Symmetric bilinear forms, superalgebras and integer matrix factorization

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 $n\ge 2$). 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.