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Does the situation 3 really appear in a meaningful way? Whenever I saw situations where 1 needed to be distinguished from 2, I recall 1 being described as a subword while 2 referred to as a subsequence.
Perhaps "Orbit harmonics and graded representations" of Garcia and Haiman would be a good reference that presents the story in this exact way with a lot of clarity and detail
@TomCopeland I am open to any kind of remarks, but as far as I am concerned, A is more general than B if B is a particular case of A. Your A is not more general, it is vaguely similar.
@TomCopeland that I see but I do not see at all how the properties "$f^{<-1>}(f(x)+f(y))$ has non-negative coefficients" and "$f^{<-1>}(f(t)-t)$ has non-negative coefficients" are related in any precise way, hence my comment.
@TomCopeland can you perhaps clarify? Do you mean "related" in some vague way or can you indicate some precise mathematical relationship (which is not at all obvious to me)?
Could you give some more specific references for problematic claims? It is not great to make this sort of claim without being precise enough. For what it's worth, I am looking at what seems to be their first paper on the topic (Math.Z. 173 (1980), 135-161), and for the zeta functions there (defined for arbitrary modules) the convergence is claimed when the real part is bigger than the dimension of the module (and not 1), see Prop.1 in that paper.
I assume that your $x$ is a vector? In that case it seems that your $L_A$ acts on vectors of homogeneous polynomials, and not on homogeneous polynomials as you claim?
This notion goes back (at least) to work of Berger and Moerdijk: arxiv.org/abs/0801.2664 (it is spelled out in the non-coloured case but readily generalises)
@LSpice in fact, what you and Qiaochu Yuan are discussing is that in the case of Lie algebras there is a very remarkable coincidence of two different universal enveloping algebras. There is the universal "multiplicative" enveloping algebra that exists for any type of algebras, as in my answer to this question dating from 2010, and the universal enveloping algebra which is the left adjoint to the "forgetful functor" from associative algebras to Lie algebras which only remembers the antisymmetrized operation $AB-BA$; this is certainly something special that does not exist in general.
It is really unclear what you are asking. The result is true over a field, not true over a ring. I have a sneaking suspicion that for whichever precise question you may ask, the answer will be obvious. For now, I do not think this post contains a precise question.
