# ag.algebraic geometry – Mumford-Tate conjecture for mixed Tate motives

Let $$X$$ be a (not necessarily smooth or proper) variety over a number field $$k$$, let be $$iota: khookrightarrow mathbb{C}$$ an embedding, and let $$bar k$$ be the algebraic closure of $$k$$ in $$mathbb{C}$$. Let $$V$$ be a subquotient of $$H^i(X_{mathbb{C}}^{text{an}}, mathbb{Q})$$ such that

1. The subquotient $$V$$ respects the mixed Hodge structure on $$H^i(X_{mathbb{C}}^{text{an}}, mathbb{Q})$$ (including the $$k$$-structure coming from the algebraic de Rham cohomology of $$X$$),
2. The subquotient $$Votimes mathbb{Q}_ell$$ of $$H^i(X_{mathbb{C}}^{text{an}}, mathbb{Q})otimes mathbb{Q}_ell=H^i(X_{bar k, text{ét}}, mathbb{Q}_ell)$$ respects the Galois action of $$text{Gal}(bar k/k)$$, and
3. $$V$$ is mixed Tate as a mixed Hodge structure (or $$Votimes mathbb{Q}_ell$$ is mixed Tate as a Galois representation).

(If you’d like, you can think of $$V$$ as the realization of a mixed Tate motive over $$k$$ under any of the various formalisms for mixed Tate motives over number fields — I think that a priori the conditions above are slightly weaker but expected to be equivalent.)

I’d like to know — is the analogue of the Mumford-Tate conjecture is known to be true for such $$V$$? Explicitly, one expects that the Lie algebra of the image of $$text{Gal}(bar k/k)to GL(Votimes mathbb{Q}_ell)$$
is the extension of scalars to $$mathbb{Q}_ell$$ of the Lie algebra of a $$mathbb{Q}$$-group, given as the Tannaka dual of the subcategory of mixed Tate Hodge structures over $$k$$, generated by $$V$$.

My feeling is that one can probably extract this from the literature by comparing ranks of various Ext groups (in one’s favorite category of mixed Tate motives and in the $$ell$$-adic and Hodge settings) but I am hoping it is written explicitly somewhere.