Series formelles croisees Original Research Article

In this paper, we generalise the well-known notion of Malcev-Neumann series with support in an ordered group G and coefficients in a field K (Neumann, 1949) to the notion of crossed Malcev-Neumann series associated to a morphism σ : G → Aut(K) and a 2-cocycle α. We first prove that the ring KM[[G, σ, α]] of those series is still a division ring and (with some additional assumptions) that the rational ones s = ∑gεGs(g)g verify: If all the “monomials” s(g)g are in a same subdivision ring Δ of KM[[G, σ, α]], then so does s itself. We then use those results to compute some centralisers in division rings of fractions of skew polynomial rings in several variables and quantum spaces.