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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2018
    • (edited Jan 17th 2018)

    trivia: is there established notation for the set of formal power series in XX that have vanishing coefficient of X 0X^0 (vanishing constant term)?

    • CommentRowNumber2.
    • CommentAuthorSimonWillerton
    • CommentTimeJan 17th 2018

    Like X[[X]]X\mathbb{R}[[X]]?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2018
    • (edited Jan 18th 2018)

    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 BV-BRST)[[,g,j]] LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]


    PolyObs(E BV-BRST) mc(())[[g,j]]. 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 gg or jj (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 BV-BRST)[[,g,j]]g,j 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 BV-BRST)[[]]{{g,j}} LocObs(E_{\text{BV-BRST}})[ [ \hbar] ] \{\{g,j\}\}

    to indicate the same, more succinctly.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2018

    so I settled now for writing

    LocObs(E BV-BRST)[[,g,j]]g,j LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j ] ]\langle g,j \rangle


    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeJan 19th 2018
    • (edited Jan 19th 2018)

    Angle brackets for variables, like Cx,y\mathbf{C}\langle x, y\rangle are in algebra the standard notation for the ring of noncommutative polynomials in variables x,yx,y with coefficients in C\mathbf{C}. Double angle brackets, like Cx,y\mathbf{C}\langle \langle x, y\rangle \rangle mean noncommutative power series in variables x,yx,y.

    You might mean and it is true that g,j\langle g, j\rangle may mean the ideal generated by gg and jj (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 Rg,jR\langle g, j\rangle (except as in nonconfusing case, as a step of an internal calculation, where RI=IR\cdot I = I for unital rings and RIIR\cdot I\subset I for nonunital) to mean the ideal at hand within RR, but only g,jR\langle g, j\rangle\subset R. Of course, when the ideal is principal, then the notation as the one suggested above, XC[[x]]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 nn-th term of filtration, then one first introduces the definition and the notation for the filtration at place, e.g. F RF^\bullet R, so for example F 1RF^1 R would be the first term in the defined filtration of the ring RR. 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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2018

    I was just thinking of the common notation for linear spans (e.g. here)

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2018
    • (edited Jan 19th 2018)

    Just to notice that these two comments are of course compatible:

    Cx,y\mathbf{C}\langle x, y\rangle are in algebra the standard notation for the ring of noncommutative polynomials in variables x,yx,y with coefficients in C\mathbf{C}


    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 LocObsLocObs is crucially not an algebra, but just a vector space, I suppose the notation I chose isn’t too bad