On the decomposability of bilinear spaces of dimension four

TL;DR

This paper investigates the decomposability of four-dimensional bilinear spaces and related algebras, revealing key differences from symmetric cases and the limitations of cohomological invariants.

Contribution

It demonstrates that decomposability problems differ between non-symmetric bilinear spaces and split central simple algebras, and shows cohomological invariants are insufficient for detection.

Findings

Decomposability problems are not equivalent for non-symmetric bilinear spaces and degree four algebras.

Cohomological invariants do not detect decomposability in general.

Determinant detects decomposability for split central simple algebras of degree four.

Abstract

In this paper, we study the problem of decomposability of bilinear spaces of dimension four without symmetry, as well as the problem of decomposability of split central simple algebras of degree four with an anti-automorphism. In particular, we show that, contrary to the case of symmetric or skew-symmetric bilinear spaces, these two problems are not equivalent. We will also prove that cohomological invariants do not detect decomposability of bilinear spaces of dimension four in general, whereas the determinant does for split central simple algebras of degree four with an anti-automorphism.