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nLab > Latest Changes: Pontrjagin ring

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2019

    starting something

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2019

    added brief pointer to homological group completion theorem

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 3rd 2023
    • (edited Jul 3rd 2023)

    added this pointer:

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    • CommentTimeJul 4th 2023

    added pointer to:

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    • CommentTimeJul 4th 2023

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    • CommentTimeJul 4th 2023

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    • CommentTimeJul 4th 2023
    • (edited Jul 4th 2023)

    added pointer to

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    • CommentTimeJul 5th 2023

    added pointer to:

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    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2023

    am starting to list references relating the Pontryagin product to quantum cohomology: here

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    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2023

    added pointer to:

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    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2023

    added (here) brief statement on the Whitehead product as the commutator of the Pontrjagin product, under the Hurewicz homomorphism, from

    diff, v10, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2023

    adding references on the Adams-Hilton model:

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    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2023
    • (edited Nov 23rd 2023)

    added pointer to:

    reviewed in:

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    • CommentAuthorUrs
    • CommentTimeNov 23rd 2023

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    • CommentAuthorUrs
    • CommentTimeNov 25th 2023

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    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2023

    have spelled out a couple of simple examples of rational Pontrjagin algebras: here

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    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2023

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    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2023

    added pointer to:

    which gives a full lifting of Milnor & Moore 1965 (Appendix), equipping the rational Pontrjagin algebra with A ∞A_\infty-algebra structure and identifying it with the universal envelope of the Whitehead L-infinity algebra.

    diff, v22, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJul 8th 2024

    added pointer to:

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    • CommentRowNumber20.
    • CommentAuthorperezl.alonso
    • CommentTimeAug 26th 2024

    In e.g. Introduction to Hypothesis H you highlight that the quantum states are not only a ring but actually a Hopf algebra, where I presume the comultiplication is given by the dual of the cup product provided by the fact that integral cohomology is multiplicative. Is this something particular to integral cohomology, or will the homology always form a Hopf algebra whenever the corresponding cohomology theory is multiplicative?

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2024

    What makes this happen is the fact that the space being evaluated on is assumed to be loop space, hence in particular an H-space.

    Cohomology of any space makes an algebra, and homology of any space makes a coalgebra (with coefficients a field, at least). But if the space itself has a product structure which makes it an H-space, then homology inherits that (via pushforward) and becomes in addition an algebra in a way that is compatible with its general coalgebra structure.

    • CommentRowNumber22.
    • CommentAuthorperezl.alonso
    • CommentTimeAug 27th 2024

    On the same topic, what exactly is higher'' as inhigher observables” about the Pontryagin-Hopf algebra? It seems weird to me that one gets a Hopf algebra (hence a 3-vector space) regardless of whether we are talking about qm or higher-dimensional qft. Why don’t we see even higher-modules for higher-dim’l qft?

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2024
    • (edited Aug 27th 2024)

    The “higher” refers to the higher degree nn of the homology groups H n(X;ℂ)H_n(X; \mathbb{C}) constituting the graded object Obs •≔H •(X;ℂ)\mathrm{Obs}_\bullet \coloneqq H_\bullet(X;\mathbb{C}).

    An ordinary observable is just a compactly supported function on the phase space. This is a “topological observable” if it is locally constant, in which case it constitutes an element of H 0(X;ℂ)H_0(X; \mathbb{C}). This way, H 0(X;ℂ)H_0(X;\mathbb{C}) is seen as the space of ordinary (non-higher) topological observables, and H ≥1(X;ℂ)H_{\geq 1}(X;\mathbb{C}) as the spaces of “higher topological observables”.

    I am not sure yet what really to make, physics-wise, of the full fact that, for XX a loop space, Obs •Obs_\bullet is a graded Hopf algebra:

    What matters at face value is that Obs •Obs_\bullet is a star-algebra (since any algebra of quantum observables ought to be a star-algebra), which uses of the Hopf algebra structure only the product and the antipode.

    On a very speculative note: Elsewhere we noticed the numerical coincidence that Hopf algebras may be understood as representing 3-vector spaces, and that 3-Hilbert spaces would also govern the would-be extended worldvolume QFT of a membrane.

    If this is more than numerology (at this point I really don’t know if it is) then it would suggestively match the observation that algebras of observables of the form H •(ΩY;ℂ)H_\bullet(\Omega Y; \mathbb{C}) reflect a form of “topological discrete light cone quantization”: This is because the star-operation on a star-algebra of observables expresses time reversal, and the star-operation on H •(ΩY;ℂ)H_\bullet(\Omega Y; \mathbb{C}) is given by complex conjugation of coefficients (as for ordinary time reversal) accompanied by inversion of loops (as appropriate for simultaneous evolution along a compactified dimension).

    • CommentRowNumber24.
    • CommentAuthorperezl.alonso
    • CommentTimeAug 27th 2024

    Got it.

    I am not sure either what that n=3n=3 means. If I follow the argument correctly, all this really works for any kk-sphere, since S kS^k is a co-H-space, and so we always get a n=3n=3 vector space regardless of kk. And to make this more puzzling, if we also consider S 1S^1 as a H-space (by identifying it with the Lie group U(1)U(1)) then it seems to me (but should be checked) that H •(ΩY,ℂ)H_{\bullet}(\Omega Y,\mathbb{C}) inherits a second comultiplication, which somewhat sounds like (but not quite) the setting for n=4n=4-vector spaces as trialgebras (or rather cotrialgebras in the language of Pfeiffer 04). Though this case is a bit more peculiar because of course only a few select spheres also have a H-space structure.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2024
    • (edited Aug 27th 2024)

    True, these considerations are independent of the 4-sphere.

    I am thinking of this as a parallel chain of evidence:

    On the one hand there is evidence that the flux-quantization of M-theory fields is controlled by 4-Cohomotopy

    On the other hand there is (less, currently) evidence that the actual quantization of (flux-quantized) M-theory fields is controlled by the Pontrjagin algebra of the field moduli spaces.

    (Together this gives Pontrjagin algebras of 4-Cohomotopy cocycle spaces, but the two items can be considered separately.)

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