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• W4321351198 abstract "Let $mathbf{T}$ be a neutral Tannakian category over a field of characteristic zero with unit object $mathbf{1}$, and equipped with a filtration $W_cdot$ similar to the weight filtration on mixed motives. Let $M$ be an object of $mathbf{T}$, and $underline{mathfrak{u}}(M)subset W_{-1}underline{Hom}(M,M)$ the Lie algebra of the kernel of the natural surjection from the fundamental group of $M$ to the fundamental group of $Gr^WM$. A result of Deligne gives a characterization of $underline{mathfrak{u}}(M)$ in terms of the extensions $0longrightarrow W_pM longrightarrow M longrightarrow M/W_pM longrightarrow 0$: it states that $underline{mathfrak{u}}(M)$ is the smallest subobject of $W_{-1}underline{Hom}(M,M)$ such that the sum of the aforementioned extensions, considered as extensions of $mathbf{1}$ by $W_{-1}underline{Hom}(M,M)$, is the pushforward of an extension of $mathbf{1}$ by $underline{mathfrak{u}}(M)$. In this article, we study each of the above-mentioned extensions individually in relation to $underline{mathfrak{u}}(M)$. Among other things, we obtain a refinement of Deligne's result, where we give a sufficient condition for when an individual extension $0longrightarrow W_pM longrightarrow M longrightarrow M/W_pM longrightarrow 0$ is the pushforward of an extension of $mathbf{1}$ by $underline{mathfrak{u}}(M)$. 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 $underline{mathfrak{u}}(M)= W_{-1}underline{Hom}(M,M)$). Using Grothedieck's formalism of textit{extensions panach'{e}es} 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 $mathbb{Q}$ with three weights and large unipotent radicals." @default.
• W4321351198 created "2023-02-20" @default.
• W4321351198 creator A5029338276 @default.
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• W4321351198 date "2023-03-24" @default.
• W4321351198 modified "2023-10-18" @default.
• W4321351198 title "On unipotent radicals of motivic Galois groups" @default.
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• W4321351198 doi "https://doi.org/10.2140/ant.2023.17.165" @default.
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