The Linear, Projective and Symplectic Groups

As formerly, whenever V is a variety, defined over a field k, we denote by V_k the set of points of V, rational over k; a vectorspace of dimension d over k can always be denoted by R_k, where R is an affine space of dimension d in the sense of algebraic geometry. In particular, any algebra over k can be so written; the obvious extension to R of the multiplication-law on the algebra R_k makes R into an algebra-variety, defined over k (which means that the multiplication-law on R is defined over k). The given algebra R_k over k is absolutely semisimple if and only if R is so, i.e. if and only if R is isomorphic (over the universal domain) to a direct sum of matrix algebras.