Pfister Involutions

The question of the existence of an analogue, in the framework of central simple algebras with involution, of the notion of Pfister form is raised. In particular, algebras with orthogonal involution which split as a tensor product of quaternion algebras with involution are studied. It is proven that, up to degree 16, over any extension over which the algebra splits, the involution is adjoint to a Pfister form. Moreover, coho-mological invariants of those algebras with involution are discussed. An involution on a central simple algebra is nothing but a twisted form of a symmetric or alternating bilinear form up to a scalar factor ([KMRT98], ch. 1). Hence the theory of central simple algebras with involution naturally appears as an extension of the theory of quadratic forms, which is an important source of inspiration for this subject. We do not have, for algebras with involution, such a nice algebraic theory as for quadratic forms, since orthogonal sums are not always defined, and are not unique when defined [Dej95]. Nevertheless, in view of the fundamental role played by Pfister forms in the theory of quadratic forms, and also of the nice properties they share, it seems natural to try and find out whether an analogous notion exists in the setting of algebras with involution. The main purpose of this paper is to raise this question, which was originally posed by David Tao [Tao]; this is done in §2. In particular, this leads to the consideration of algebras with orthogonal involution which split as a tensor product of r quaternion algebras with involution. One central question is then the following: consider such a product of quaternions with involution, and assume the algebra is split. Is the corresponding involu-tion adjoint to a Pfister form? The answer is positive up to r = 5. A survey of this question is given in §2.4. In §4, we give a direct proof of this fact for r = 4. Before that, we study in §3, the existence of cohomological invariants for some of the algebras with involution which can naturally be considered as generalisations of Pfister quadratic forms.