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arXiv · 2609.00215

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

Abraham Orinda·Geoffrey Mboya·Arvince Ogendi
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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 $\alpha(X) = [G,X]$ on $\g$, equipping $(\g, [\cdot,\cdot], \alpha)$ with a Hom--Lie algebra structure. We compute all structure constants explicitly in terms of the strongly regular graph parameters $(k_1,\lambda_1,\mu_1,k_2,\lambda_2,\mu_2)$ 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.

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Abraham Orinda, Geoffrey Mboya, Arvince Ogendi. 2026-08-31. Hom--Lie Algebras and Explicit MSS Partition Bounds for $(6,3)$ Biangular Frames. https://arxiv.org/abs/2609.00215

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