The structure of symmetric tensor powers of composition algebras.

Let be a composition algebra which is either the Hamilton quaternion algebra ℍ or the Cayley octonion algebra over ℝ. In a previous work, the nth symmetric power Sym n of is shown to be a direct sum of central simple algebras, corresponding to the partitions of n of length 2 , such that the component corresponding to the partition (m , n − m) is isomorphic to the component T 2 m − n of Sym 2 m − n corresponding to the partition (2 m − n , 0) of 2 m − n. In this work, we study the building blocks T n of these decompositions. We show that the "local" structure of , i.e. the complex-like subfields of , determine both the complement of T n in Sym n and the trace map of T n , induced from the trace map of . We also derive a recursive trace formula on the T n 's. We use the "local-global" results to define positive definite symmetric bilinear forms on the vector space ⊕ n = 0 ∞ T n , which has a natural structure of a commutative and associative algebra. Finally, the structure of the central simple algebra T n is described. [ABSTRACT FROM AUTHOR]