like n, n, n ∈ ⊗ 3 C n 2 , some interesting tensors also live in ⊗ 3 C n 2 −1 , for example, the structure tensor of sl n studied in [5].…”

“…• det e (6) 1 , e 6) 2 , e (6) 3 , e (6) 4 , e (6) 5 , e (6) 6 =(−1) 3×inv(1,2,3,4)+2×inv(1,4,2,5,3,6)+2×inv (1,2,3,4,5,6) =1.…”