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Symmetric bilinear forms, superalgebras and integer matrix factorization

Dan Fretwell, Jenny Roberts

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.

Symmetric bilinear forms, superalgebras and integer matrix factorization

Abstract

We construct and investigate certain (unbalanced) superalgebra structures on , with a field of characteristic and a finite dimensional -vector space (of dimension ). These structures are induced by a choice of non-degenerate symmetric bilinear form on and a choice of non-zero base vector . After exploring the construction further, we apply our results to certain questions concerning integer matrix factorization and isometry of integral lattices.
Paper Structure (7 sections, 11 theorems, 69 equations)

This paper contains 7 sections, 11 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Symmetric bilinear forms and superalgebras
  3. The subspace $E^{(0)}(V)$
  4. The subspace $E^{(1)}(B,w)$
  5. The endomorphism superalgebra induced by $B$.
  6. Further results
  7. Application to integer matrix factorization

Key Result

Lemma 2.1

$\phi \in E^{(0)}(B,w)$ if and only if there exists $\lambda\in K$ such that

Theorems & Definitions (31)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 21 more