The algebraic groups leading to the Roth inequalities

Connectedness of the Tannakian group attached to semi-stable multiple filtrations

“…When L is a separable closure of the base field K and M is infinite, we have shown in the appendix of our previous paper [6] that any affine algebraic group scheme a dense subgroup of which is generated by tori defined over K appears (up to isomorphism) as a quotient of Aut v ss 0 (KY LY M). Applying the method of [6] to tori defined over L, we observe that any affine algebraic group which fills the necessity in Corollary 3.2 really appears (up to isomorphism) as a quotient of Aut v ss 0 (KY LY M). In this way, Corollary 3.2 presents a kind of characterization for linear algebraic groups which can be isomorphic to quotients of Aut v ss 0 (KY LY M).…”

“…The group S is defined over an arbitrary field K, solvable, and generated by two split tori. According to [6,Theorem A.14], the solvable group S appears (up to isomorphism) as a quotient of Aut v ss 0 (KY LY M) provided at least the cardinality of the index set M is greater than three. Thus the affine group scheme Aut v ss 0 (KY LY M) is not pro-reductive and the Tannakian category g ss 0 (KY LY M) is not poly-stable in that case.…”

“…In our previous paper [6], we have particularly shown that any connected reductive group over K appears (up to isomorphism) in many ways as a quotient group scheme of the affine group scheme Aut v ss 0 (KY LY M) when the cardinality of M is infinite. Denoting by v K (G) the forgetful tensor functor of the tensor category Rep K (G) of finite dimensional representations over K of an affine group scheme G over K onto the tensor category of finite dimensional vector spaces over K, this result is differently stated that when G is connected reductive, there exists a fully faithful tensor functor…”

“…In Section 2, a proof of Theorem 1.2 is given for an arbitrary (finite or infinite) GALOIS extension L. It might be useful and natural to bring in a finite e Âtale K-algebra L. But we would like in the present paper to stick to our original setting [5,6] bearing Diophantine approximation in mind. A little deviation is that the index set M is any non-empty set.…”