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trivia: is there established notation for the set of formal power series in $X$ that have vanishing coefficient of $X^0$ (vanishing constant term)?
Like $X\mathbb{R}[[X]]$?
So I am asking regarding the entry S-matrix, concerning this def. and others.
This involves complex vector spaces of observables that presently already go by the names
$LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$and
$PolyObs(E_{\text{BV-BRST}})_{mc }((\hbar))[ [ g,j ] ] \,.$But for the first I should really be using the subspace of power series where each term is at least linear in $g$ or $j$ (or so I came to think, I feel like the relevant literature might not be taking proper care here).
To indicate this I could write explicitly something like
$LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ] \otimes \mathbb{C}\langle g,j\rangle$But this is becoming notationally awkward. I was hoping that maybe there’d be some established variant of the brackets to indicate the same idea, maybe something like
$LocObs(E_{\text{BV-BRST}})[ [ \hbar] ] \{\{g,j\}\}$to indicate the same, more succinctly.
so I settled now for writing
$LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j ] ]\langle g,j \rangle$(here)
Angle brackets for variables, like $\mathbf{C}\langle x, y\rangle$ are in algebra the standard notation for the ring of noncommutative polynomials in variables $x,y$ with coefficients in $\mathbf{C}$. Double angle brackets, like $\mathbf{C}\langle \langle x, y\rangle \rangle$ mean noncommutative power series in variables $x,y$.
You might mean and it is true that $\langle g, j\rangle$ may mean the ideal generated by $g$ and $j$ (in any algebraic context, commutative or noncommutative) and then the ring is assumed from the context (not part of the notation), this usage of $\langle, \rangle$ is just an auxiliary notation within the proofs, internal statements (like steps in a definition of some ring) etc. not a complete notation. Thus in algebra one does not ever write $R\langle g, j\rangle$ (except as in nonconfusing case, as a step of an internal calculation, where $R\cdot I = I$ for unital rings and $R\cdot I\subset I$ for nonunital) to mean the ideal at hand within $R$, but only $\langle g, j\rangle\subset R$. Of course, when the ideal is principal, then the notation as the one suggested above, $X\mathbf{C}[[x]]$, is preferrable.
But there is a notation which can be somewhat more fitting to your case at hand. In the case when in descending filtration one takes elements in the $n$-th term of filtration, then one first introduces the definition and the notation for the filtration at place, e.g. $F^\bullet R$, so for example $F^1 R$ would be the first term in the defined filtration of the ring $R$. If I understood you right this is the case you have at place.
I am sorry I do not have a fully satisfactory solution for this notational problem, but from the point of view of an algebraist this is the best I can suggest.
I was just thinking of the common notation for linear spans (e.g. here)
Just to notice that these two comments are of course compatible:
$\mathbf{C}\langle x, y\rangle$ are in algebra the standard notation for the ring of noncommutative polynomials in variables $x,y$ with coefficients in $\mathbf{C}$
and
common notation for linear spans (e.g. here
in both cases we have the free construction on the given generators (algebras in the first case, vector spaces in the second).
Since my $LocObs$ is crucially not an algebra, but just a vector space, I suppose the notation I chose isn’t too bad
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