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 A1,A2 whose span forms a three-dimensional commutative algebra, and adjoining the commutator [A1,A2] produces a three-dimensional Lie algebra g. The Gram matrix G = I + c1 A1 + c2 A2 induces a derivation α(X) = [G,X] on g, equipping (g, [·,·], α) with a Hom--Lie algebra structure. We compute all structure constants explicitly in terms of the strongly regular graph parameters (k1,λ1,μ1,k2,λ2,μ2) and the frame angles (c1,c2), 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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BibTeX
@misc{orinda2026homlie,
title = {{Hom--Lie Algebras and Explicit MSS Partition Bounds for $(6,3)$ Biangular Frames}},
author = {Abraham Orinda and Geoffrey Mboya and Arvince Ogendi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00215},
eprint = {2609.00215},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00215},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hom--Lie Algebras and Explicit MSS Partition Bounds for (6,3) Biangular Frames
AU - Abraham Orinda
AU - Geoffrey Mboya
AU - Arvince Ogendi
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00215
UR - https://arxiv.org/abs/2609.00215
AB - 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, [·,·], 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.
ER -