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@Jason a friend pointed me to this post as well: mathoverflow.net/questions/193/… So I guess this answers in the affirmative that $S_l$ is algebraic, but is there a hope of a dimension bound as in my second question?
Of course I should have thought about that, thank you. Is it possible the question is interesting if we insist that $\mu_n \in K$? I will edit my question.
