Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The discriminant Pfister form of an algebra with involution of capacity four

Karim Johannes Becher, Nicolas Grenier-Boley, Jean-Pierre Tignol

Abstract

To an orthogonal or unitary involution on a central simple algebra of degree 4, or to a symplectic involution on a central simple algebra of degree 8, we associate a Pfister form that characterises the decomposability of the algebra with involution. In this way we obtain a unified approach to known decomposability criteria for several cases, and a new result for symplectic involutions on degree $8$ algebras in characteristic 2.

The discriminant Pfister form of an algebra with involution of capacity four

Abstract

To an orthogonal or unitary involution on a central simple algebra of degree 4, or to a symplectic involution on a central simple algebra of degree 8, we associate a Pfister form that characterises the decomposability of the algebra with involution. In this way we obtain a unified approach to known decomposability criteria for several cases, and a new result for symplectic involutions on degree algebras in characteristic 2.

Paper Structure

This paper contains 8 sections, 32 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Algebras with involution
  3. Capacity
  4. Decomposability
  5. Hyperbolic involutions and quaternion factors
  6. Hyperbolicity in capacity four
  7. Construction of the discriminant from a biquadratic algebra
  8. Determination of the discriminant Pfister form

Key Result

Proposition 2.2

Set $d=\mathsf{deg} A$, hence $\mathsf{dim}_F A=[\mathop{\mathrm{\mathsf{Z}}}\nolimits(A):F]\cdot d^2$. If $1\in\mathop{\mathrm{\mathsf{Symd}}}\nolimits(\sigma)$, then If $1\notin\mathop{\mathrm{\mathsf{Symd}}}\nolimits(\sigma)$, then $\mathsf{dim}_F\mathop{\mathrm{\mathsf{Symd}}}\nolimits(\sigma)=\frac{d(d-1)}{2}$.

Theorems & Definitions (37)

  • Proposition 2.2
  • Remark 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 4.1
  • Corollary 4.2
  • Corollary 4.3
  • Proposition 4.4
  • Proposition 5.1
  • ...and 27 more