On the decomposability of bilinear spaces of dimension four
Grégory Berhuy
TL;DR
The paper investigates decomposability for bilinear spaces of dimension four without symmetry and for split central simple algebras of degree four with an anti-automorphism. It develops a robust asymmetry framework and a correspondence between similarity classes and symmetric elements in the centralizer, revealing that decomposability is governed by different obstructions than in the symmetric case. A key result (Theorem thmadjoint) shows that, in the split case, (\mathscr{L}(V'), σ_{b'}) is decomposable exactly when b' has a decomposable asymmetry and trivial determinant, yielding a positive answer to Question 1 and a negative one for Question 3. The work also constructs a cohomological invariant \tilde{ν} detecting decomposability in the generic asymmetry case, while demonstrating fundamental limits: cohomological invariants in general do not suffice to detect decomposability across all fields and asymmetries, highlighting the nuanced interplay between determinants, asymmetries, and Galois cohomology in this setting.
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.