Bilinear maps and graphs

Let nu be a linear space of arbitrary dimension and over an arbitrary base field F, endowed with a bilinear map f : nu x nu -> nu. A basis B = {vi}(i is an element of I) of nu is an f-basis if for any i,j is an element of I we have that f (nu(i), nu(j)) E F nu(k) for some k is an element of I. We associate to any triplet (nu, f B) an adequate graph (V, E). By arguing on this graph we show that nu decomposes as a direct sum of strongly f-invariant linear subspaces, each one being associated to one connected component of (V, E). Also the B-semisimplicity and the B-simplicity of nu are characterized in terms of the associated graph. (C) 2018 Elsevier B.V. All rights reserved.