Unified Maxwell-Einstein and Yang-Mills-Einstein Supergravity Theories in Five Dimensions

Unified N=2 Maxwell-Einstein supergravity theories (MESGTs) are supergravity theories in which all the vector fields, including the graviphoton, transform in an irreducible representation of a simple global symmetry group of the Lagrangian. As was established long time ago, in five dimensions there exist only four unified Maxwell-Einstein supergravity theories whose target manifolds are symmetric spaces. These theories are defined by the four simple Euclidean Jordan algebras of degree three. In this paper, we show that, in addition to these four unified MESGTs with symmetric target spaces, there exist three infinite families of unified MESGTs as well as another exceptional one. These novel unified MESGTs are defined by non-compact (Minkowskian) Jordan algebras, and their target spaces are in general neither symmetric nor homogeneous. The members of one of these three infinite families can be gauged in such a way as to obtain an infinite family of unified N=2 Yang-Mills-Einstein supergravity theories, in which all vector fields transform in the adjoint representation of a simple gauge group of the type SU(N,1). The corresponding gaugings in the other two infinite families lead to Yang-Mills-Einstein supergravity theories coupled to tensor multiplets.Comment: Latex 2e, 28 pages. v2: reference added, footnote 14 enlarge

A series of algebras generalizing the octonions and Hurwitz-Radon identity

International audienceWe study non-associative twisted group algebras over (ℤ2)n with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion. We present two applications of the constructed algebras and the developed technique. The first application is a simple explicit formula for the following famous square identity: (a21+⋯+a2N)(b21+⋯+b2ρ(N))=c21+⋯+c2N , where c k are bilinear functions of the a i and b j and where ρ(N) is the Hurwitz-Radon function. The second application is the relation to Moufang loops and, in particular, to the code loops. To illustrate this relation, we provide an explicit coordinate formula for the factor set of the Parker loop

Alternative loop rings

For the past ten years, alternative loop rings have intrigued mathematicians from a wide cross-section of modern algebra. As a consequence, the theory of alternative loop rings has grown tremendously. One of the main developments is the complete characterization of loops which have an alternative but not associative, loop ring. Furthermore, there is a very close relationship between the algebraic structures of loop rings and of group rings over 2-groups. Another major topic of research is the study of the unit loop of the integral loop ring. Here the interaction between loop rings and group r