On unipotent radicals of motivic Galois groups

Let T be a neutral Tannakian category over a field of characteristic zero with unit object 1, and equipped with a filtration W • similar to the weight filtration on mixed motives.Let M be an object of T , and u(M) ⊂ W -1 Hom(M, M) the Lie algebra of the kernel of the natural surjection from the fundamental group of M to the fundamental group of Gr W M. A result of Deligne gives a characterization of u(M) in terms of the extensions 0 → W p M → M → M/W p M → 0: it states that u(M) is the smallest subobject of W -1 Hom(M, M) such that the sum of the aforementioned extensions, considered as extensions of 1 by W -1 Hom(M, M), is the pushforward of an extension of 1 by u(M).We study each of the abovementioned extensions individually in relation to u(M).Among other things, we obtain a refinement of Deligne's result, where we give a sufficient condition for when an individual extension 0 → W p M → M → M/W p M → 0 is the pushforward of an extension of 1 by u(M).In the second half of the paper, we give an application to mixed motives whose unipotent radical of the motivic Galois group is as large as possible (i.e., with u(M) = W -1 Hom(M, M)).Using Grothendieck's formalism of extensions panachées we prove a classification result for such motives.Specializing to the category of mixed Tate motives we obtain a classification result for 3-dimensional mixed Tate motives over ‫ޑ‬ with three weights and large unipotent radicals.