Series formelles croisees

Abstract 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 K M [[ G , σ , α ]] of those series is still a division ring and (with some additional assumptions) that the rational ones s = ∑ gϵG s ( g ) g verify: If all the “monomials” s ( g ) g are in a same subdivision ring Δ of K M [[ 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.