Symmetric bilinear forms, superalgebras and integer matrix factorization

TL;DR

This paper explores superalgebra structures induced by symmetric bilinear forms on vector spaces and applies these findings to problems in integer matrix factorization and lattice isometry.

Contribution

It introduces new superalgebra structures on endomorphism algebras based on symmetric bilinear forms and applies these to integer matrix factorization and lattice isometry problems.

Findings

Constructed superalgebra structures on End(V) using symmetric bilinear forms.

Applied superalgebra theory to integer matrix factorization.

Connected superalgebra structures to isometry of integral lattices.

Abstract

We construct and investigate certain (unbalanced) superalgebra structures on $\text{End}_K(V)$, with $K$ a field of characteristic $0$ and $V$ a finite dimensional $K$-vector space (of dimension $n\geq 2$). These structures are induced by a choice of non-degenerate symmetric bilinear form $B$ on $V$ and a choice of non-zero base vector $w\in V$. After exploring the construction further, we apply our results to certain questions concerning integer matrix factorization and isometry of integral lattices.