Jolanta Słomińska

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=99625)

## Selected writings

On the [[equivariant Chern character]]:

* [[Jolanta Słomińska]], *On the Equivariant Chern homomorphism*, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys. **24** (1976) 909-913

On the [[Dold-Kan correspondence]]:

* [[Jolanta Słomińska]], _Dold-Kan type theorems and Morita equivalences of functor categories_, Journal of Algebra **274** 1 (2004) 118-137 $[$[doi:10.1016/j.jalgebra.2003.10.025](https://doi.org/10.1016/j.jalgebra.2003.10.025)$]$

On [[Hochschild homology]] and [[cyclic homology]]:

* [[Jolanta Słomińska]], _Decompositions of the category of noncommutative sets and Hochschild and cyclic homology_, Cent. Eur. J. Math. __1__ (2003), no. 3, 327–331 ([doi:10.2478/BF02475213](http://dx.doi.org/10.2478/BF02475213), [MR2004f:16011](http://www.ams.org/mathscinet-getitem?mr=1992895))

category: people

[[!redirects J. Słomińska]]

[[!redirects Jolanta Slominska]]

[[!redirects J. Slominska]]


Jon Awbrey

[[!redirects JA]]

* [JA @ Mathstodon](https://mathstodon.xyz/@Inquiry)

* [JA @ WordPress](https://inquiryintoinquiry.com/)

* [JA @ OEIS Wiki](https://oeis.org/wiki/User:Jon_Awbrey)

* [JA @ InterSciWiki](https://web.archive.org/web/20191124081501/http://intersci.ss.uci.edu/wiki/index.php/User%3AJon_Awbrey)

# Algebra #

* [[multigrade operator]]
* [[parametric operator]]

# Arithmetic #

* [[Riffs and Rotes]]

# Combinatorics #

* [graph theory](http://ncatlab.org/nlab/revision/graph+theory+%3E+history/9)

# Computation #

* [[propositions as types in combinatory algebra]]

# Inquiry #

* [[inquiry]]
* [[inquiry driven system]]
* [[Peirce's logic of information]]
* [[semiotic information]]

# Logic #

* [[boolean domain]]
* [[boolean function]]
* [[boolean-valued function]]
* [[cactus language]]
* [[continuous predicate]]
* [[differential logic]]
* [[differential propositional calculus]]
* [higher order proposition](http://ncatlab.org/nlab/revision/higher-order+logic/47)
* [[hypostatic abstraction]]
* [[logical graph]]
* [[minimal negation operator]]
* [[Peirce's law]]
* [[praeclarum theorema]]
* [[propositional calculus]]
* [[propositional equation reasoning system]]
* [relation theory](http://ncatlab.org/nlab/revision/relation+theory+%3E+history/35)
* [[semiotic equivalence relation]]
* [[sign relation]]
* [triadic relation](http://ncatlab.org/nlab/revision/ternary+relation/17)
* [universe of discourse](http://ncatlab.org/nlab/revision/type/3)

# Miscellaneous #

* [[Charles Sanders Peirce]]
* [[precursors]]
* [syntax, semantics, pragmatics](http://ncatlab.org/nlab/revision/syntax%2C+semantics%2C+pragmatics+>+history/1)

# Utilities #

* [Symbol Sandbox](http://ncatlab.org/nlab/revision/Symbol+Sandbox+%3E+history/12)

# Gallery #

__NB.__ I'll be using this area to test remote images and upload local images that I use in articles.

Templates ---

    [[mypic.jpg|alt text:pic]]

    ![alt text](/nlab/files/mypic.jpg)

    http://ncatlab.org/nlab/files/mypic.jpg

    <img alt="mytext" src="/nlab/files/mypic.jpg" width="500" />

Upload Queue ---

$\varnothing$

***

category: people


Jon Beck

Jon Mock Beck was a [[category theory|category theorist]] whose thesis (under the supervision of [[Sammy Eilenberg]]) and subsequent work with [[Mike Barr]] have been highly influential in the development of the subject.  He died in March 2006.

* [Wikipedia article](http://en.wikipedia.org/wiki/Jonathan_Mock_Beck)

* [thesis](http://www.tac.mta.ca/tac/reprints/articles/2/tr2abs.html)

## Selected writings

Introducing the [[monadicity theorem]]:

* {#Beck68} [[Jon Beck]], untitled manuscript, 1968 ([[Beck_MonadicityTheorem.pdf:file]])

On [[monads]] in [[universal algebra]] and ([[cohomology|co-]])[[homology]]-theory:

* [[H. Applegate]], [[M. Barr]], [[J. Beck]], [[F. W. Lawvere]], [[F. E. J. Linton]], [[E. Manes]], [[M. Tierney]], [[F. Ulmer]]: _[[Seminar on Triples and Categorical Homology Theory]]_, ETH 1966/67, edited by [[Beno Eckmann]] and [[Myles Tierney]], **[[LNM 80]]**, Springer (1969), reprinted as: Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

Particularly on [[distributive laws]]:

* [[Jon Beck]], *Distributive Laws*, in: *[[Seminar on Triples and Categorical Homology Theory]]*, ETH 1966/67, Lecture Notes in Mathemativs, Springer (1969), Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

and on [[monadic cohomology]] via [[canonical resolutions]]:

* [[Michael Barr]], [[Jon Beck]], _Homology and Standard Constructions_, in: _[[Seminar on Triples and Categorical Homology Theory]]_, Lecture Notes in Maths. **80**, Springer (1969), Reprints in Theory and Applications of Categories __18__ (2008) 186-248 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

category:people

[[!redirects J. M. Beck]]
[[!redirects J. Beck]]
[[!redirects Jonathan Mock Beck]]
[[!redirects Jonathan Beck]]
[[!redirects Jon Mock Beck]]


Jon Magne Leinaas

* [Wikipedia entry](https://en.wikipedia.org/wiki/Jon_Magne_Leinaas)

## Selected writings

Early discussion of what came to be known as [[anyon]] [[braid group statistics]], and preconceiving the role of the [[configuration space of points]]:

* {#LeinaasMyrheim77} [[Jon Magne Leinaas]], [[Jan Myrheim]], _On the theory of identical particles_, _К теории тождествениых частиц_, Nuovo Cim B 37, 1–23 (1977) ([doi:10.1007/BF02727953](https://doi.org/10.1007/BF02727953))

category: people

[[!redirects J. M. Leinaas]]


Jon McCammond

* [institute page](https://www.math.ucsb.edu/people/jon-mccammond)

* [personal page](http://web.math.ucsb.edu/~mccammon/)

## Selected writings

On [[finitely presented group|finite presentations]] of the [[pure braid group]]:

* [[Dan Margalit]], [[Jon McCammond]], *Geometric presentations for the pure braid group*,  Journal of Knot Theory and Its Ramifications **18** 01 (2009) 1-20 [[arXiv:math/0603204](https://arxiv.org/abs/math/0603204), [doi:10.1142/S0218216509006859](https://doi.org/10.1142/S0218216509006859)]

category: people


Jon Sterling

> see at *[[Jonathan Sterling]]*


Jonas Frey

* [webpage](https://sites.google.com/site/jonasfreysite/)

## related $n$Lab entries

* [[realizability topos]]

category: people


Jonatan Lenells

* [Institute page](https://www.kth.se/profile/jlenells)

* [GoogleScholar page](https://scholar.google.com/citations?user=JQy0LtIAAAAJ&hl=en)

## Selected writings

On [[perturbative quantum field theory|perturbative]] [[Chern-Simons theory with complex gauge group]]:

* {#DGLZ09} [[Tudor Dimofte]], [[Sergei Gukov]], [[Jonatan Lenells]], [[Don Zagier]], _Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group_, Commun. Num. Theor. Phys. **3** (2009) 363-443 [[arXiv:0903.2472](http://arxiv.org/abs/0903.2472), [doi:10.4310/CNTP.2009.v3.n2.a4](https://dx.doi.org/10.4310/CNTP.2009.v3.n2.a4)]

category: people


Jonathan A. Campbell

* [personal page](http://www.jonathanacampbell.com/)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=181219)

## Selected writings

On the [[Steenrod algebra]] and the [[Adams spectral sequence]] (with motivation from [[functorial quantum field theory]]):

* [[Agnès Beaudry]], [[Jonathan A. Campbell]], *A Guide for Computing Stable Homotopy Groups*, in *Topology and Quantum Theory in Interaction*, Contemporary Mathematics **718**, Amer. Math. Soc. (2018)  [[arXiv:1801.07530](https://arxiv.org/abs/1801.07530), [ams:conm/718](https://bookstore.ams.org/view?ProductCode=CONM/718)]

category: people

[[!redirects Jonathan Campbell]]


Jonathan A. Jones

* [Research group page](https://www.physics.ox.ac.uk/our-people/jjones)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Jonathan_A._Jones)

## Selected writing

On [[quantum computers]] based on [[nuclear magnetic resonance]]-technology:

* [[Jonathan A. Jones]], *Quantum Computing and Nuclear Magnetic Resonance*, PhysChemComm **11** (2001) [[doi:10.1039/b103231n](https://doi.org/10.1039/b103231n), [arXiv:quant-ph/0106067](https://arxiv.org/abs/quant-ph/0106067)]

* [[Jonathan A. Jones]], *Quantum Computing with NMR*, Prog. NMR Spectrosc. **59** (2011) 91-120 [[doi:10.1016/j.pnmrs.2010.11.001](https://doi.org/10.1016/j.pnmrs.2010.11.001), [arXiv:1011.1382](https://arxiv.org/abs/1011.1382)]

[[!redirects Jonathan Jones]]

category: people


Jonathan Atteia

* [Institute page](https://www.uni-due.de/physik/twist/jonathan_atteia.php)

## Selected writings

On the behaviour of [[graphene]]-like [[topological phases of matter]] ([[Haldane model]]) under irradiation:

* {#Atteia16} [[Jonathan Atteia]], *Topology and electronic transport in Dirac systems under irradiation* (2016) $[$[tel:02426217](https://tel.archives-ouvertes.fr/tel-02426217), [pdf](https://tel.archives-ouvertes.fr/tel-02426217/document)$]$

category: people


Jonathan Bagger

* [webpage](http://www.pha.jhu.edu/~bagger/)

## Selected writings

On multiple [[M2-branes]] and the [[BLG model]]/[[ABJM model]] of their [[worldvolume]] theory:

* {#BaggerLambert06} [[Jonathan Bagger]], [[Neil Lambert]], _Modeling Multiple M2's_, Phys. Rev. D75, 045020 (2007). ([hep-th/0611108](http://arxiv.org/abs/hep-th/0611108)).

* [[Jonathan Bagger]], [[Neil Lambert]], _Gauge Symmetry and Supersymmetry of Multiple M2-Branes_, Phys. Rev. D77, 065008 (2008). ([arXiv:0711.0955](http://arXiv.org/abs/0711.0955)).

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

category: people


Jonathan Bain

* [webpage](http://ls.poly.edu/~jbain/)

## related pages

* [[philosophy of physics]]

category: people


Jonathan Barmak

* [webpage](http://mate.dm.uba.ar/~jbarmak/)

## related $n$Lab entries:

* [[finite topological space]]

* [[Quillen's theorem A]]

##Publications

* Jonathan Barmak, _Algebraic Topology of Finite Topological Spaces and Applications_, Lecture Notes in Mathematics Vol. 2032. Springer (2011).

* Jonathan Barmak and [[Gabriel Minian|G. Minian]], _Minimal finite models_ J. Homotopy Relat. Struct. 2 (2007), 127-140. ([ArXiv](https://arxiv.org/pdf/math/0611156.pdf))

category: people


Jonathan Beardsley

Assistant Professor of mathematics at the University of Nevada, Reno.

* [Website](https://www.jonathanbeardsley.com/)

## Selected writings

On [[Ravenel's spectra]] $X(n)$:

* [[Jonathan Beardsley]], _A Theorem on Multiplicative Cell Attachments with an Application to Ravenel's $X(n)$ Spectra_, Journal of Homotopy and Related Structures volume 14, pages 611–624 (2019) ([arXiv:1708.03042](https://arxiv.org/abs/1708.03042), [doi:10.1007/s40062-018-0222-6](https://link.springer.com/article/10.1007/s40062-018-0222-6))

* [[Jonathan Beardsley]], _Topological Hochschild homology of $X(n)$_ ([arXiv:1708.09486](https://arxiv.org/abs/1708.09486))

On [[configuration spaces of points]] and a possible [[Galois group]] over the [[sphere spectrum]]:

* {#MoravaBeardsly17} [[Jack Morava]], [[Jonathan Beardsley]], _Toward a Galois theory of the integers over the sphere spectrum_,  Journal of Geometry and Physics Volume 131, September 2018, Pages 41-51 ([arXiv:1710.05992](https://arxiv.org/abs/1710.05992), [doi:10.1016/j.geomphys.2018.04.011](https://doi.org/10.1016/j.geomphys.2018.04.011))

On the [[enriched Grothendieck construction]]:

* {#BeardsleyWong18} [[Jonathan Beardsley]], [[Liang Ze Wong]], _The Enriched Grothendieck Construction_, Advances in Mathematics **344** (2019) 234-261 [[arXiv:1804.03829](https://arxiv.org/abs/1804.03829), [doi:10.1016/j.aim.2018.12.009](https://doi.org/10.1016/j.aim.2018.12.009)]

* {#BeardsleyWong19} [[Jonathan Beardsley]], [[Liang Ze Wong]], *The operadic nerve, relative nerve and the Grothendieck construction*, Theory and Applications of Categories, **34** 13 (2019) 349-374 [[tac:34-13](http://www.tac.mta.ca/tac/volumes/34/13/34-13abs.html)]

On [[G-structure]] for $G = Br_\infty$ the infinite [[braid group]]:

* [[Jonathan Beardsley]], *On Braids and Cobordism Theories*, Glasgow (2022) [notes: [pdf](https://www.jonathanbeardsley.com/GlasgowNotes2022.pdf), [[Beardsley-BraidsAndCobordism.pdf:file]]]

## Related entries

* [[Grothendieck construction]]

category: people

[[!redirects Jon Beardsley]]
[[!redirects Jon Beardsley]]


Jonathan Block

* [website](http://www.math.upenn.edu/~blockj/)

## Selected writings

On [[infinity-local systems|$\infty$-local systems]] in the sense of [[(infinity,1)-vector bundles|$(\infty,1)$-vector bundles]] with [[flat infinity-connections|flat $\infty$-connections]]:

* [[Jonathan Block]], [[Aaron M. Smith]], *The higher Riemann--Hilbert correspondence*, Advances in Mathematics
**252** (2014) 382-405 [[arXiv:0908.2843](https://arxiv.org/abs/0908.2843), [doi:10.1016/j.aim.2013.11.001](https://doi.org/10.1016/j.aim.2013.11.001)]

## Related entries

* [[Koszul-Malgrange theorem]]

* [[curved dg-algebra]]

* [[iterated integral]]

category: people


Jonathan Campbell

* [personal page](http://www.jonathanacampbell.com/)

## Selected writings

On the [[Dennis trace]] on [[algebraic K-theory]] and relating to [[bicategorical trace]]:

* {#CLMPZ20} [[Jonathan Campbell]], [[John Lind]], [[Cary Malkiewich]], [[Kate Ponto]], [[Inna Zakharevich]], _K-theory of endomorphisms, the TR-trace, and zeta functions_ [[arXiv:2005.04334](https://arxiv.org/abs/2005.04334)]

category: people


Jonathan Dimock

* [webpage](http://www.math.buffalo.edu/~dimock/)

## related $n$Lab pages

* [[AQFT]]

* [[wave front set]]

* [[lattice gauge theory]]

category: people

[[!redirects Jon Dimock]]


Jonathan Evans

* [webpage](http://www.damtp.cam.ac.uk/people/j.m.evans/)

## Writings

* _Aspects of Yang-Mills theory_, Lecture notes on [[Yang-Mills theory]] ([web](http://www.homepages.ucl.ac.uk/~ucahjde/yangmills.htm))

category: people


Jonathan Gleason

Jonathan Gleason (1989-2018) received his Bachelor’s degree from the University of Chicago, his Master’s from Cambridge University in Cambridge, England, and at the time of his death was a fifth year Grad student at the University of California, Berkeley.

* [Obituary page](https://www.sunsetfuneralandcemetery.com/obituaries/Jonathan-Gleason-32965/)

* [Physics.SE page](https://physics.stackexchange.com/users/3397/jonathan-gleason)

In two unpublished notes (see [below](#SelectedWritings)), Gleason was concerned with the motivation of the mathematical foundations of [[AQFT|algebraic]] [[quantum mechanics]]:

> [[Gleason (2011)](#Gleason11), [p. 12](https://ncatlab.org/nlab/files/GleasonFAlgebraic.pdf#page=12)]:

> "In [Mackey (1963)](quantum+mechanics#Mackey63), immediately after stating what he believes to be an unmotivated axiom that invokes in some way the usual Hilbert space formulation of quantum mechanics, the author states the following:

> 'Ideally one would like to have a list of physically plausible assumptions from which one could deduce [this axiom]. [...]'

> The problem he is referring to is a problem that has bothered me ever since I first learned quantum mechanics, and while I do not think the argument we have just given is as strong an argument as Mackey is referring to, in my opinion, it is damn well close, and indeed, close enough that this author will finally be able to sleep at night."

## Selected writings
 {#SelectedWritings}

On the foundations of [[quantum mechanics]]/[[quantum probability theory]] conceptualized [[AQFT|algebraically]] in close analogy to [[classical mechanics]]/[[probability theory]] (with [[algebras of quantum observables]] understood as [[star-algebras]] and [[quantum states]] understood as [[states on a star-algebra]]):

* {#Gleason09} [[Jonathan Gleason]], *The $C^*$-algebraic formalism of quantum mechanics* (2009) [[[Gleason09.pdf:file]], [[GleasonAlgebraic.pdf:file]]]

* {#Gleason11} [[Jonathan Gleason]], *From Classical to Quantum: The $F^\ast$-Algebraic Approach*, contribution to *[VIGRE REU 2011](http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2011.html)*, Chicago (2011) [[pdf](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Gleason.pdf), [[GleasonFAlgebraic.pdf:file]]]

## Related entries

* [[quantum mechanics]]

* [[quantum probability]]

* [[algebraic quantum field theory]]

category: people


Jonathan Grattage

* [personal page](http://jonathan.grattage.free.fr/jon/)

## Selected writings

Introducing the [[functional programming language|functional]] [[quantum programming language]] *[[QML]]*:

* {#AltenkirchGrattage05} [[Thorsten Altenkirch]], [[Jonathan Grattage]], _A functional quantum programming language_, Logic in Computer Science, 2005. Proceedings. 20th Annual IEEE Symposium on, 26-29 June 2005 Page(s):249 - 258 ([arXiv:quant-ph/0409065](https://arxiv.org/abs/quant-ph/0409065), [doi:10.1109/LICS.2005.1](https://doi.org/10.1109/LICS.2005.1), [pdf](http://www.cs.nott.ac.uk/~txa/publ/qml.pdf))

* [[Jonathan Grattage]], *A functional quantum programming language*, PhD thesis, Nottingham (2006) [[eprints:10250](http://eprints.nottingham.ac.uk/10250), [pdf](https://eprints.nottingham.ac.uk/10250/1/thesis.pdf)]

Review:

* [[Jonathan Grattage]], _An overview of QML with a concrete implementation in Haskell_, Quantum Physics and Logic 2008 ([arXiv:0806.2735](https://arxiv.org/abs/0806.2735))

* [[Jonathan Grattage]] with [[Thorsten Altenkirch]], *Compiling a functional quantum programming language* (talk slides) [[pdf](http://www.cs.nott.ac.uk/~pszgmh/bctcs-slides/grattage.pdf), [[Grattage-CompilingFunctionalQPL.pdf:file]]]

category: people


Jonathan Heckman

* [webpage](http://www.jjheckman.com)

## Selected writings

On [[M-theory on S1/G_HW times H/G_ADE]] and [[D=6 N=(1,0) SCFT]]:

* {#DHTV14} [[Michele Del Zotto]], [[Jonathan Heckman]], [[Alessandro Tomasiello]], [[Cumrun Vafa]], _6d Conformal Matter_, JHEP02(2015)054 [[arXiv:1407.6359](https://arxiv.org/abs/1407.6359)]

On the [[cosmological constant]] in [[F-theory on Spin(7)-manifolds]] ([[Witten's Dark Fantasy]]):

* [[Jonathan Heckman]], Craig Lawrie, Ling Lin, [[Gianluca Zoccarato]], _F-theory and Dark Energy_, Fortschritte der Physik  ([arXiv:1811.01959](https://arxiv.org/abs/1811.01959), [doi:10.1002/prop.201900057]( https://doi.org/10.1002/prop.201900057))

* [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Jeremy Sakstein, [[Gianluca Zoccarato]], _Pixelated Dark Energy_ ([arXiv:1901.10489](https://arxiv.org/abs/1901.10489))

Comprehensive review of expected [[top-down model building|top-down constructions]] of [[D=6 N=(2,0) SCFT|$D=6$ $\mathcal{N}=(2,0)$ SCFTs]] and [[D=6 N=(1,0) SCFT|$D=6$ $\mathcal{N}=(1,0)$ SCFTs]]

* [[Jonathan J. Heckman]], [[Tom Rudelius]], *Top Down Approach to 6D SCFTs*, J. Phys. A: Math. Theor. **52** 9 (2019) 093001 [[arXiv:1805.06467](https://arxiv.org/abs/1805.06467), [doi:10.1088/1751-8121/aafc81](https://doi.org/10.1088/1751-8121/aafc81)]

Relating [[M-theory on Spin(7)-manifolds]] with [[F-theory on Spin(7)-manifolds]] via [[Higgs bundles]]:

* {#CHRTZ20} [[Mirjam Cvetic]], [[Jonathan Heckman]], Thomas B. Rochais, Ethan Torres, [[Gianluca Zoccarato]], _Geometric Unification of Higgs Bundle Vacua_ ([arXiv:2003.13682](https://arxiv.org/abs/2003.13682))

On the [[swampland cobordism conjecture]]:

* [[Markus Dierigl]], [[Jonathan Heckman]], _On the Swampland Cobordism Conjecture and Non-Abelian Duality Groups_, Physical Review D 2021 ([arXiv:2012.00013](https://arxiv.org/abs/2012.00013))

## Related $n$Lab entries

* [[SCFT]], [[string theory]], [[F-theory]].

category: people

[[!redirects Jonathan J. Heckman]]


Jonathan M. Corson

[[!redirects J. M. Corson]]
[[!redirects J. M. Corson]]
Jon Corson is a professor in the University of Alabama, whose research is in low dimensional topology and geometric ring theory. He helped develop the theory of [[complexes of groups]].

* [webpage](https://math.ua.edu/profile/jon-corson/)

category:people


Jonathan M. Robbins

* [Institute page](https://www.bristol.ac.uk/people/person/Jonathan-Robbins-60101278-f877-4e14-946f-262f3b95d5ae/)

* [GoogleScholar page](https://scholar.google.com/citations?user=mUzvUTwAAAAJ&hl=en)

## Selected writings

A geometric proof of the [[spin-statistics theorem]] (leading to the [[Atiyah-Sutcliffe conjecture]]):

* {#BerryRobbins97} [[Michael V. Berry]], [[Jonathan M. Robbins]], *Indistinguishability for quantum particles: spin, statistics and the geometric phase*, Proceedings of the Royal Society A **453** 1963 (1997) 1771-1790 ([doi:10.1098/rspa.1997.0096](https://doi.org/10.1098/rspa.1997.0096))

category: people

[[!redirects Jonathan Robbins]]

[[!redirects J. M. Robbins]]


Jonathan Oppenheim

* [personal page](https://www.ucl.ac.uk/oppenheim/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Jonathan_Oppenheim)

## Selected writings

Discussion of "[[noisy quantum operations]]" in the particular sense of [[quantum channels]] that admit an [environmental representation](quantum+channel#QuantumChannelsAndDecoherence) where the environment/bath system $\mathscr{B}$ is in its *maximally* [[mixed quantum state]]:

* {#HorodeckiHorodeckiOppenheim03} [[Michal Horodecki]], [[Pawel Horodecki]], [[Jonathan Oppenheim]], *Reversible transformations from pure to mixed states, and the unique measure of information*, Phys. Rev. A 67 062104 (2003) [[doi:10.1103/PhysRevA.67.062104](https://doi.org/10.1103/PhysRevA.67.062104), [arXiv;quant-ph/0212019](https://arxiv.org/abs/quant-ph/0212019)]

category: people


Jonathan Pridham

Jonathan P. Pridham is a mathematician based in Edinburgh.

* [webpage](https://www.maths.ed.ac.uk/~jpridham/)

* [publications](https://www.maths.ed.ac.uk/~jpridham/#preprints)

## related $n$Lab entries

* [[model structure for L-infinity algebras]]

* [[geometric infinity-stacks]]

* [[Representability Theorems]]

category: people

[[!redirects Pridham]]
[[!redirects Jon Pridham]]
[[!redirects J. P. Pridham]]


Jonathan Rosenberg

_Jonathan M. Rosenberg_ is a mathematician working on [[noncommutative geometry]], [[algebraic K-theory]], [[operator algebras]] and applications to [[string theory]] ([[D-brane charge]], [[T-duality]], [[orientifolds]]).

He is an editor of the [Journal of K-theory](https://www.cambridge.org/core/journals/journal-of-k-theory).

* [homepage](http://www-users.math.umd.edu/~jmr)
* [selected publications](http://www-users.math.umd.edu/~jmr/jmr_pub.html)

> Note: do not confuse with [[Alexander Rosenberg]], who works mainly in categorical approach to [[noncommutative algebraic geometry]], but who also started in [[functional analysis]].

## Selected writings

On [[T-duality]] and specifically [[topological T-duality]]:

* [[Jonathan Rosenberg]], *Topology, $C^*$-algebras, and string duality*, Regional Conference Series in Mathematics **111**, Amer. Math. Soc. (2009)  [[doi:10.1090/cbms/111](https://doi.org/10.1090/cbms/111), [ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:05632075&type=pdf&format=complete)]

On [[D-brane charge]] in [[orientifolds]] formulated in [[KR-theory]]:

* {#DMR13} [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], _T-duality For Orientifolds and Twisted KR-theory_ ([arXiv:1306.1779](http://arxiv.org/abs/1306.1779))

* {#DMR14} [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], _String theory on elliptic curve orientifolds and KR-theory_ ([arXiv:1402.4885](http://arxiv.org/abs/1402.4885))

On [[twisted cohomology]]:

* [[Jonathan Rosenberg]], *Twisted cohomology*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.03966](https://arxiv.org/abs/2401.03966)]

category: people

[[!redirects Jonathan M. Rosenberg]]
[[!redirects J. Rosenberg]]
[[!redirects J. M. Rosenberg]]


Jonathan Rubin

* [personal page](https://www.math.ucla.edu/~jrubin/)

## Selected writings

On [[equivariant homotopy theory]] and [[Elmendorf's theorem]] via [[enriched model categories]]:

* {#GuillouMayRubin13} [[Bertrand Guillou]], [[Peter May]], [[Jonathan Rubin]], _Enriched model categories in equivariant contexts_, Homology, Homotopy and Applications 21 (1), 2019 ([arXiv:1307.4488](https://arxiv.org/abs/1307.4488), [arXiv:10.4310/HHA.2019.v21.n1.a10](https://dx.doi.org/10.4310/HHA.2019.v21.n1.a10))

* [[Jonathan Rubin]], _Elmendorf constructions for G-categories and G-posets_ ([arXiv:2006.08876](https://arxiv.org/abs/2006.08876))

category: people


Jonathan Scott

* [Institute page](https://facultyprofile.csuohio.edu/csufacultyprofile/detail.cfm?FacultyID=J_A_SCOTT3)

## Selected writings

On the [[Pontrjagin ring]]-structure on the [[homology of loop spaces]]:

* [[Jonathan A. Scott]], _Algebraic Structure in the Loop Space Homology Bockstein Spectral Sequence_,  Transactions of the American Mathematical Society **354** 8 (2002) 3075-3084 [[jstor:3073034](https://www.jstor.org/stable/3073034)]

On the [[Adams-Hilton model]] for the [[Pontrjagin ring]]-structure on the [[singular chain complex]] of a [[based loop space]]:

* [[Kathryn Hess]], [[Paul-Eugène Parent]], [[Jonathan Scott]], [[Andrew Tonks]], *A canonical enriched Adams-Hilton model for simplicial sets*, Advances in Mathematics **207** 2 (2006) 847-875 [[doi:10.1016/j.aim.2006.01.013](https://doi.org/10.1016/j.aim.2006.01.013), [arXiv:math/0408216](https://arxiv.org/abs/math/0408216)]

On [[category theory|category theoretic]] aspects of [[persistence objects]] (as used in [[topological data analysis]]:

* {#BubenikScott14} [[Peter Bubenik]], [[Jonathan Scott]] _Categorification of Persistent Homology_, Discrete & Computational Geometry **51** (2014) 600-627 [[doi:10.1007/s00454-014-9573-x](https://doi.org/10.1007/s00454-014-9573-x)]

category: people

[[!redirects Jonathan A. Scott]]


Jonathan Seldin

* [webpage](http://www.cs.uleth.ca/~seldin/)

## Selected writings

On [[lambda-calculus]] and [[combinator logic]]:

* [[J. Roger Hindley]], [[Jonathan P. Seldin]], *Lambda-Calculus and Combinators -- An Introduction*, Cambridge University Press (2008) [[doi:10.1017/CBO9780511809835](https://doi.org/10.1017/CBO9780511809835)]

category: people

[[!redirects J. Seldin]]

[[!redirects Jonathan P. Seldin]]


Jonathan Smith

* [webpage](https://www.cis.upenn.edu/~jms/)

## Selected writings

Introducing the [[functional programming language|functional]] [[quantum programming language]] [[Quipper]]:

* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: A Scalable Quantum Programming Language_, ACM SIGPLAN Notices 48(6):333-342, 2013 ([arXiv:1304.3390](https://arxiv.org/abs/1304.3390))

* [[Jonathan Smith]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: Concrete Resource Estimation in Quantum Algorithms_, QAPL 2014 ([arXiv:1412.0625](https://arxiv.org/abs/1412.0625))

Exposition of the general idea of [[quantum programming languages]] for [[classically controlled quantum computation]] with an eye towards the [[Quipper]]-language:

* [[Benoît Valiron]], [[Neil J. Ross]], [[Peter Selinger]], [[D. Scott Alexander]], [[Jonathan M. Smith]], *Programming the quantum future*, Communications of the ACM **58** 8 (2015) 52–61 [[doi:10.1145/2699415](https://doi.org/10.1145/2699415), [pdf](https://www.iro.umontreal.ca/~brassard/cours/6155/Langages%20de%20programmation%20quantique/Programming%20the%20Quantum%20Future.pdf),  [web](https://cacm.acm.org/magazines/2015/8/189851-programming-the-quantum-future/abstract), promo vid:[YT](https://youtu.be/mzce69oFdH0?list=PLn0nrSd4xjjbIHhktZoVlZuj2MbrBBC_f)]

category: people

[[!redirects Jonathan M. Smith]]


Jonathan Sorce

* [personal page](http://www.jonsorce.com/)

* [GoogleScholar entry](https://scholar.google.com/citations?user=V1bCscgAAAAJ&hl=en)

## Selected writings

On [[holographic tensor networks]] and [[quantum error correcting codes]] in full [[AdS-CFT]]:

* [[Ning Bao]], [[Geoffrey Penington]], [[Jonathan Sorce]], [[Aron C. Wall]], _Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT_,  JHEP 2019:69 ([arXiv:1812.01171](https://arxiv.org/abs/1812.01171))

* [[Ning Bao]], [[Geoffrey Penington]], [[Jonathan Sorce]], [[Aron C. Wall]], _Holographic Tensor Networks in Full AdS/CFT_ ([arXiv:1902.10157](https://arxiv.org/abs/1902.10157))

On [[von Neumann algebra factors]] as algebras of [[quantum observables]] in [[quantum physics]] and [[quantum field theory]]:

* [[Jonathan Sorce]], *Notes on the type classification of von Neumann algebras* [[arXiv:2302.01958](https://arxiv.org/abs/2302.01958)]

category: people


Jonathan Sterling

* [website](http://www.jonmsterling.com/)

## Selected writings

On [[guarded recursion]]:

* [[Jonathan Sterling]], [[Robert Harper]], *Guarded Computational Type Theory*, LICS '18 (2018) 879–888 [[arxiv:1804.09098](https://arxiv.org/abs/1804.09098), [doi:10.1145/3209108.3209153](https://doi.org/10.1145/3209108.3209153)]

On [[categorical models of dependent types]] via [[locally Cartesian closed categories]]:

* [[Daniel Gratzer]], [[Jonathan Sterling]], *Syntactic categories for dependent type theory: sketching and adequacy* [[arXiv:2012.10783](https://arxiv.org/abs/2012.10783)]

On [[synthetic Tait computability theory]]:

* [[Jonathan Sterling]], _First Steps in Synthetic Tait Computability: The Objective Metatheory of Cubical Type Theory_ (2021) [[PhD thesis](https://www.jonmsterling.com/pdfs/sterling:2021:thesis.pdf)]

* [[Jonathan Sterling]], _Naïve logical relations in synthetic Tait computability_ (2022) [[pdf](https://www.jonmsterling.com/papers/sterling:2022:wg6.pdf)]

On [[XTT]]:

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _Cubical syntax for reflection-free extensional equality_. In Herman Geuvers, editor, _4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)_, volume 131 of _Leibniz International Proceedings in Informatics (LIPIcs)_, pages 31:1-31:25. ([arXiv:1904.08562](https://arxiv.org/abs/1904.08562), [doi:10.4230/LIPIcs.FCSD.2019.31](https://doi.org/10.4230/LIPIcs.FCSD.2019.31))

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _A Cubical Language for Bishop Sets_, Logical Methods in Computer Science, 18 (1), 2022. ([arXiv:2003.01491](https://arxiv.org/abs/2003.01491)).

## Related entries

* [[meaning explanation]]

* [[cubical type theory]]

* [[XTT]]

category: people

[[!redirects Jon Sterling]]
[[!redirects Jon Sterling]]

[[!redirects Jonathan Sterling]]


Jonathan Weinberger

* [Personal page](https://sites.google.com/view/jonathanweinberger)

## Selected writings

On [[synthetic (infinity,1)-category theory|synthetic $(\infty,1)$-category theory]] via [[simplicial type theory]]:

* {#BuchholtzWeinberger21} [[Ulrik Buchholtz]], [[Jonathan Weinberger]], *Synthetic fibered $(\infty,1)$-category theory* [[arXiv:2105.01724](https://arxiv.org/abs/2105.01724), [talk slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Weinberger-2022-01-20-HoTTEST.pdf)]

* [[Jonathan Weinberger]], *A Synthetic Perspective on $(\infty,1)$-Category Theory: Fibrational and Semantic Aspects* [[arXiv:2202.13132](https://arxiv.org/abs/2202.13132)]. PhD Thesis, TU Darmstadt, Germany, 2022.

* [[Jonathan Weinberger]], *Strict stability of extension types* [[arXiv:2203.07194](https://arxiv.org/abs/2203.07194)]

* [[Jonathan Weinberger]], *Two-sided cartesian fibrations of synthetic $(\infty,1)$-categories* [[arXiv:2204.00938](https://arxiv.org/abs/2204.00938)]

* [[Jonathan Weinberger]], *Internal sums for synthetic fibered $(\infty,1)$-categories* [[arXiv:2205.00386](https://arxiv.org/abs/2205.00386)]

On [[cubical-type model category|model structures on cubical sets]] and [[univalence axiom|univalent universes]]:

* [[Thomas Streicher]], [[Jonathan Weinberger]]: *Simplicial sets inside cubical sets* [[arXiv:1911.09594](https://arxiv.org/abs/1911.0959)], [Theory and Applications of Categories, Vol. 37, 2021, No. 10, pp 276–286](http://www.tac.mta.ca/tac/volumes/37/10/37-10abs.html)

Formalization of the [[(infinity,1)-Yoneda lemma|$(\infty,1)$-Yoneda lemma]] via [[simplicial homotopy type theory]] (in [[Rzk]]):

* [[Nikolai Kudasov]], [[Emily Riehl]], [[Jonathan Weinberger]]. *Formalizing the $\infty$-categorical Yoneda lemma* (2023) [[arXiv:2309.08340](https://arxiv.org/abs/2309.08340)]

category: people


Jonathan Wise

* [personal page](https://math.colorado.edu/~jonathan.wise/)

* [institute page](https://www.colorado.edu/math/jonathan-wise)

* [GoogleScholar page](https://scholar.google.com/citations?user=TI-lQKQAAAAJ&hl=en)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=128696)

## Selected writings

On the [[snake lemma]] (and the [[salamander lemma]]):

* {#Wise} [[Jonathan Wise]], *A non-elementary proof of the snake lemma* (2011, 2023) [[pdf](https://math.colorado.edu/%7Ejonathan.wise/papers/snake.pdf), [[Wise-SnakeLemma.pdf:file]] [MO:a/7531](https://mathoverflow.net/a/7531/381)]

category: people


Jonathan Woolf

* [personal webpage](http://pcwww.liv.ac.uk/~jonwoolf/)

* [staff webpage with publication list](https://www.liverpool.ac.uk/mathematical-sciences/staff/jonathan-woolf/publications/)

category:people
[[!redirects J. Woolf]]


Jonathon Funk

* [webpage](https://www.qcc.cuny.edu/mathCS/people/jfunk.html)

category:people


Jones polynomial

+-- {: .rightHandSide}
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###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

The _Jones polynomial_ is a [[knot invariant]]. It is a special case of the [[HOMFLY-PT polynomial]]. See there for more details.

## Properties

### Relation to 3d Chern-Simons theory

In ([Witten 89](#Witten89)) it was shown that the Jones polynomial as a polynomial in $q$ is equivalently the [[partition function]] of $SU(2)$-[[Chern-Simons theory]] with a [[Wilson loop]] specified by the given [[knot]] as a function of the exponentiated [[level (Chern-Simons theory)|level]] of the Chern-Simons theory. Extensive lecture notes on this are in ([Witten 13a](#Witten13a)).

### Relation to 4d super Yang-Mills theory

Later in ([Witten 11](#Witten11)) this identification was further refined to a correspondence between [[Khovanov homology]] and observables in 4-dimensional [[super Yang-Mills theory]]. Extensive lectures notes on this are in ([Witten 13b](#Witten13b)).

## Related concepts

* [[colored Jones polynomial]]

## References

### General

See also:

* Wikipedia, _[Jones polynomial](https://en.m.wikipedia.org/wiki/Jones_polynomial)_

An [[algorithm]] for computing the Jones polynomial on a [[topological quantum computer]] based on [[anyon statistics]]:

* [[Samuel J. Lomonaco]], [[Louis Kauffman]], _Topological Quantum Computing and the Jones Polynomial_, Proc. SPIE 6244, Quantum Information and Computation IV, 62440Z (12 May 2006) ([arXiv:quant-ph/0605004](https://arxiv.org/abs/quant-ph/0605004))

[[!include Jones polynomial as Wilson loop observable -- references]]

category: knot theory


Jones polynomial as Wilson loop observable -- references

### Jones polynomial as Wilson loop observables
  {#ReferencesJonesPolynomialAsWilsonLoopObservables}

The identification of the [[Jones polynomial]] with [[Wilson loop observables]] in [[Chern-Simons theory]] is due to

* {#Witten89} [[Edward Witten]], _Quantum field theory and the Jones polynomial_, Commun. Math. Phys. 121,351-399 (1989) ([euclid.cmp/1104178138](https://projecteuclid.org/euclid.cmp/1104178138))

see also

* [[Jürg Fröhlich]], C. King, _The Chern-Simons theory and knot polynomials_, Comm. Math. Phys.
Volume 126, Number 1 (1989), 167-199 ([euclid.cmp/1104179728](https://projecteuclid.org/euclid.cmp/1104179728))

Lecture notes:

* {#Witten13a} [[Edward Witten]], _A New Look At The Jones Polynomial of a Knot_, Clay Conference, Oxford, October 1, 2013 ([[WittenOnJonesPolynomial2013.pdf:file]])

The [[categorification]] of this relation to an identification of [[Khovanov homology]] with [[observables]] in [[D=4 super Yang-Mills theory]]:

* [[Edward Witten]], _Khovanov homology and gauge theory_, [arxiv/1108.3103](http://arxiv.org/abs/1108.3103)

* {#Witten11} [[Edward Witten]], _Fivebranes and Knots_ ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216))

Lecture notes:

* {#Witten13b} [[Edward Witten]], _Khovanov Homology And Gauge Theory_, Clay Conference, Oxford, October 2, 2013 ([[WittenOnKhovanovHomology2013.pdf:file]])

See also

* [[Vaughan Jones]], _Index for subfactors_, Invent. Math. __72__, I (I983); _A polynomial invariant for links via yon Neumann algebras_, Bull. AMS __12__, 103 (1985); _Hecke algebra representations of braid groups and link polynomials_, Ann. Math. __126__, 335 (1987)


Jones' theorem

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#### Higher algebra
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#Contents#
* table of contents
{:toc}

## Idea

Let $X$ be a [[simply connected topological space|simply connected]] [[topological space]].

The [[ordinary cohomology]] $H^\bullet$ of its free loop space is the [[Hochschild homology]] $HH_\bullet$ of its [[singular cohomology|singular chains]] $C^\bullet(X)$:

$$
  H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( C^\bullet(X) )
  \,.
$$

Moreover the $S^1$-equivariant cohomology of the loop space, hence the ordinary cohomology of the [[cyclic loop space]] $\mathcal{L}X/^h S^1$ is the [[cyclic homology]] $HC_\bullet$ of the singular chains:

$$
  H^\bullet(\mathcal{L}X/^h S^1)
    \simeq
  HC_\bullet( C^\bullet(X) )
$$

([Loday 11](#Loday11))

If the [[coefficients]] are [[rational numbers|rational]], and $X$ is of [[finite type]] then this may be computed by the _[[Sullivan model for free loop spaces]]_, see there the section on _[Relation to Hochschild homology](Sullivan+model+of+free+loop+space#RelationToHochschildHomology)_.

In the special case that the [[topological space]] $X$ carries the structure of a [[smooth manifold]], then the singular cochains on $X$ are equivalent to the [[dgc-algebra]] of [[differential forms]] (the [[de Rham algebra]]) and hence in this case the statement becomes that

$$
  H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( \Omega^\bullet(X) )
  \,.
$$

$$
  H^\bullet(\mathcal{L}X/^h S^1)
    \simeq
  HC_\bullet( \Omega^\bullet(X) )
  \,.
$$

This is known as _[[Jones' theorem]]_ ([Jones 87](#Jones87))

An [[(infinity,1)-category theory|infinity-category theoretic]] proof of this fact is indicated at _[Hochschild cohomology -- Jones' theorem](Hochschild+cohomology#JonesTheorem)_.

## References

* {#Jones87} [[John D.S. Jones]], _Cyclic homology and equivariant homology_, Invent. Math. __87__, 403-423 (1987) ([pdf](https://math.berkeley.edu/~nadler/jones.pdf))

* {#Loday11} [[Jean-Louis Loday]], _Free loop space and homology_ ([arXiv:1110.0405](https://arxiv.org/abs/1110.0405))

[[!redirects Jones theorem]]


Joonho Kim

* [webpage](https://www.ias.edu/scholars/joonho-kim)

## Selected writings

On [[E-strings]] and emergence of [[SU(2)]]  [[flavor (particle physics)|flavor]]-[[chiral symmetry|symmetry]] on [[M5-branes]] in [[heterotic M-theory]] (in the [[D=6 N=(1,0) SCFT]] on [[small instantons]] in [[heterotic string theory]]):

* {#GHKKLV15} [[Abhijit Gadde]], [[Babak Haghighat]], [[Joonho Kim]], [[Seok Kim]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _6d String Chains_, JHEP **1802** (2018) 143 [[arXiv:1504.04614](https://arxiv.org/abs/1504.04614)]

On [[black holes in string theory]]:

* {#CKJN18} Sunjin Choi, [[Joonho Kim]], [[Seok Kim]], June Nahmgoong, _Large AdS black holes from QFT_ ([arxiv:1810.12067](https://arxiv.org/abs/1810.12067))

category: people


Joost Nuiten

Joost Nuiten is a postdoc in mathematics at Université de Montpellier

Joost Nuiten did his PhD student in mathematics at Utrecht University with [[Ieke Moerdijk]].

Nuiten did his Master in mathematics and physics at Utrecht University with [[Urs Schreiber]].

## Texts

* bachelor thesis: _[[schreiber:bachelor thesis Nuiten|Bohrification of local nets of observables]]_, EPTCS 95, 2012, pp. 211-218 ([arXiv:1109.1397](http://arxiv.org/abs/1109.1397))

  On the [[Bohr topos]] formulation of [[local quantum field theory]] ([[AQFT]]).

* master thesis: _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_  (August 2013)

  On the [[quantization]] of [[boundary field theory|boundary]] [[local prequantum field theory]] by [[motivic quantization]].

Further preprints and publications:

* [[Ieke Moerdijk]], [[Joost Nuiten]], _Minimal fibrations of dendroidal sets_ ([arXiv:1509.01073](http://arxiv.org/abs/1509.01073))

  on [[minimal dendroidal fibrations]]

* [[Yonatan Harpaz]], [[Joost Nuiten]], [[Matan Prasma]], _Tangent categories of algebras over operads_ ([arXiv:1612.02607](https://arxiv.org/abs/1612.02607))

* _The abstract cotangent complex and Quillen cohomology of enriched categories_ ([arXiv:1612.02608](https://arxiv.org/abs/1612.02608))

  on [[model category]] presentations for [[tangent (infinity,1)-categories]] and the [[cotangent complex]]

* [[Joost Nuiten]], _Localizing infinity-categories with hypercovers_ ([arXiv:1612.03800](https://arxiv.org/abs/1612.03800))

  on [[localization]] of [[(infinity,1)-categories]]

category: people


Joost Slingerland

* [personal page](http://www.thphys.nuim.ie/staff/joost/)

* [institute page](https://www.maynoothuniversity.ie/people/joost-slingerland)

## Selected writings

On potential [[quantum measurement]] of non-abelian [[anyons]]:

* {#BondersonShtengelSlingerland07} [[Parsa Bonderson]], [[Kirill Shtengel]], [[Joost Slingerland]], *Decoherence of Anyonic Charge in Interferometry Measurements*, Phys. Rev. Lett. **98** (2007) 070401 [[doi:10.1103/PhysRevLett.98.070401](https://doi.org/10.1103/PhysRevLett.98.070401), [arXiv:quant-ph/0608119](https://arxiv.org/abs/quant-ph/0608119)]

* [[Parsa Bonderson]], [[Kirill Shtengel]], [[Joost Slingerland]], *Interferometry of non-Abelian Anyons*, Annals Phys. **323** (2008) 2709-2755 [[doi:10.1016/j.aop.2008.01.012](https://doi.org/10.1016/j.aop.2008.01.012), [arXiv:0707.4206](https://arxiv.org/abs/0707.4206)]

On [[vortex]]$\;$[[anyons]] in [[Bose-Einstein condensates]]:

* {#MPSS19} Thomas Mawson, Timothy Petersen, [[Joost Slingerland]], [[Tapio Simula]], *Braiding and fusion of non-Abelian vortex anyons*, Phys. Rev. Lett. **123** (2019) 140404 [[doi:10.1103/PhysRevLett.123.140404](https://doi.org/10.1103/PhysRevLett.123.140404)]

category: people


Joost Vercruysse

Joost Vercruysse works at the Université Libre de Bruxelles

His research ares include [[Hopf algebra]]s and [[Hopf algebroid]]s, [[coring]]s and [[coalgebra]]s, [[Hopf-Galois extension|Hopf-Galois]] theory and Galois descent, [[noncommutative geometry]] and quantum groups.

* [Homepage](http://homepages.vub.ac.be/~jvercruy/Homepage.html)

##Selected publications and preprints

* [[Gabriella Böhm]], Joost Vercruysse, _Morita theory for coring extensions and cleft bicomodules_, Advances in Mathematics __209__:2 (2007) 611-648 [doi/10.1016/j.aim.2006.05.010](https://doi.org/10.1016/j.aim.2006.05.010); _Corrigendum_, Adv. Math. __221__:2 (2009) 682-686 [doi/10.1016/j.aim.2008.11.018](https://doi.org/10.1016/j.aim.2008.11.018)
* [[Stefaan Caenepeel]], Joost Vercruysse, Shuanhong Wang, _Morita theory for corings and cleft entwining structures_, J. Algebra __276__1 (2004) 210-235 [doi/10.1016/j.jalgebra.2004.02.002](https://doi.org/10.1016/j.jalgebra.2004.02.002)

* [[T. Brzeziński]], A. Vazquez Marquez and J. Vercruysse, _The Eilenberg-Moore category and a Beck-type theorem for a Morita context_, Applied Categorical Structures, 19, (2011), 821–858.

category: people, algebra

[[!redirects J. Vercruysse]]


Jordan algebra

+-- {: .rightHandSide}
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#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Quantum physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A _Jordan algebra_ is an algebra that may not be [[associativity|associative]], but is [[commutative algebra|commutative]], subject to some further conditions which are modeled after the archetypical example: for $(A, \cdot)$ any [[associative algebra]], equipping it with the _symmetrized product_

$$
  x \circ y \coloneqq \frac{1}{2}(x y + y x)
$$

makes $(A, \circ)$ a Jordan algebra. It is this relation that originally motivated the notion in discussion of [[quantum mechanics]], for the symmetrized product and hence the Jordan algebra structure of the [[algebra of observables]] of a [[quantum mechanics|quantum mechanical]] system is what remains when one ignores the otherwise all-important [[commutators]] and hence the [[Hamiltonian]] flows on observables.

But later Jordan algebras have been studied largely for their own sake.

More recently, the [[Bohr topos]] associated to a noncommutative [[algebra of observables]] was found to really see the underlying Jordan algebra structure. See at _[[Bohr topos]]_ and _[[poset of commutative subalgebras]]_ for more on this.

## Definition

A **Jordan algebra** is a [[commutative algebra|commutative]] [[nonassociative algebra]] $J$ satisfying the __Jordan identity__ $(x y) (x x) = x (y (x x))$ for all $x,y$ in $J$.

It follows (via a nontrivial argument) that $J$ is [[power-associative algebra|power-associative]], and the Jordan identity generalizes to
$$ (x^m y) x^n = x^m (y x^n) $$
for [[natural numbers]] $m, n \geq 1$ (and, trivially, for $m, n \geq 0$ if there is an [[identity element]]).

Thus, we may equivalently define a Jordan algebra to be a commutative power-associative algebra $J$ such that for any $x \in J$, the operations of multiplication by powers $x^n$ ($n \ge 1$) all commute with each other.

If $k$ is a [[field]] whose [[characteristic]] is not $2$ (or is any [[commutative ring]] in which $2$ is invertible), then to any [[associative algebra|associative]] $k$-algebra $A$ with product $\cdot$, one associates a Jordan $k$-algebra with the same underlying vector space and whose Jordan product $\circ$ is given by

$$x\circ y \stackrel{def}{=} \frac{x\cdot y + y \cdot x}{2}.$$

Such Jordan algebras are called __special__ Jordan algebras; all others are called __[[exceptional Jordan algebra|exceptional]]__.

## Formally real Jordan algebras and their origin in quantum physics
 {#OriginInQuantumPhysics}

Jordan algebras had their origin in the study of the foundations of [[quantum mechanics|quantum theory]].  In 1932, [[Pascual Jordan]] tried to isolate some axioms that an 'algebra of [[observable]]s' should satisfy ([Jordan 32](#Jordan32)).

The unadorned phrase 'algebra' usually signals an [[associative algebra]], but this is not the kind of algebra Jordan was led to.  In both [[classical mechanics|classical]] and [[quantum
mechanics]], [[observables]] are closed under addition and multiplication by [[real number|real]] scalars.  In classical mechanics we can also multiply observables, but in quantum mechanics this becomes problematic.  After
all, given two [[bounded linear operator|bounded]] [[self-adjoint]] [[linear operators]] on a complex [[Hilbert space]], their product is self-adjoint if and only if they commute.

However, in quantum mechanics one can still raise an observable to a power and obtain another observable.  From squaring and taking real linear combinations, one can construct a commutative product using the [[polarization identity]]:

$$
  x \circ y \coloneqq \frac{1}{2}((x+y)^2 - x^2 - y^2)
                  = \frac{1}{2}(x y + y x)
  \,.
$$

This is sometimes called the _anti-commutator_ (or more precisely, half the anti-commutator). Notice that it is analogous to the more famous [[commutator]]

$$
  [x,y] \coloneqq x y - y x
$$

and that both together recover the full algebra of observables in that

$$
  x y = \frac{1}{2}[x,y] + x \circ y
$$

for all $x,y$. (A [[JLB-algebra]] is a [[Banach space]] equipped with the compatible structures of both a [[Jordan algebra]] and a [[Lie algebra]], and these are equivalent to $C^*$-[[C-star-algebra|algebras]] in just this way, using $\circ$ and ${\bullet} \coloneqq \frac{1}{2} \mathrm{i} [{-},{-}]$ as the operations.)

From the point of view of [[deformation quantization]] of [[Poisson manifolds]], one can read this as follows: the deformation quantization of a Poisson manifold $(X,\{-,-\})$ breaks up into two pieces:

1. the [[Poisson bracket]] $\{-,-\}$ on $C^\infty(X)$ deforms to the [[commutator]];

2. the pointwise multiplication on $C^\infty(X)$ deforms into the Jordan algebra structure.

This perspective on deformation quantization making the role of Jordan algebras explicit is mentioned for instance in ([Bates-Weinstein, p. 80](#BatesWeinstein)).

The symmetrized product $(-) \circ (-)$ is not associative, in general, but it is power-associative: any way of parenthesizing a product of copies of the same observable $x$ gives the same result.  This led Jordan to define what is now called a **formally real Jordan algebra**: a commutative and power-associative algebra $J$ satisfying
$$  x_1^2 + \cdots + x_n^2 = 0 \quad \implies \quad x_1 = \cdots = x_n = 0  $$
for all $n$.  The last condition (as in any [[formally real algebra]]) gives $J$ a partial ordering: if we write $x \le y$ when the element $y - x$ is a sum of squares, it says that
$$x \le y \; \& \; y \le x \quad \implies \quad x = y\,.$$
So, in a formally real Jordan algebra we can reasonably talk about one observable being 'greater' than another.

In fact the Jordan identity $(x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x)) $ is a _consequence_ of the above definition of formally real Jordan algebra.  So, every formally real Jordan algebra is a Jordan algebra (but not conversely).

For more on this see also at _[order-theoretic structure in quantum mechanics -- Relation to non-commutative geometry](order-theoretic+structure+in+quantum+mechanics#RelationToTheNonCommutativePhaseSpace)_.

## Examples

### Classification of formally real Jordan algebras
{#frc}

In 1934, Jordan published a paper with von Neumann and Wigner classifying finite-dimensional formally real Jordan algebras ([Jordan-vonNeumann-Wigner 34](#JordanvNeumannWigner34)).

They began by defining an **ideal** of a formally real
Jordan algebra $J$ to be a linear subspace $S \subseteq J$ such that $x \in S$ implies $x \circ y \in S$ for all $y \in J$.  Next they defined $J$ to be **simple** when its only ideals were $\{0\}$ and $J$ itself.  Then they proved that any finite-dimensional formally Jordan algebra is a direct sum of simple ones.

This reduced the classification problem to the task of classifying _simple_ finite-dimensional formally real Jordan algebras.  There are four families of these, and one exception:

* The $n \times n$ self-adjoint real matrices, $\mathfrak{h}_n(\mathbb{R})$,
with the product $a \circ b = {1\over 2}(a b + b a)$.

* The $n \times n$ self-adjoint complex matrices, $\mathfrak{h}_n(\mathbb{C})$,
with the product $a \circ b = {1\over 2}(a b + b a)$.

* The $n \times n$ self-adjoint quaternionic matrices, $\mathfrak{h}_n(\mathbb{H})$,
with the product $a \circ b = {1\over 2}(a b + b a)$.

* The $n \times n$ self-adjoint octonionic matrices, $\mathfrak{h}_n(\mathbb{O})$,
with the product $a \circ b = {1\over 2}(a b + b a)$, where $n \le 3$.

* The space $\mathbb{R}^n \oplus \mathbb{R}$ with the product $(x,t) \circ (x', t') = (t x' + t' x, x \cdot x' + t t').$

Here we say a square matrix with entries in the $\ast$-algebra $A$ is **hermitian** if it equals its conjugate transpose.   (Note that $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ are all $\ast$-algebras.)

Because the octonions are an [[alternative algebra]] but not associative, we cannot go beyond $3 \times 3$ matrices and still get a Jordan algebra.  The $1 \times 1$ self-adjoint octonionic matrices are just the real numbers, and the $2 \times 2$ ones are isomorphic to the spin factor $\mathbb{R}^9 \oplus \mathbb{R}$.   The $3 \times 3$ self-adjoint octonionic matrices form the [[Albert algebra]].

Jordan algebras in the fifth family are called **spin
factors**.   This family has some overlaps with the others.  Most notably:

* The Jordan algebra of $2 \times 2$ self-adjoint real matrices is isomorphic to the spin factor $\mathbb{R}^2 \oplus \mathbb{R}$.

* The Jordan algebra of $2 \times 2$ self-adjoint complex matrices is isomorphic to the spin factor $\mathbb{R}^3 \oplus \mathbb{R}$.

* The Jordan algebra of $2 \times 2$ self-adjoint quaternionic matrices is isomorphic to the spin factor $\mathbb{R}^5 \oplus \mathbb{R}$.

* The Jordan algebra of $2 \times 2$ self-adjoint octonionic matrices is isomorphic to the spin factor $\mathbb{R}^9 \oplus \mathbb{R}$.

Because the spin factor $\mathbb{R}^n \oplus \mathbb{R}$ can be identified with $(n+1)$-dimensional [[Minkowski space]], this sets up a relation between the real numbers, complex numbers, quaternions and octonions and Minkowski space in 3,4,6 and 10 dimensions --- a pattern which becomes important in [[string theory]].  For more details, see  [[division algebras and supersymmetry]].

* [[John Baez]], The octonions, Sec. 3.3: $\mathbb{O}P^1$ and Lorentzian geometry.  ([web](http://math.ucr.edu/home/baez/octonions/node11.html))

* [[John Baez]] and [[John Huerta]], Division algebras and supersymmetry I.  ([arXiv](http://arxiv.org/abs/0909.0551))

In 1983, Zelmanov drastically generalized the result of Jordan, von Neumann and Wigner by classifying all simple Jordan algebras, including infinite-dimensional ones ([Zelmanov 83](#Zelmanov83)).

### Exceptional Jordan algebras

Among the exceptional Jordan algebras over the [[real numbers]], there is a remarkable $27$-dimensional example: the **[[Albert algebra]]** $\mathbb{Al}$ of [[self-adjoint matrix|self-adjoint]] $3\times 3$ [[matrix|matrices]] over the [[octonion|octonions]] with the same formula as above for the product in terms of [[matrix product]]. Notice that the octonions and their matrices do *not* form associative algebras, but only [[alternative algebras]], so the Jordan identity for the Albert algebra is not automatic (it does not hold for all alternative algebras) but is a consequence of more special circumstances.

## Self-dual homogeneous convex cones

The formalism of Jordan algebras seems rather removed from the
actual practice of [[physics]], because in quantum theory we hardly ever take two observables $a$ and $b$ and form their Jordan product
${1\over 2}(a b + b a)$.  As hinted in the previous section, it is better to think of this operation as derived from the process of _squaring_ an observable, which is something we actually do.  But
still, one must ask: can we see the classification of finite-dimensional formally real Jordan algebras, and thus the special role of [[normed division algebras]], as arising from some [[axioms]] more closely  tied to [[quantum theory]] as physicists usually practice it?

One answer involves the Koecher--Vinberg classification of  self-dual homogeneous [[convex cone]]s.
Consider first the case of ordinary [[quantum theory]].  If a quantum
system has the [[Hilbert space]] $\mathbb{C}^n$, [[observables]] are described by
self-adjoint $n \times n$ complex [[matrix|matrices]]: elements of the Jordan algebra $\mathfrak{h}_n(\mathbb{C})$.  But matrices of this form that are nonnegative and have [[trace]] 1 also play another role.  They are called **[[density matrix|density matrices]]**, and they describe [[states]] of our quantum system: not just pure [[states]], but also more general mixed states.  The idea is that any density matrix $\rho \in \mathfrak{h}_n(\mathbb{C})$ allows us to define [[expectation value]]s of [[observable]]s $a \in \mathfrak{h}_n(\mathbb{C})$ via

\[
  \langle a \rangle = \tr(\rho a)
  \,.
\]

The map sending observables to their expectation values is
real-linear.  The fact that $\rho$ is nonnegative is equivalent to

\[
  a \ge 0 \; \implies \; \langle a \rangle \ge 0
\]

and the fact that $\rho$ has trace 1 is equivalent to

\[
  \langle 1 \rangle = 1
  \,.
\]

All of this generalizes to an arbitrary finite-dimensional formally real Jordan algebra $J$.  Any such algebra automatically has an [[identity]] element.  This lets us define a **[[state]]** on $J$ to be a [[linear functional]] $\langle \cdot \rangle : J \to \mathbb{R}$ that is

* **nonnegative**:
  $$
    a \ge 0  \implies \langle a \rangle \ge 0
  $$

* and **normalized**:

  $$
    \langle 1 \rangle = 1
    \,.
  $$

But in fact, there is a [[bijection|bijective]] correspondence between
linear functionals on $J$ and elements of $J$.  The reason is that
every finite-dimensional Jordan algebra has a **[[trace]]**

$$
  \mathrm{tr} : J \to \mathbb{R}
$$

defined so that $\mathrm{tr}(a)$ is the trace of the linear operator 'multiplication by $a$'.  Such a Jordan algebra is then formally real if and only if

$$
  \langle a, b \rangle = \mathrm{tr}(a \circ b)
$$

is a real-valued [[inner product]]. So, when $J$ is a finite-dimensional formally real Jordan algebra, any linear functional
$\langle \cdot \rangle : J \to \mathbb{R}$ can be written as

$$
  \langle a \rangle = \mathrm{tr}(\rho \circ a)
$$

for a unique element $\rho \in J$.  Conversely, every element $\rho
\in J$ gives a linear functional by this formula.  While not obvious,
it is true that the linear functional $\langle \cdot \rangle$ is
nonnegative if and only if $\rho \ge 0$ in terms of the ordering on
$J$.  More obviously, $\langle \cdot \rangle$ is normalized if and
only if $\mathrm{tr}(\rho) = 1$.  So, states can be identified with certain special observables: namely, those observables $\rho \in J$ with $\rho \ge 0$ and $\mathrm{tr}(\rho) = 1$.

These ideas help motivate an important theorem of Koecher and Vinberg.  The idea is to axiomatize the situation we we have just described, in a way that does not mention the Jordan product in $J$, but instead emphasizes:

* the [[isomorphism]] between $J$ and its [[dual vector space|dual space]]

* the fact that 'positive' elements of $J$ form a [[cone]].

To find appropriate axioms, suppose $J$ is a finite-dimensional
formally real Jordan algebra.  Then seven facts are always
true.

1. The set of positive observables

   $$
     C = \{a \in A \colon a \gt 0\}
     \,.
   $$

   is a **[[cone]]**: that is, $a \in C$ implies that every positive
multiple of $a$ is also in $C$.

1. This cone is **[[convex space|convex]]**:

   if $a,b \in C$ then any linear combination $x a + (1-x) b$ with
$0 \le x \le 1$ also lies in $C$.

1. It is an open set.

1. It is **regular**, meaning that if $a$ and $-a$ are both in the closure $\overline{C}$, then $a = 0$.

   This condition may seem obscure, but if we note that

   $$
     \overline{C} = \{ a \in J \colon a \ge 0 \}
   $$

   we see that $C$ being regular simply means

   $$
    a \ge 0 \; and \; -a \ge 0 \quad \implies \quad a = 0 ,
   $$

   a perfectly plausible assumption.

1. Recall that $J$ has an inner product; this is what lets us
identify linear functionals on $J$ with elements of $J$.  This also
lets us define the **dual cone**

   $$
     C^* = \{  a \in J \colon \forall b \in A \; \; \langle a,b \rangle \gt 0 \}
   $$

   which one can check is indeed a cone.

   The fifth fact about $C$ is that it is **self-dual**, meaning $C = C^*$.

   This formalizes the fact that states may be identified with special observables.

1. $C$ is **homogeneous**: given any two points $a,b \in C$, there is a real-linear transformation $T : A \to A$ mapping $C$ to itself in a [[bijection|bijective]] way, with  the property that $T a = b$.  This says that cone $C$ is highly symmetrical: no point of $C$ is any 'better' than any other, at  least if we only consider the linear structure of the space $A$, ignoring the Jordan product and the trace.

1. From another viewpoint, however, there is a very special point of $C$, namely the identity $1$ of our Jordan algebra.  And this brings us to our seventh and final fact: the cone $C$ is **pointed** meaning
that it is equipped with a distinguished element (in this case $1 \in
C$).

In short: when $J$ is a finite-dimensional formally real Jordan
algebra, $C$ is a pointed homogeneous self-dual regular open convex
cone.  All the elements $a \in J$ are positive observables, but
certain special ones, namely those with $\langle a, 1 \rangle = 1$,
can also be viewed as states.

In fact, there is a category of pointed homogeneous self-dual regular
open convex cones, where:

* An object is a finite-dimensional real inner product space
$V$ equipped with a pointed homogeneous self-dual regular open convex cone $C \subset V$.

* A morphism from one object, say $(V,C)$, to another, say $(V',C')$,
is a linear map $T : V \to V'$ preserving the inner product and
mapping $C$ into $C'$.

Now for the payoff.  The work of Koecher and Vinberg, nicely explained in Koecher's  Minnesota notes ([Koecher](#Koecher)) shows that:

+-- {: .num_theorem}
###### Theorem

The [[category]] of pointed homogeneous self-dual
regular open convex cones is [[equivalence of categories|equivalent]] to the category of finite-dimensional formally real Jordan algebras.

=--

This means that the theorem of Jordan, von Neumann and Wigner also classifies the pointed homogeneous self-dual regular convex cones!

+-- {: .num_theorem}
###### Theorem

Every pointed homogeneous self-dual regular open convex cone is [[isomorphic]] to a [[direct sum]] of those on this list:

* the cone of positive elements in $\mathfrak{h}_n(\mathbb{R})$,

* the cone of positive elements in $\mathfrak{h}_n(\mathbb{C})$,

* the cone of positive elements in $\mathfrak{h}_n(\mathbb{H})$,

* the cone of positive elements in $\mathfrak{h}_3(\mathbb{O})$,

* the future lightcone in $\mathbb{R}^n \oplus \mathbb{R}$.

=--

Some of this deserves a bit of explanation.  For $\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}$, an element $T \in \mathfrak{h}_n(\mathbb{K})$ is **positive** if and only if the corresponding operator $T : \mathbb{K}^n \to \mathbb{K}^n$ has
$$            \langle v, T v \rangle \gt 0  $$
for all nonzero $v \in \mathbb{K}^n$.  A similar trick works for defining positive elements of $\mathfrak{h}_3(\mathbb{O})$, but we do not need the details here. We say an element $(x,t) \in \mathbb{R}^n \oplus \mathbb{R}$ lies in the **future lightcone** if $t \gt 0$ and $t^2 - x \cdot x \gt 0$.  This of course fits in nicely with the idea that the spin factors are connected to Minkowski spacetimes.  Finally, there is an obvious notion of direct sum for Euclidean spaces with cones, where the direct sum of $(V,C)$ and $(V',C')$ is $V \oplus V'$ equipped with the cone
$$             C \oplus C' = \{(v,v') \in V\oplus V' \colon \;
v \in C, v' \in C' \} . $$

In short: finite-dimensional formally real Jordan algebras arise fairly naturally as observables starting from a formalism where nonnegative observables form a cone, as long as we insist on some properties of this cone.

## Relation to commutative subalgebras

For every associative algebra there is its [[semilattice of commutative subalgebras]]  $ComSub(A)$. At least for $A, B$ [[von Neumann algebra]]s without type $I_2$ [[von Neumann algebra factor]]-subfactors, the isomorphisms $ComSub(A) \to ComSub(B)$ correspond to isomorphisms between the corresponding Jordan algebras $A_J \to B_J$.

For more details see [[semilattice of commutative subalgebras]].

## Related concepts

* [[Jordan-Lie-Banach algebra]]

* [[JBW-algebra]], [[JBW-algebraic quantum mechanics]]

* [[Kantor-Koecher-Tits construction]]

* [[order-theoretic structure in quantum mechanics]]

  * [[Alfsen-Shultz theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

* [[W-algebra]]

* [[Jordan superalgebra]]

* [[exceptional Jordan algebra]]/[[Albert algebra]]

## References

The original articles:

* {#Jordan32} [[Pascual Jordan]], *Über eine Klasse nichtassociativer hyperkomplexer Algebren*, Nachr. Ges. Wiss. Göttingen (1932) 569-575 [[eudml:59403](https://eudml.org/doc/59403), [[Jordan-Algebren.pdf:file]]]

* {#JordanvNeumannWigner34} [[Pascual Jordan]], [[John von Neumann]], [[Eugene Wigner]], *On an algebraic generalization of the quantum mechanical formalism*, Ann. Math. **35** (1934) 29-64 [[jstor:1968117](https://www.jstor.org/stable/1968117), [doi:10.1007/978-3-662-02781-3_21](https://doi.org/10.1007/978-3-662-02781-3_21)]

Textbooks:

*  Harald Hanche-Olsen and Erling Stormer: _Jordan Operator Algebras_, Pitman, 1984.  ([web](http://www.math.ntnu.no/~hanche/joa/))

* Nathan Jacobson, _Structure and Representations of Jordan Algebras_, American Mathematical Society, 1968.

* Kevin McCrimmon, _A Taste of Jordan Algebras_, Springer, 2006.  ([pdf](http://math.nsc.ru/LBRT/a1/files/mccrimmon.pdf))

* {#SpringerVeltkamp00} [[Tonny Springer]], [[Ferdinand Veldkamp]], Chapter 5 of _Octonions, Jordan Algebras, and Exceptional Groups_, Springer Monographs in Mathematics, 2000.

* Harald Upmeier, _Jordan Algebras in Analysis, Operator Theory, and
Quantum Mechanics_, AMS, 1987.

Introductions and surveys include:

* Kevin McCrimmon, _Jordan algebras and their applications_,
_Bull. Amer. Math. Soc._ **84** (1978), 612--627.  ([AMS website](http://www.ams.org/bull/1978-84-04/S0002-990\
4-1978-14503-0/home.html)) and ([Project Euclid website](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540925)).

* {#Koecher} Max Koecher, _The Minnesota Notes on Jordan Algebras and
Their Applications_, eds. Aloys Krieg and Sebastican Walcher, _Lecture Notes in Mathematics_ **1710**, Springer, Berlin, 1999. doi:[10.1007/BFb0096285](http://dx.doi.org/10.1007/BFb0096285) (paywalled)

* [[Paul Townsend]], _The Jordan formulation of quantum mechanics: a review_ ([arXiv:1612.09228](https://arxiv.org/abs/1612.09228))

More on the physical motivation for regarding any [[algebra of quantum observables]] (just) as a Jordan algebra:

* [[John Baez]], *Jordan algebras*, Section 4 of: *Getting to the Bottom of Noether's Theorem* [[arXiv:2006.14741](https://arxiv.org/abs/2006.14741)]

See also:

* Wikipedia, *[Jordan algebra](https://en.wikipedia.org/wiki/Jordan_algebra)*

* {#Zelmanov83} E. I. Zelmanov, On prime Jordan algebras. II,
_Sibirsk Mat. J._ **24** (1983), 89-104.

Remarks on Jordan algebras as [[algebras of observables]] in quantum physics are for instance in

* {#BatesWeinstein} Sean Bates, [[Alan Weinstein]], p. 80 of: _[[Lectures on the geometry of quantization]]_, Berkeley Mathematics Lecture Notes series, American Mathematical Society (1997) [[ams:bmln-8](https://bookstore.ams.org/bmln-8), [pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf), [[BatesWeinstein-Lectures.pdf:file]]]

Discussion of [[spectral triples]] over Jordan algebras in the [[Connes-Lott model]]:

* [[Latham Boyle]], [[Shane Farnsworth]], _The standard model, the Pati-Salam model, and "Jordan geometry"_ ([arxiv:1910.11888](https://arxiv.org/abs/1910.11888))

* [[Shane Farnsworth]], _The geometry of physical observables_ ([arXiv:2003.01708](https://arxiv.org/abs/2003.01708))

* Fabien Besnard, Shane Farnsworth, *Particle models from special Jordan backgrounds and spectral triples* $[$[arXiv:2206.07039](https://arxiv.org/abs/2206.07039)$]$

[[!redirects Jordan algebra]]
[[!redirects Jordan algebras]]

[[!redirects formally real Jordan algebra]]
[[!redirects formally real Jordan algebras]]

[[!redirects special Jordan algebra]]
[[!redirects special Jordan algebras]]


Jordan content

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

A *Jordan content* is a kind of *[[content (measure theory)|content]]* which may be used to define the [[Riemann integral]] of [[indicator functions]] over a [[bounded set]], in the same way that [[Lebesgue measures]] are used to define the [[Lebesgue integral]] of [[indicator functions]].

## See also

* [[content (measure theory)]]

* [[Riemann integration]]

* [[Lebesgue measure]]

## References

* [[James Munkres]], *Analysis on Manifolds*, Westview Press (1991) ISBN:0-201-31596-3

* Wikipedia, *[Jordan content](https://en.wikipedia.org/wiki/Jordan_content)*

[[!redirects Jordan contents]]
[[!redirects Jordan measure]]
[[!redirects Jordan measures]]


Jordan curve

# Jordan curves
* table of contents
{: toc}

## Definition

A __continuous simple closed [[curve]]__, or __Jordan curve__, in a [[topological space]] (or [[convergence space]], [[locale]], etc) $X$ is the [[image]] of a [[continuous map|continuous]] [[injection]] to $X$ from the [[unit circle]] $S^1$.  (This map itself is a continuous [[parametrization]] of the curve.)

The word 'continuous' is generally assumed, so that one speaks simply of a __simple closed curve__.  (If 'closed' is removed, then the domain is taken to be the [[unit interval]] $B^1$ instead of $S^1$.  If 'simple' is removed, then the map is no longer assumed injective.  If the image alone is not sufficient data, then the word 'parametrized' may be added to indicate the map itself while thinking of the map as its image.)

Similarly, a __Jordan [[surface]]__ in $X$ is the image of a continuous injection to $X$ from the [[unit sphere]] $S^2$.  This can be generalized to higher-dimensional [[spheres]] or other domains, so long as there is an appropriate term to use in place of 'curve' and 'surface'.  In particular, if $X$ has [[dimension]] $n$ (in some understood sense), then a __Jordan [[hypersurface]]__ in $X$ is the image of a continuous injection to $X$ from $S^{n-1}$.

### In cohesive homotopy type theory

In [[cohesive homotopy type theory]], let the [[continuum]] [[line object]] $\mathbb{A}$ be a [[commutative ring]] such that the [[shape]] of $\mathbb{A}$ is contractible $\mathrm{isContr}(\esh(\mathbb{A}))$.

A Jordan curve is a type $J$ whose shape is equivalent to the [[circle type]]: $\esh(J) \simeq S^1$. Similarly, a Jordan surface is a type $J$ whose shape is equivalent to the sphere type $\esh(J) \simeq S^2$, and a $n$-dimensional Jordan hypersurface is a type $J$ whose shape is equivalent to the $(n-1)$-sphere type $S^{n-1}$.

Commonly seen examples include in [[Euclidean geometry]] where $\mathbb{A}$ is the [[Dedekind real numbers]] $\mathbb{R}$, and [[algebraic geometry]] where $\mathbb{A}$ is the [[affine line]] $\mathbb{A}^1$.

## Related entries

* [[Cauchy integral formula]] (about Jordan curves in the [[complex plane]] $\mathbb{C}$)

* [[Jordan curve theorem]] (about Jordan curves in the [[Cartesian plane]] $\mathbb{R}^2$)

* [[Jordan--Brouwer separation theorem]] (about Jordan hypersurfaces in any [[Cartesian space]] $\mathbb{R}^n$)

## References

* Eric Weisstein, _[Jordan curve](http://mathworld.wolfram.com/JordanCurve.html)_

[[!redirects Jordan curve]]
[[!redirects Jordan curves]]
[[!redirects simple closed curve]]
[[!redirects simple closed curves]]
[[!redirects continuous simple closed curve]]
[[!redirects continuous simple closed curves]]


Jordan curve theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The _Jordan curve theorem_ is a basic fact in [[topology]]. It stands out as having a statement that intuitively seems completely obvious, but which turns out to require a non-trivial [[formal proof]].

The theorem states that every [[continuous function|continuous]] [[loop]] (where a loop is a closed [[curve]]) in the [[Euclidean plane]] which does not intersect itself (a _[[Jordan curve]]_) divides the [[plane]] into two [[disjoint subsets]] (the [[connected components]] of the curve\'s [[complement]]), a [[bounded subspace|bounded]] region inside the curve, and an unbounded region outside of it, each of which has the original curve as its [[boundary]].

## In constructive mathematics

Given a notion of real number $\mathbb{R}$ in constructive mathematics, a [[Jordan curve]] consists of a [[set]] $J$ with a specific [[injection]] $i:J \to \mathbb{R}^2$ into the Euclidean plane $\mathbb{R}^2$, and a [[homeomorphism]] $h:\mathbb{S}^1 \to \mathrm{im}(i)$ from the topological [[unit circle]] $\mathbb{S}^1$ to the [[image]] of the subset injection.

...

section under construction

...

In [[constructive mathematics]], the Jordan curve theorem is stated as

\begin{theorem}
Given a Jordan curve $J$, two points $a$ and $b$ can be constructed off of $J$, such that given any point $c$ off of $J$, we can construct a polygonal path that is bounded away from $J$, and that joins $c$ to one of $a$ or $b$, and any polygonal path joining $a$ and $b$ comes arbitrarily close to $J$.
\end{theorem}

## In cohesive homotopy type theory

In cohesive homotopy type theory, the Jordan curve theorem says that:

Let $J$ be a Jordan curve in the [[Euclidean plane]] $J \subseteq \mathbb{R}^2$ with injection $i:J \hookrightarrow \mathbb{R}^2$, and let $\mathbb{R}^2 \setminus J$ be the set of all points in $\mathbb{R}^2$ which is away from all points in the Jordan curve $J$.
$$\mathbb{R}^2 \setminus J \coloneqq \sum_{x:R^2} \prod_{a:J} \vert x - i(a) \vert \gt 0$$
The shape of $\mathbb{R}^2 \setminus J$ is equivalent to $\mathbb{1} + S^1$, where $\mathbb{1}$ is the [[unit type]] and $S^1$ is the [[circle type]].
$$\esh(\mathbb{R}^2 \setminus J) \simeq \mathbb{1} + S^1$$

## Generalization

The __Jordan--Brouwer separation theorem__ states that, in the [[Cartesian space]] $\mathbb{R}^n$, every simple closed continuous [[hypersurface]] (defined as the [[image]] of a [[continuous map|continuous]] [[injection]] from the [[sphere]] $S^{n-1}$) has an [[exterior]] with two [[connected components]], one [[bounded subspace|bounded]] (the inside of the hypersurface) and one unbounded, each of which has the original hypersurface as its [[boundary]].  (Furthermore, the continuous map defining the hypersurface can be extended to an [[automorphism|auto]]-[[homeomorphism]] of $\mathbb{R}^n$ under which the inside and outside appear as the images of the inside and outside of the sphere.)

## Related theorems

* [[Tietze extension theorem]]

* [[topological invariance of dimension]]

* [[Brouwer's fixed point theorem]]

## References

*  G. Berg, W. Julian, R. Mines, F. Richman, *The constructive Jordan curve theorem*,  Rocky Mountain J. Math. 5(2): 225-236 (Spring 1975). ([DOI:10.1216/RMJ-1975-5-2-225](https://doi.org/10.1216/rmj-1975-5-2-225))

* Wikipedia, _[Jordan curve theorem](https://en.wikipedia.org/wiki/Jordan_curve_theorem)_

[[!redirects Jordan curve theorem]]
[[!redirects Jordan Curve Theorem]]

[[!redirects Jordan-Brouwer theorem]]
[[!redirects Jordan-Brouwer Theorem]]
[[!redirects Jordan–Brouwer theorem]]
[[!redirects Jordan–Brouwer Theorem]]
[[!redirects Jordan--Brouwer theorem]]
[[!redirects Jordan--Brouwer Theorem]]
[[!redirects Jordan-Brouwer separation theorem]]
[[!redirects Jordan-Brouwer Separation Theorem]]
[[!redirects Jordan–Brouwer separation theorem]]
[[!redirects Jordan–Brouwer Separation Theorem]]
[[!redirects Jordan--Brouwer separation theorem]]
[[!redirects Jordan--Brouwer Separation Theorem]]

[[!redirects Jordan's curve theorem]]


Jordan S. Ellenberg

* [personal page](https://people.math.wisc.edu/~ellenberg/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Jordan_Ellenberg)

## Selected writings

Introducing the notion of *[[FI-modules]]* ([[FinInj|$FinSet_{inj}$]]-[[representations]]):

* [[Thomas Church]], [[Jordan S. Ellenberg]], [[Benson Farb]], *FI-modules and stability for representations of symmetric groups*, Duke Math. J. **164** 9 (2015) 1833-1910 [[arXiv:1204.4533](https://arxiv.org/abs/1204.4533), [doi:10.1215/00127094-3120274](https://doi.org/10.1215/00127094-3120274)]

* [[Thomas Church]], [[Jordan S. Ellenberg]], [[Benson Farb]], [[Rohit Nagpal]], *FI-modules over Noetherian rings*, Geom. Topol. **18** (2014) 2951-2984 [[arXiv:1210.1854](https://arxiv.org/abs/1210.1854), [doi:10.2140/gt.2014.18.2951](https://doi.org/10.2140/gt.2014.18.2951)]

category: people

[[!redirects Jordan Ellenberg]]


Jordan superalgebra

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#### Super-Algebra and Super-Geometry
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[[!include supergeometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The notion of _Jordan superalgebra_ is the [[internalization|analog]] in [[superalgebra]]/[[supergeometry]]
of that of *[[Jordan algebra]]*:

A Jordan superalgebra is a [[supercommutative superalgebra]]
with [[underlying]] $\mathbb{Z}/2$-[[graded algebra]] $J = J_0 \oplus J_1$, where:

* $J_0$ is an ordinary [[Jordan algebra]],

* $J_1$ is a $J_0$-[[bimodule]] with a “[[Lie bracket]]-like” product into $J_0$,

* satisfying a super Jordan identity.

## Classification

Simple Jordan superalgebras over an [[algebraically closed field]] of [[characteristic 0]] were classified by Kac ([Kac 77](#Kac77)). The only [[exceptional structure|exceptional]] finite-dimensional example is the 10-dimensional Jordan superalgebra $K_{10}$.

## References

* {#Kac77} [[Victor Kac]], _Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras_, Comm. Algebra 5(1977), 1375–1400, ([doi: 10.1080/00927877708822224]( https://doi.org/10.1080/00927877708822224))

* [[Isaiah Kantor]], _Connections between Poisson brackets and Jordan and Lie superalgebras, Lie theory, differential equations and representation theory_, Montreal (1989), 213–225.

* [[Irving Kaplansky]], _Graded Jordan algebras I_ ([preprint](https://www1.osu.cz/~zusmanovich/links/files/kaplansky/gja.pdf)).

* Consuelo Martínez, _Simplicity of Jordan Superalgebras and Relations with Lie Structures_,  Irish Math. Soc. Bulletin 50(2003), 97–116 ([pdf](http://www.irishmathsoc.org/bull50/R5005.pdf))

* M. E. Martin, _Classification of three-dimensional Jordan superalgebras_ ([arXiv:1708.01963](https://arxiv.org/abs/1708.01963))

* M. L. Racine, E. I. Zel’manov, _An octonionic construction of the Kac superalgebra $K_10$_,
Proc. Amer. Math. Soc. 143 (2015), 1075-1083, ([pdf](https://www.ams.org/journals/proc/2015-143-03/S0002-9939-2014-12327-6/S0002-9939-2014-12327-6.pdf)).

[[!redirects super Jordan algebra]]
[[!redirects Jordan superalgebras]]


Jordan Watts

* [webpage](http://people.cst.cmich.edu/watts1j/index.php)

## Selected writings

* {#Watts15} [[Jordan Watts]], _The Differential Structure of an Orbifold_, Rocky Mountain Journal of Mathematics, Vol. 47, No. 1 (2017), pp. 289-327 ([arXiv:1503.01740](https://arxiv.org/abs/1503.01740))

  (on [[orbifolds]] regarded as naive quotient [[diffeological spaces]])

## Related $n$Lab entries

* [[orbifold]]

* [[diffeological space]]

category: people


Jordan Williamson

* [personal page](https://sites.google.com/view/jordanwilliamson/home)

## Selected writings

On ([[monoidal localization|monoidal]]) [[Bousfield localization of model categories|Bousfield localization]] of [[stable model categories]]:

* [[Luca Pol]], [[Jordan Williamson]], *The Left Localization Principle, completions, and cofree $G$-spectra*, J. Pure Appl. Algebra **224** 11 (2020) 106408 [[arXiv:1910.01410](https://arxiv.org/abs/1910.01410), [doi:10.1016/j.jpaa.2020.106408](https://doi.org/10.1016/j.jpaa.2020.106408)]

category: people


Jordan-Banach algebra

# Jordan--Banach algebras
* table of contents
{: toc}

## Idea

Jordan--Banach algebras, $JB$-algebras, and the like fill out the following grand [[analogy]]:

| | | |
|-|-|-|
| [[associative algebra|associative]] $*$-[[star-algebra|algebra]] | [[Jordan algebra]] | [[Jordan–Lie algebra]] |
| associative [[Banach algebra|Banach]] $*$-algebra | Jordan--Banach algebra | [[Jordan–Lie–Banach algebra]] |
| $C^*$-[[C-star-algebra|algebra]] | $JB$-algebra | $JLB$-[[JLB-algebra|algebra]] |
| [[von Neumann algebra]] | $JBW$-algebra | $JLBW$-[[JLBW-algebra|algebra]] |

Just as a [[Jordan algebra]] that happens to be associative is the same thing as an [[associative algebra|associative]] $*$-[[star-algebra|algebra]] with trivial [[involution]] (aka simply an associative algebra) that happens to be [[commutative algebra|commutative]], the analogous result holds in the lower rows.

One can also consider Jordan $*$-algebras and the like, but the interesting thing is that important results about $C^*$-[[C-star-algebra|algebras]] have analogues already for $JB$-algebras. Instead of an involution, we can add a compatible [[Lie algebra]] structure to a Jordan algebra; then even *without* assuming associativity or commutativity, a [[Jordan–Lie algebra]] over the [[real numbers]] is the same thing as an associative $*$-algebra over the [[complex numbers]], up to [[equivalence of categories]], and this extends to lower rows.

The right column is discussed at [[Jordan–Lie–Banach algebra]]; here we discuss the middle column (assuming the top row and left column as known).

## Definitions

Let $A$ be a [[Banach space]], typically over the [[real numbers]], but potentially over any [[topological field]] (or possibly even more general).  We will generally assume the real numbers, and some theorems may rely on this, or at least on the divisibility of $2$ in the [[ground field]].

+-- {: .num_defn}
###### Definition

The Banach space $A$ becomes a __Jordan--Banach algebra__ if it is equipped with a [[binary operation]] $A \times A \to A$, called the _Jordan multiplication_ and often written infix as $\circ$, satisfying these identities:

* [[short map|shortness]]: ${\|x \circ y\|} \leq {\|x\|} {\|y\|}$,
* [[bilinear map|bilinearity]]: $(a x + y) \circ (b u + v) = a b (x \circ u) + a (x \circ v) + b (y \circ u) + y \circ v$,
* commutativity: $x \circ y = y \circ x$,
* Jordan identity: $(x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x))$.
=--

For motivation of the first two, see [[Banach algebra]]; for motivation of the last three, see [[Jordan algebra]].

+-- {: .num_defn}
###### Definition

The Jordan--Banach algebra $A$ is __unital__ if the Jordan multiplication has an [[identity element]] in its unit ball, usually denoted $1$:

* $1 \circ x = x$,
* $x \circ 1 = x$,
* ${\|1\|} \leq 1$.
=--

People might state the last clause as ${\|1\|} = 1$, which follows (using shortness of the multiplication) from the existence of any $x \ne 0$.  However, ${\|1\|} = 0$ in the [[trivial ring|trivial algebra]], and this should be allowed.

+-- {: .num_defn}
###### Definition

The Jordan--Banach algebra $A$ is a __$JB$-algebra__ if it satisfies these identities:

* __$B$-identity__: ${\|x \circ x\|} = {\|x\|^2}$,
* _[[formally real algebra|positivity]]_: ${\|x \circ x\|} \leq {\|x \circ x + y \circ y\|}$.
=--

Shortness of the multiplication follows from the $B$-identity (via the [[polarization identities]] and the triangle identity), so it may be left out of a direct definition of $JB$-algebras; the same goes for the norm of $1$ in the unital case.  (Compare the analogous results for $C^*$-[[C-star-algebra|algebras]].)  Conversely, given shortness of multiplication (or even of squaring), these two identities may be combined into the single inequality
$$ {\|x\|}^2 \leq {\|x \circ x + y \circ y\|} .$$

Also, positivity implies that $A$ is [[formally real algebra|formally real]]: if $x^2 + y^2 = 0$, then $x, y = 0$ (and so on for any number of terms).  Given this, the Jordan identity is equivalent to [[power-associative algebra|power-associativity]] (which it implies regardless) in a direct definition of $JB$-algebra.

+-- {: .num_defn}
###### Definition

The $JB$-algebra $A$ is a __$JBW$-algebra__ if it is unital and if its [[underlying]] Banach space has a [[predual]] $A_*$.
=--

See [[von Neumann algebra]] for motivation of the predual.  As with $C^*$-algebras, the predual (if it exists) is [[unique up to unique isomorphism]].

+-- {: .num_defn}
###### Definition

A __[[homomorphism]]__ from a Jordan--Banach algebra $A$ to a Jordan--Banach algebra $B$ is a [[bounded linear map]] $T\colon A \to B$ of Banach spaces (everywhere defined) such that $T(x \circ y) = T(x) \circ T(y)$ always holds.  If $A$ and $B$ are unital, then the homomorphism is __unital__ if $T(1) = 1$.
=--

To get the right notion of [[isomorphism]], the [[morphisms]] in the [[category]] of Jordan--Banach algebras should be the *[[short linear map|short]]* homomorphisms (see [Ban#morphisms](Ban#morphisms) for discussion).  However, if $A$ and $B$ are $JB$-algebras, then every homomorphism is short (as with $C^*$-algebras); in fact, we do not even have to assume that $T$ is bounded.  Similarly, a morphism of unital Jordan--Banach algebras should be unital.

+-- {: .num_defn}
###### Definition

Given a Jordan--Banach algebra $A$ and a [[Hilbert space]] $H$, a __[[representation]]__ $\pi$ of $A$ on $H$ is a homomorphism from $A$ to the algebra of [[Hermitian operators]] on $H$ (which is a $JB$-algebra, in fact a $JBW$-algebra, under the symmetrized product, as described in the examples).  Such a representation is __[[faithful representation|faithful]]__ if it\'s an [[injective function]].
=--

Again, we should really look only at the short representations (which is automatic for a $JB$-algebra) and look especially at the unital representations of a unital Jordan--Banach algebra.

Recall that a Jordan algebra is __[[special Jordan algebra|special]]__ if it has a faithful representation on a [[vector space]] (which follows if it has any injective homomorphism to the Jordanization of any associative algebra).

+-- {: .num_defn}
###### Definition

A __$JC$-algebra__ is $JB$-algebra that has a faithful representation on a Hilbert space.  A __$JW$-algebra__ is a $JBW$-algebra that\'s also a $JC$-algebra.
=--

This follows if the $JB$-algebra has any injective homomorphism to the algebra of [[Hermitian operators]] in any $C^*$-algebra, by the theorem that every abstract $C^*$-algebra may be made concrete.  But this theorem does *not* itself have an analogue for $JB$-algebras; the [[Albert algebra]] $\mathfrak{h}_3(\mathbb{O})$ is a $JB$-algebra (even a $JBW$-algebra) that is not a $JC$-algebra.

This is probably unfortunate terminology (compare '$B^*$-algebra' vs '$C^*$-algebra' and '$W^*$-algebra' vs 'von Neumann algebra'); it would probably be better just to call such $JB$-algebras __special__ (but somebody might think that this just means that the underlying Jordan algebra is special, which is weaker).  We do need some term, however, thanks to the Albert algebra.

## Examples

The main example of a $JB$-algebra is the algebra of [[self-adjoint operators]] in a $C^*$-[[C-star-algebra|algebra]].  For a $JBW$-algebra, try the algebra of self-adjoint operators in a [[von Neumann algebra]].  In particular, the algebra of Hermitian operators on a [[Hilbert space]] is a $JBW$-algebra, in fact a $JW$-algebra.  Still more particularly, the [[trivial ring|trivial algebra]] (which is the algebra of self-adjoint operators on the zero Hilbert space) is a $JW$-algebra (although it won\'t fit definitions by authors who require ${\|1\|} = 1$).

Every $JB$-algebra is [[formally real algebra|formally real]]; conversely, all of the formally real Jordan algebras in [[finite-dimensional space|finite dimensions]] are $JBW$-algebras, and all of the special ones are $JW$-algebras.  This leaves the [[Albert algebra]] $\mathfrak{h}_3(O)$ as the basic example of a $JBW$-algebra that is not a $JW$-algebra.  (See [Jordan algebra#frc](Jordan+algebra#frc) for the classification of the finite-dimensional formally real Jordan algebras.)

## Properties

$JB$-algebras have nice properties like those of $C^*$-algebras, and $JBW$-algebras have nicer properties like those of von Neumann algebras.  They are generally proved in the analogous ways.  Some properties are different, however.

An associative $JB$-algebra is the same thing as a commutative $C^*$-algebra with trivial involution, which (over the [[real numbers]]) is in turn the same thing as the algebra of (real-valued) [[continuous maps]] vanishing at infinity on a [[local compactum]] (which is a [[compactum]] iff the algebra is unital, and then every continuous map vanishes at infinity).

Like any Jordan algebra, a $JB$-algebra $A$ is [[power-associative algebra|power-associative]], so each element $x$ generates an associative (and of course commutative) [[subalgebra]] and hence a local compactum.  In a unital $JB$-algebra, each element generates an associative unital subalgebra and hence a compactum $Spec(x)$.  Any [[continuous map]] $f\colon Spec(x) \to \mathbb{R}$ therefore defines an element $f(x)$ of $A$.  More generally, any associative unital subalgebra $X$ generates a compactum $Spec(X)$, its [[spectrum]], and any continuous map on $Spec(X)$ defines an element of $A$ (in fact belonging to $X$).  Thus we have a [[functional calculus]] on $JB$-algebras.  In a $JBW$-algebra, we may instead interpret the spectrum as a [[localizable measure space]], with $X$ identified as the algebra of [[essentially bounded function|essentially bounded]] [[measurable functions]] (modulo [[almost equality]]) on the spectrum, so that the functional calculus extends to measurable functions.

Since every $JB$-algebra $A$ is [[formally real algebra|formally real]], it comes equipped with a [[partial order]]: $x \leq y$ iff $y - x$ is a sum of squares.

The [[order-theoretic structure in quantum mechanics]] fixes the $JB$-algebra structure of a $C^*$-algebra, but not the $JLB$-algebra structure.

## Related concepts

* [[JBW-algebraic quantum mechanics]]

## References

Basic stuff is at

* [Jordan operator algebra](https://en.wikipedia.org/wiki/Jordan_operator_algebra) on the English Wikipedia;

and most of that appears to be from

* Harald Hanche-Olsen and Erling Størmer (1984); _Jordan operator algebras_; Monographs and Studies in Mathematics 21 (Pitman); [web](http://www.math.ntnu.no/~hanche/joa/),

which I have only begun to read.

There is also

* Harald Upmeier (1987); _Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics_; CBMS Regional Conference Series in Mathematics 67 (AMS),

of which a lot is already on pages 1--4 (the only ones that Google Books would show me).

## Related pages

* [[Jordan algebra]]
* **Jordan--Banach algebra**
* [[Jordan–Lie–Banach algebra]]

[[!redirects Jordan Banach algebra]]
[[!redirects Jordan Banach algebras]]
[[!redirects Jordan-Banach algebra]]
[[!redirects Jordan-Banach algebras]]
[[!redirects Jordan–Banach algebra]]
[[!redirects Jordan–Banach algebras]]
[[!redirects Jordan--Banach algebra]]
[[!redirects Jordan--Banach algebras]]

[[!redirects JB-algebra]]
[[!redirects JB-algebras]]
[[!redirects JB algebra]]
[[!redirects JB algebras]]

[[!redirects JBW-algebra]]
[[!redirects JBW-algebras]]
[[!redirects JBW algebra]]
[[!redirects JBW algebras]]

[[!redirects JC-algebra]]
[[!redirects JC-algebras]]
[[!redirects JC algebra]]
[[!redirects JC algebras]]

[[!redirects JW-algebra]]
[[!redirects JW-algebras]]
[[!redirects JW algebra]]
[[!redirects JW algebras]]

[[!redirects B-identity]]


Jordan-Hölder theorem

## Idea

The _Jordan-Hölder theorem_ says that every _[[composition series]]_ of a given [[group]], and every _[[Jordan-Hölder sequence]]_ on a given object in an [[abelian category]], has the same length, and the same [[simple object|simple]] factors, up to [[permutation]]. In particular says that the [[length of an object]] in an abelian category is well defined.

More generally, a form of the theorem holds in any [[homological category]].

## References

* [[David Speyer]], _The Jordan-Hölder theorem_ ([pdf](http://www.math.lsa.umich.edu/~speyer/594_2013/JordanHolder.pdf))

[[!redirects Jordan-Holder theorem]]

[[!redirects Jordan–Hölder theorem]]
[[!redirects Jordan–Holder theorem]]