Hom--Lie Algebras and Explicit MSS Partition Bounds for $(6,3)$ Biangular Frames

Abstract

We study the algebraic structure of $(6,3)$ biangular Parseval frames. The two-distance property yields adjacency matrices $A_1,A_2$ whose span forms a three-dimensional commutative algebra, and adjoining the commutator $[A_1,A_2]$ produces a three-dimensional Lie algebra $\g$. The Gram matrix $G = I + c_1 A_1 + c_2 A_2$ induces a derivation $α(X) = [G,X]$ on $\g$, equipping $(\g, [\cdot,\cdot], α)$ with a Hom--Lie algebra structure. We compute all structure constants explicitly in terms of the strongly regular graph parameters $(k_1,λ_1,μ_1,k_2,λ_2,μ_2)$ and the frame angles $(c_1,c_2)$, and use this framework to derive explicit bounds for the partial frame operators arising in Marcus--Spielman--Srivastava (MSS) partitions. These bounds depend directly on the structure constants and refine the universal MSS estimate in the regime of highly unbalanced partitions.