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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2012

    wanted to be able to say sum and have a pointer to somewhere.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeOct 26th 2012
    • (edited Oct 26th 2012)

    People use word sum whenever they denote a binary operation additively, even sometimes in nonabelian examples, while it is a rule of thumb that the additive notation is used mainly in commutative cases. Complementary, in many abelian groups, the multiplication is denoted multiplicatively and called multiplication, despite that it is commutative.

    I made some changes partly reflecting this. One should also treat the case of tracks and Baues-Wirsching cohomology where also the additive operation used in the story is noncommutative.

    Maybe we should have a redirect additive operation ?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2012

    What’s an example of a non-commutative sum that you have in mind?

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeOct 26th 2012
    • (edited Oct 26th 2012)

    For example in group field theory, people use addition of momenta which are considered as coefficients in front of Lie algebra generators, this means that one has to look at how the exponents add up when they belong to a Lie algebra. Of course, the answer can be inferred with the usage of Hausdorff series. See e.g.

    • Sam Halliday, Richard J. Szabo, Noncommutative Field Theory on Homogeneous Gravitational Waves, http://arxiv.org/abs/hep-th/0602036 and references therein.

    Another case is so-called BRAIDED ADDITION of Shahn Majid, very important in quantum group theory. It is related to some construction also in work of Gurevich and then Majid and Lyubashenko about combining R-matrices into more complicated R-matrices (solutions of quantum Yang.Baxter equation), which is not diagonal combination into bigger matrices.

    For additive operation in works of Baues and Wirsching look at their papers or book by Baues on algebraic homotopy.

    These are the main three examples I had in mind.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2012

    Where in hep-th/0602036 is a noncommutative sum? Please give me the page number.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeOct 26th 2012
    • (edited Oct 26th 2012)

    Well, the notation with sum in a box is used on page 17 following earlier references in this field which call this operation Fourier/group/quantum “addition” while it is just a logarithm of group multiplication of exponentials, whose the essence is in fact contained in function DD which is in formula 3.11 below. Function DD becomes usual some if the parameter of the Lie bracket gets smaller (i.e. for abelian Lie algebra). The terminology and notation is used more in references more focused on group field theory, see formula (34) in http://arxiv.org/abs/0903.3475

    See also pages 4 and 5 of presentation http://fqxi.org/tools/download/__details/Girelli%20Azores%20Talk.pdf where it comes even as an argument in delta function expressions.

    The terminology here comes from the fact that the Fourier mode addition for usual plane waves modifies by small amount coming from noncommutativity in Lie algebra. So it can be considered as a correction to the addition of monenta.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeOct 26th 2012
    • (edited Oct 26th 2012)

    See also Freidel and Majid, formula 71, where a nontrivial formula for the addition (in notation \oplus) is given: http://arxiv.org/abs/hep-th/0601004 Definition is in formula 70.

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeOct 26th 2012
    • (edited Oct 26th 2012)

    It is called quantum addition because in terms of commutative variables and star product, one replaces multiplication for plane waves with star product what amount of replacing addition of momenta with quantum addition. So it is as quantum as star product is.

    Thus

    exp(ikx)exp(iqx)=exp(i(kq)x) exp (i k x) \star exp (i q x) = exp (i (k \oplus q) x)

    (k,q,xk,q,x vector quantities). If \star is usual multiplication then \oplus is ++, while if it is a star product, it is deformed and noncommutative along with the product.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeOct 27th 2012

    I’ve seen additive notation used when writing cocycle equations for nonabelian group cohomology.

    • CommentRowNumber10.
    • CommentAuthorTim_Porter
    • CommentTimeOct 27th 2012

    Ronnie often uses additive notation in omega groupoids, etc., even though the composition is non-commutative.

    • CommentRowNumber11.
    • CommentAuthorTom Leinster
    • CommentTimeOct 27th 2012

    Addition of ordinals is noncommutative. That’s the only example I know.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2012
    • (edited Oct 29th 2012)

    Ah, well.

    Most of the examples of “non-commutative sums” given above are examples of composition – of flows, of morphisms, of intervals.

    I’d rather not let such heavy abuse of terminology dominate an nnLab entry on the notion of sum. I’d rather have the nnLab entry say prominently “sums are commutative” and then much less prominently add a remark that some authors have and will abuse notation. I am always for highlighting the pattern over the exception.

    But I won’t fight further about it.

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeOct 29th 2012
    • (edited Oct 29th 2012)

    But what is wrong in saying something like “sum is the name for many examples of binary operations, in the cases which are by convention considereda additive and denoted by ++, what is in most examples the case of commutative binary operation” ? Multiplication of integers is commutative, but by convention it is not called addition. So we must say somewhere that it is the matter of convention in particalar examples, and not of universal rule, what we will call addition.

    Every associative binary operation may be considered concatenation or composition.

    I do not see any use in making big simplifications about something as simple as notion of sum. It is better to be up to the point and include all major possibilities and subtleties. Somebody who does not know from experience how most sums like will anyway have no use. And for elementary schoolers word commutative is already too much anyway.

    In fact the current wording is quite clear, for people who are at level to use anything in nnLab.

    • CommentRowNumber14.
    • CommentAuthorTobyBartels
    • CommentTimeOct 30th 2012

    I don't see how addition of ordinals is an abuse of notation; it's a straightforward generalisation of addition of natural numbers. There is a symmetrised notion of addition of ordinals (which I've seen denoted with ‘\oplus’ and Wikipedia says is denoted ‘#\#’), but that is less fundamental and (IME) less used.

    • CommentRowNumber15.
    • CommentAuthorTobyBartels
    • CommentTimeOct 30th 2012

    I made addition a redirect and put it prominently at the top. (Actually, I'd be inclined to rename the entry addition since an analogous entry would have to be called multiplication and not product.) I split the first example into two and added to them slightly.

    I also did some rephrasing of the first example that people may wish to argue about. (^_^)

    • CommentRowNumber16.
    • CommentAuthorzskoda
    • CommentTimeOct 31st 2012
    • (edited Oct 31st 2012)

    I like the Toby’s changes, except maybe about the integrals (various kinds of integrals do not need a topology in general, which is emphasised there).

    • CommentRowNumber17.
    • CommentAuthorTobyBartels
    • CommentTimeNov 1st 2012

    That's true.

    • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeNov 3rd 2012

    I added a link to biproduct and direct sum.

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