Hauteurs de sous-espaces sur les corps non commutatifs

We study heights of subspaces of D N where D is a finite-dimensional rational division algebra and N a positive integer. We define them in terms of volumes of Euclidean lattices by extending a formula of W. Schmidt so that we recover the classical height if D is commutative. We review basic properties, prove a Siegel Lemma over D, a duality theorem and a new formula for the degree of certain abelian varieties. We further give matrix versions and compare our notion with the height defined through algebraic groups by J. Franke, Y. Manin and Y. Tschinkel