Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: Hermitian form

Bottom of Page

1 to 11 of 11

  1.  
    • 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

  1.