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nLab > Latest Changes: Hermitian form

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2022

    I have added an observation (here) that complex Hermitian inner product spaces ℋ\mathcal{H} may be regarded as (ℤ/2↷ℂ)(\mathbb{Z}/2 \curvearrowright \mathbb{C})-modules of the form ℋ⊕ℋ *\mathcal{H} \oplus \mathcal{H}^\ast in the topos of ℤ/2\mathbb{Z}/2-sets.

    diff, v6, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2022

    added (here) a characterization of hermitian operators, in this fashion

    diff, v7, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2023
    • (edited Aug 31st 2023)

    I have added some textbook references

    and some historical references,

    such as to von Neumann’s original definition of Hermitian forms (1930, 1932)

    Von Neumann’s is the modern definition – was he the first to state it this way?

    I gather from

    • Jürgen Elstrodt, Fritz Grunewald, Jens Mennicke: Integral Binary Hermitian Forms, Ch. 9 in Groups Acting on Hyperbolic Space, Springer (1998) [doi:10.1007/978-3-662-03626-6_9]

    that the concept is named after

    but it is not easy to recognize a Hermitian form in there (admittedly, I have only scanned over the article, so far)

    Similarly for

    • Luigi Bianchi, Forme definite di Hermite, §24 in: Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî, Mathematische Annalen 40 (1892) 332–412 [doi:10.1007/BF01443558]

    which may be the earliest coinage of the term “Hermitian form” (?)

    diff, v9, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 5th 2023

    I have now written out more text in the section on reagarding Hermitian/unitary complex modules as complex Euclidean/orthogonal “Real complex modules” internal to the category of real vector spaces with involutions.

    Also adjusted some of the terminology, for better flow and thus renamed the sub-section:

    As Real complex modules.

    diff, v13, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 23rd 2023

    added pointer to:

    • Max Karoubi, §1 in: Le théorème de périodicité en K-théorie hermitienne, Quanta of Maths 1, AMS and Clay Math Institute Publications (2010) [arXiv:0810.4707]

    diff, v19, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 25th 2023

    I see now that the embedding of fin-dim Hermitian spaces into self-dual ℤ/2↷ℂ\mathbb{Z}/2 \curvearrowright \mathbb{C}-modules which I described (here) is essentially what is known in the context of Hermitian K-theory as the hyperbolic functor

    Or rather, what I consider here is the hyperbolic functor equipped with ℤ/2×ℤ/2\mathbb{Z}/2 \times \mathbb{Z}/2-equivariance. I see now that this is known as establishing an equivalence between KR-theory and topological Hermitian K-theory.

    The two references for this equivalences that I am (now) aware of (here) are:

    and

    both of which are rather terse, each in its own way.

    So I am not surprised that my construction here is known, on the contrary, I was surprised that I couldn’t find something that elementary discussed in the literature. And with the above two references it still takes some squinting to see what’s going on.

    diff, v22, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 6th 2023

    added (here) original references expressing hermitian forms via dagger-compact structure

    diff, v23, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 10th 2023
    • (edited Nov 10th 2023)

    added (here) statement and proof of a more pronounced way of phrasing the expression of Hermitian forms as inner products internal to Real vector bundles:

    Passing a real inner product space (𝒱,g)(\mathscr{V}, g) equipped with an isometric complex structure JJ through the equivalence of real vector spaces with Real vector bundles over the point yields the corresponding Hermitian form ⟨−|−⟩≡g(−,−)+ig(J(−),v)\langle - \vert - \rangle \,\equiv\, g(-,-) + \mathrm{i} g\big(J(-), v\big), as shown here.

    diff, v25, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2023

    added pointer to:

    diff, v27, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2023

    added pointer to:

    diff, v28, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2023

    We now have a (very) brief note on the story I was getting at above, at Quantum and Reality.

1 to 11 of 11

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