Alexander Polyakov

* [Wikipedia entry](http://en.wikipedia.org/wiki/Alexander_Markovich_Polyakov)

* [old personal page](https://phy.princeton.edu/people/alexander-polyakov) at Princeton (archived [screenshot](https://ncatlab.org/nlab/files/APolyakovOldHomepage.jpg))

  > "My main interests this year [1993?] were directed towards [[string theory]] of [[quark]] [[confinement]]. The problem is to find the string Lagrangian for the Faraday's "[[flux tube|lines of force]]", which would reproduce perturbative corrections from the [[Yang-Mills theory]] to the Coulomb law at small distances and would give permanent [[confinement]] of quarks at large distances."

  > (cf. *[[Polyakov gauge-string duality]]*)

* Valery Pokrovsky, *Hidden Sasha Polyakov’s life in Statistical and Condensed Matter Physics*, in: *Polyakov’s String: Twenty Five Years After*, Proceedings of an International Workshop at Chernogolovka (2005) 2-15 [[arXiv:hep-th/0510214](https://arxiv.org/abs/hep-th/0510214)]

## Selected writings

Discussing what came to be called the *[[Polyakov action]]* for the [[bosonic string]] and the resulting [[Liouville theory]], in the non-critical case:

* [[Alexander Polyakov]], *Quantum geometry of bosonic strings*, Phys. Lett. B **103** (1981) 207-210 [doi:10.1016/0370-2693(81)90743-7, [pdf](http://qft.itp.ac.ru/polyakov-2.pdf)]

and for the [[superstring]]:

* [[Alexander Polyakov]], *Quantum geometry of fermionic strings*, Physics Letters B **103** 3 (1981) 211-213 [doi:10.1016/0370-2693(81)90744-9]

Introducing [[conformal field theory]]:

* [[Alexander Belavin]], [[Alexander Polyakov]], [[Alexander Zamolodchikov]], _Infinite conformal symmetry in two–dimensional quantum field theory_, Nuclear Physics B Volume 241, Issue 2, 23 July 1984, Pages 333-380 (doi:10.1016/0550-3213(84)90052-X)

Proposing the identification of [[flux tubes]] in [[confinement|confined]] [[QCD]] with the [[strings]] of [[string theory]], hence of the [[holographic principle]] in what came to be known as the [[AdS-QCD correspondence]] (see *[[Polyakov gauge-string duality]]*):

* [[Alexander Polyakov]], *String representations and hidden symmetries for gauge fields*, Physics Letters B **82** 2 (1979) 247-250 [doi:10.1016/0370-2693(79)90747-0]

* [[Alexander Polyakov]], *Gauge fields as rings of glue*, Nuclear Physics B **164** (1980) 171-188 [doi:10.1016/0550-3213(80)90507-6]

* [[Alexander Polyakov]], *Gauge Fields and Strings*, Routledge, Taylor and Francis (1987, 2021) [[doi:10.1201/9780203755082](https://doi.org/10.1201/9780203755082), [oapen:20.500.12657/50871](https://library.oapen.org/handle/20.500.12657/50871)]

* {#Polyakov98} [[Alexander Polyakov]], *String Theory and Quark Confinement*, Nucl. Phys. Proc. Suppl. **68** (1998) 1-8 [[arXiv:hep-th/9711002](https://arxiv.org/abs/hep-th/9711002), doi:10.1016/S0920-5632(98)00135-2]

* [[Alexander Polyakov]], *The wall of the cave*, Int. J. Mod. Phys. A **14** (1999) 645-658 [[arXiv:hep-th/9809057](https://arxiv.org/abs/hep-th/9809057), [doi:10.1142/S0217751X99000324](https://doi.org/10.1142/S0217751X99000324)]

eventually leading to the rules of the [[AdS-CFT correspondence]]:

* {#GubserKlebanovPolyakov98} [[Steven Gubser]], [[Igor Klebanov]], [[Alexander Polyakov]], *Gauge theory correlators from non-critical string theory*, Physics Letters B **428** 105-114 (1998) [[hep-th/9802109](http://arxiv.org/abs/hep-th/9802109), doi:10.1016/S0370-2693(98)00377-3]

> Relations between gauge fields and strings present an old, fascinating and unanswered question. The full answer to this question is of great importance for theoretical physics. It will provide us with a theory of quark confinement by explaining the dynamics of color-electric fluxes.

* [[Alexander Polyakov]], [[Vyacheslav Rychkov]] *Gauge fields-strings duality and the loop equation*, Nucl. Phys. B **581** (2000) 116-134 $[$[arXiv:hep-th/0002106](https://arxiv.org/abs/hep-th/0002106), doi:10.1016/S0550-3213(00)00183-8$]$

and specifically between [[single trace operators]] and [[superstring]]-excitations:

* {#Polyakov02} [[Alexander Polyakov]], _Gauge Fields and Space-Time_, Int. J. Mod. Phys. A17S1 (2002) 119-136 ([arXiv:hep-th/0110196](https://arxiv.org/abs/hep-th/0110196)):

> The picture which slowly arises from the above considerations is that of the space-time gradually disappearing in the regions of large curvature. The natural description in this case is provided by a gauge theory in which the basic objects are the texts formed from the gauge-invariant words. The theory provides us with the expectation values assigned to the various texts, words and sentences.

> These expectation values can be calculated either from the gauge theory or from the strongly coupled 2d sigma model. The coupling in this model is proportional to the target space curvature. This target space can be interpreted as a usual continuous space-time only when the curvature is small. As we increase the coupling, this interpretation becomes more and more fuzzy and finally completely meaningless.

* [[Steven Gubser]], [[Igor Klebanov]], [[Alexander Polyakov]], *A semi-classical limit of the gauge/string correspondence*, Nucl. Phys. B **636** (2002) 99-114 [[arXiv:hep-th/0204051](https://arxiv.org/abs/hep-th/0204051), doi:10.1016/S0550-3213(02)00373-5]

Historical reminiscences:

* [[Alexander Polyakov]], *Confinement and Liberation*, in [[Gerardus ’t Hooft]] (ed.) *50 Years of Yang-Mills Theory* (2005) 311-329 [[arXiv:hep-th/0407209](https://arxiv.org/abs/hep-th/0407209), [doi:10.1142/9789812567147_0013](https://doi.org/10.1142/9789812567147_0013), [doi:10.1142/5601](https://doi.org/10.1142/5601)]

* {#Polyakov07} [[Alexander M. Polyakov]], *Beyond Space-Time*, in *The Quantum Structure of Space and Time*, Proceedings of the 23rd Solvay Conference on Physics, World Scientific (2007) [[arXiv:hep-th/0602011](https://arxiv.org/abs/hep-th/0602011), [pdf](http://www.solvayinstitutes.be/pdf/Proceedings_Physics/23rd_Solvay_Physics.pdf)]

* {#Polyakov08} [[Alexander M. Polyakov]], *From Quarks to Strings* [[arXiv:0812.0183](https://arxiv.org/abs/0812.0183)]

  published as *Quarks, strings and beyond*, section 44 in: [[Paolo Di Vecchia]] et al. (ed.), *The Birth of String Theory*, Cambridge University Press (2012) 544-551 [[doi:10.1017/CBO9780511977725.048](https://doi.org/10.1017/CBO9780511977725.048)]

  > "By the end of ’77 it was clear to me that I needed a new strategy [for understanding confinement] and I became convinced that the way to go was the [[gauge/string duality]]. [...]"

  > "I kept thinking about [[gauge/string duality|gauge/strings dualities]]. Soon after the Liouville mode was discovered it became clear to many people including myself that its natural interpretation is that random surfaces in 4d are described by the strings flying in 5d with the Liouville field playing the role of the fifth dimension. The precise meaning of this statement is that the wave function of the general string state depends on the four center of mass coordinates and also on the fifth, the Liouville one. In the case of minimal models this extra dimension is related to the matrix eigenvalues and the resulting space is flat."

  > "Since this 5d space must contain the flat 4d subspace in which the gauge theory resides, the natural ansatz for the metric is just the Friedman universe with a certain warp factor. This factor must be determined from the conditions of conformal symmetry on the world sheet. Its dependence on the Liouville mode must be related to the renormalization group flow. As a result we arrive at a fascinating picture -- our 4d world is a projection of a more fundamental 5d string theory. [...]"

  > "At this point I was certain that I have found the right language for the [[gauge/string duality]]. I attended various conferences, telling people that it is possible to describe gauge theories by solving Einstein-like equations (coming from the conformal symmetry on the world sheet) in five dimensions. The impact of my talks was close to zero. That was not unusual and didn’t bother me much. What really caused me to delay the publication ([Polyakov 1998](#Polyakov98)) for a couple of years was my inability to derive the asymptotic freedom from my equations. At this point I should have noticed the paper of [Klebanov 1997](https://arxiv.org/abs/hep-th/9702076) in which he related [[D3 branes]] described by the supersymmetric Yang Mills theory to the same object described by supergravity. Unfortunately I wrongly thought that the paper is related to matrix theory and I was skeptical about this subject. As a result I have missed this paper which would provide me with a nice special case of my program. This special case was presented little later in full generality by Juan Maldacena ([Maldacena 1997](AdS-CFT+correspondence#Maldacena97a)) and his work opened the flood gates."

## Related entries

* [[Polyakov gauge-string duality]]

* [[QCD]], [[confinement]]

* [[Polyakov action]], [[harmonic map]]

* [[conformal bootstrap]]

* [[string theory]]

* [[AdS-CFT correspondence]]

* [[AdS-QCD correspondence]]

* [[Polyakov gauge-string duality -- references]]

category: people

[[!redirects Alexander M. Polyakov]]


Alexander polynomial

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#### Knot theory
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[[!include knot theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The *Alexander polynomial* ([Alexander 1928](#Alexander28)) is a [[polynomial]] [[knot invariant|invariant]] related to [[braid representation]]-theory (cf. the *[[Burau representation]]*).

There are several ways to look at thid invariant, some of these use the [[knot group]] previously defined by [[Max Dehn]], but there are also various combinatorial methods derived from Alexander's original one. One of the best known methods is via [[Fox derivatives]] and is described in the classical text by [[Richard Crowell]] and [[Ralph Fox]].

(...)

Consider some 3-manifold given as a [[surface]] [[fiber bundle]] over the circle (notice the _[[virtually fibered conjecture]]_).
For a [[fiber]] surface $T$, the translation of the fibre around the base-space circle determines an element in the mapping-class group of $T$, a homeomorphism $h\colon T \to T$ well defined up to isotopy; this element is called the _holonomy_ of the fiber surface; the _Alexander polynomial_ is the [[characteristic polynomial]] of the map the holonomy  induces on $H_1(T)$.

([Stallings 87](#Stallings87))

## Properties

### Analogue in  number theory

See [Sikora 01, analogy 2.2 (10)](#Sikora01)) for the comparison in [[arithmetic topology]], where Alexander-Fox theory is the analog of [[Iwasawa theory]] ([Morishita, section 7](#Morishita)).

In [Remark 3.3 of Sugiyama 04](#Sugiyama04), the Alexander polynomial is described as the L-function of the knot complement, taken there with the trivial represenation. As such it resembles the local zeta function of a curve.

## Related concepts

* [[Burau representation]]

## References

The original article:

* {#Alexander28} [[James W. Alexander]], *Topological invariants of knots and links*, Trans. Amer. Math. Soc. **30** (1928) 275-306 $[$[doi:10.1090/S0002-9947-1928-1501429-1](https://doi.org/10.1090/S0002-9947-1928-1501429-1)$]$

Textbook accounts:

* [[R. H. Crowell]] and [[R. H. Fox]], *Introduction to Knot Theory* Springer, Graduate Texts **57** (1963)

* [[Nick Gilbert]], [[Tim Porter]], *Knots and surfaces*, Oxford University Press (1994) [[ISBN:9780198514909](https://global.oup.com/academic/product/knots-and-surfaces-9780198514909?cc=de&lang=en&)]

See also:

* {#Stallings87} [[John Stallings]], _Constructions of fibered knots and links_, Proceedings of Symposia in Pure Mathematics **32** (1987) $[$[pdf](http://www.maths.ed.ac.uk/~aar/papers/stallfib2.pdf)$]$

* Wikipedia, *[Alexander polynomial](https://en.m.wikipedia.org/wiki/Alexander_polynomial)*

An analogue in number theory is the *Iwasawa polynomial*. Cf. for number theoretic analogies

* [[Barry Mazur]], _Remarks on the Alexander polynomial_, [pdf](http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf)
* {#Morishita} Masanori Morishita, _Analogies between Knots and Primes, 3-Manifolds and Number Rings_, ([arxiv](http://arxiv.org/abs/0904.3399))
* Masanori Morishita, Knots and primes: an introduction to arithmetic topology, Springer 2012, chapter 12
* Ken-ichi Sugiyama, _The properties of an L-function from a geometric point of view_, 2007 [pdf](http://geoquant2007.mi.ras.ru/sugiyama.pdf)

* {#Sugiyama04} Ken-ichi Sugiyama, _A topological $\mathrm{L}$ -function for a threefold_, 2004 [pdf](http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1376-12.pdf);

* {#Sugiyama06} Ken-ichi Sugiyama _An analog of the Iwasawa conjecture for a compact hyperbolic threefold_, [math.GT/0606010](http://arxiv.org/abs/math/0606010)

* {#Sikora01} Adam S. Sikora, _Analogies between group actions on 3-manifolds and number fields_, [arXiv:0107210](http://arxiv.org/abs/math/0107210)

Other works

* Takefumi Nosaka, _Twisted cohomology pairings of knots I; diagrammatic computation_, [arxiv/1602.01129](http://arxiv.org/abs/1602.01129); _Twisted cohomology pairings of knots II; to classical invariants_, [arxivs/1602.01131](http://arxiv.org/abs/1602.01131)

* V. Mishnyakov, A. Sleptsov, N. Tselousov, _A new symmetry of the colored Alexander polynomial_ ([arXiv:2001.10596](https://arxiv.org/abs/2001.10596))

[[!redirects Alexander polynomials]]

category:knot theory


Alexander Popov

## Selected writings

A derivation of 4d [[Skyrmions]] by [[KK-compactification]] of [[D=5 Yang-Mills theory]] on a [[closed interval]], motivated by [[M5-branes]] instead of by [[D4/D8-brane intersections]] as in the [[Sakai-Sugimoto model]]:

* [[Tatiana Ivanova]], [[Olaf Lechtenfeld]], [[Alexander Popov]], _Skyrme model from 6d $\mathcal{N}= (2,0)$ theory_, Physics Letters B Volume 783, 10 August 2018, Pages 222-226 ([arXiv:1805.07241](https://arxiv.org/abs/1805.07241))

following

* [[Tatiana Ivanova]], [[Olaf Lechtenfeld]], [[Alexander Popov]], _Non-Abelian sigma models from Yang-Mills theory compactified on a circle_, Physics Letters B Volume 781, 10 June 2018, Pages 322-326 ([arXiv:1803.07322](https://arxiv.org/abs/1803.07322))

category: people


Alexander Postnikov

*[web](http://math.mit.edu/~apost)

category: people


Alexander Reznikov

* [Wikipedia entry](https://de.wikipedia.org/wiki/Alexander_Resnikow)

## Selected writings

On [[secondary characteristic classes]] being [[torsion elements]]:

* [[Alexander Reznikov]], _Rationality of secondary classes_, J. Differential Geom., Volume 43, Number 3 (1996), 674-692. ([arXiv:dg-ga/9407007](https://arxiv.org/abs/dg-ga/9407007), [euclid:jdg/1214458328](https://projecteuclid.org/euclid.jdg/1214458328))

* [[Alexander Reznikov]], _All regulators of flat bundles are torsion_, Annals of Mathematics Second Series, Vol. 141, No. 2 (Mar., 1995), pp. 373-386 (14 pages) ([arXiv:dg-ga/9407006](https://arxiv.org/abs/dg-ga/9407006), [jstor:2118525](https://www.jstor.org/stable/2118525))

category: people


Alexander Rosenberg

There are at least two Alexander Rosenbergs (and one [[Jonathan Rosenberg]]) in mathematics:

* algebraist Alex (Alexander) Rosenberg (1926-2007) known for [[Hochschild-Kostant-Rosenberg theorem]] and for many other articles in [[algebra]] (cf. [notices](http://www.spelman.edu/~colm/alexnotices.html))

* Alexander L. Rosenberg (or Sasha Rosenberg) who was a  professor until 2012 at [Kansas State University](http://www.math.ksu.edu).

Sasha Rosenberg had defended in 1973 dissertation at Moscow State University studying [[Tannaka duality|Tannakian reconstruction theorems]], mainly using methods of [[functional analysis]]. His main goals were in [[representation theory]]. Until leaving Soviet Union around 1987, Rosenberg worked in applied mathematics; on the side however he was developing some methods in representation theory which included functional analysis and categorical and ring/module theoretic methods, and the noncommutative localization in particular. A main construction, presented at a conference at Baikal (1981), was a new spectrum of a ring, so-called left spectrum, later generalized to a [[spectrum of an abelian category]], used to prove (in the  quasicompact case) the Gabriel-Rosenberg reconstruction theorem for commutative schemes; more [[spectral theory|spectral]] constructions followed. This work outgrown into a wide ongoing work on foundations of [[noncommutative algebraic geometry]], including a natural definition of [[noncommutative scheme]]. More recently he proposed a new framework for nonabelian derived functors, including a new version of algebraic K-theory. Other related entries in $n$Lab include [[Q-category]], [[neighborhood of a topologizing subcategory]], [[sheaf on a noncommutative space]], [[spectral cookbook]].

Rosenberg's coauthors in pure mathematics works are [[Valery Lunts]] and [[Maxim Kontsevich]]. One should be warned that most of the newly released articles of Rosenberg are not at the arXiv but at the Max Planck Bonn [preprint server](http://www.mpim-bonn.mpg.de/Research/MPIM+Preprint+Series) and many of works are republished in a volume

* [Selected papers on noncommutative geometry](http://newprairiepress.org/ebooks/1), New Prairie Press 2014, which are open access

Rosenberg's available papers and online resources include:

* [web page](http://sasharosenberg.com/), [mathematical work](http://sasharosenberg.com/?page_id=7)

* 3 volume work on noncommutative geometry: _Geometry of Noncommutative 'Spaces' and Schemes_, [pdf](http://www.mpim-bonn.mpg.de/preblob/5148); _Homological algebra of noncommutative 'spaces' I._, [pdf](http://www.mpim-bonn.mpg.de/preblob/5159) (note: longer version than another MPI preprint with the same title mentioned before); _Noncommutative 'Spaces' and 'Stacks'_, [pdf](http://www.mpim-bonn.mpg.de/preblob/5149)

* video of msri lecture _Noncommutative schemes and spaces_ (Feb 2000) can be found at [msri](http://www.msri.org/publications/ln/msri/2000/interact/rosenberg/1/index.html)

* A. L. Rosenberg, _Topics in noncommutative algebraic geometry, homological algebra and K-theory_, preprint MPIM Bonn 2008-57 [pdf](http://www.mpim-bonn.mpg.de/preblob/3589)

* А. Л. Розенберг, _Теоремы двойственности для групп и алгебр Ли_, УМН, 26:6(162) (1971), 253–254, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=5287&what=fullt&option_lang=rus), _Инвариантные алгебры на компактных группах_, Матем. сб., 81(123):2 (1970), 176–184, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=sm&paperid=3368&what=fullt&option_lang=rus)

* A. L. Rosenberg, _Almost quotient categories, sheaves and localizations_, 181 p. Seminar on supermanifolds __25__, University of Stockholm, D. Leites editor, 1988 (in Russian; partial remake in English exists)

* A. L. Rosenberg, _Non-commutative affine semischemes and schemes_, Seminar on supermanifolds __26__, Dept. Math., U. Stockholm (1988)

* A. L.  Rosenberg, _Noncommutative algebraic geometry and representations of quantized algebras_, MIA __330__, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9

* A. L. Rosenberg, _Reconstruction of groups_, Selecta Math. N.S. __9__:1 (2003) [doi](http://dx.doi.org/10.1007/s00029-003-0322-x) ($n$lab remark: this paper is on a generalization of Tannaka--Krein and not of the Gabriel--Rosenberg kind of reconstruction)

* A. L. Rosenberg, _The left spectrum, the Levitzki radical, and noncommutative schemes_, Proc. Nat. Acad. Sci. U.S.A. __87__  (1990),  no. 21, 8583--8586.

* A. L. Rosenberg, _Noncommutative local algebra_,  Geom. Funct. Anal. __4__  (1994),  no. 5, 545--585.

* A. L. Rosenberg, _The existence of fiber functors_, The Gelfand Mathematical Seminars, 1996--1999,  145--154, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 2000 [doi](https://dx.doi.org/10.1007/978-1-4612-1340-6_7)

* A. L. Rosenberg, _The spectrum of abelian categories and reconstructions of schemes_, in Rings, Hopf Algebras, and Brauer groups, Lectures Notes in Pure and Appl. Math. __197__, Marcel Dekker, New York, 257--274, 1998; MR99d:18011; and Max Planck Bonn preprint _Reconstruction of Schemes_, [MPIM1996-108](http://www.mpim-bonn.mpg.de/preblob/3948) (1996).

* A. L. Rosenberg, _Spectra of noncommutative spaces_, MPIM2003-110 [ps](http://www.mpim-bonn.mpg.de/preblob/1946) [dvi](http://www.mpim-bonn.mpg.de/preblob/1945) (2003)

* A. L. Rosenberg, _Noncommutative schemes_, Compos. Math. __112__ (1998) 93--125 [doi](http://dx.doi.org/10.1023/A:1000479824211)

* V. A. Lunts, A. L. Rosenberg, _Differential operators on noncommutative rings_, Selecta Math. (N.S.)  __3__  (1997),  no. 3, 335--359 ([doi](http://dx.doi.org/10.1007/s000290050014)); sequel: _Localization for quantum groups_, Selecta Math. (N.S.) __5__ (1999), no. 1, pp. 123--159 ([doi](http://dx.doi.org/10.1007/s000290050044)).

* V. A. Lunts, A. L. Rosenberg, _Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings_, MPI 1996-53 [pdf](http://www.mpim-bonn.mpg.de/preblob/3894), II. D-Calculus in the braided case. The localization of quantized enveloping algebras, MPI 1996-76 [pdf](http://www.mpim-bonn.mpg.de/preblob/3916)

* V. A. Lunts, A. L. Rosenberg, _Kashiwara theorem for hyperbolic algebras_,
MPIM1999-82, [dvi](http://www.mpim-bonn.mpg.de/preblob/947), [ps](http://www.mpim-bonn.mpg.de/preblob/948)

* M. Kontsevich, A. Rosenberg, _Noncommutative spaces_, preprint MPI-2004-35 ([dvi](http://www.mpim-bonn.mpg.de/preblob/2303),[ps](http://www.mpim-bonn.mpg.de/preblob/2331)), _Noncommutative spaces and flat descent_, MPI-2004-36 [dvi](http://www.mpim-bonn.mpg.de/preblob/2304),[ps](http://www.mpim-bonn.mpg.de/preblob/2332), _Noncommutative stacks_, MPI-2004-37 [dvi](http://www.mpim-bonn.mpg.de/preblob/2305),[ps](http://www.mpim-bonn.mpg.de/preblob/2333)

* M. Kontsevich, A. Rosenberg, _Noncommutative smooth spaces_,  The Gelfand Mathematical Seminars, 1996--1999,  85--108, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 2000; ([arXiv:math/9812158](http://arxiv.org/abs/math/9812158))

* A. Rosenberg, [Homological algebra of noncommutative 'spaces' I](http://www.mpim-bonn.mpg.de/preblob/3623), 199 pages, preprint Max Planck, Bonn: MPIM2008-91.

* A. Rosenberg, _Underlying spaces of noncommutative schemes_, MPIM2003-111, [dvi](http://www.mpim-bonn.mpg.de/preblob/1947), [ps](http://www.mpim-bonn.mpg.de/preblob/1948)

* A. Rosenberg, _Noncommutative spaces and schemes_,
MPIM1999-84, [dvi](http://www.mpim-bonn.mpg.de/preblob/949), [ps](http://www.mpim-bonn.mpg.de/preblob/950)

[[!redirects Sasha Rosenberg]]
[[!redirects Sasha Rozenberg]]
[[!redirects A. L. Rosenberg]]
[[!redirects Rosenberg]]

category: people, algebraic geometry, noncommutative geometry


Alexander Schenkel

* [webpage](http://www.aschenkel.eu)

## Selected writings

On construction and axiomatization of [[homotopical AQFT]] via [[homotopy theory]] and [[homotopical algebra]]:

* {#BDS} [[Marco Benini]], [[Claudio Dappiaggi]], [[Alexander Schenkel]], _Quantized Abelian principal connections on Lorentzian manifolds_, Communications in Mathematical Physics 2013 ([arXiv:1303.2515](http://arxiv.org/abs/1303.2515))

* {#BeniniSchenkelSzabo15} [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], _Homotopy colimits and global observables in Abelian gauge theory_ ([arXiv:1503.08839](http://arxiv.org/abs/1503.08839))

* {#BeniniSchenkel16} [[Marco Benini]], [[Alexander Schenkel]], _Quantum field theories on categories fibered in groupoids_, Communications in Mathematical Physics November 2017, Volume 356, Issue 1, pp 19–64 ([arXiv:1610.06071](https://arxiv.org/abs/1610.06071))

* [[Simen Bruinsma]], [[Christopher J. Fewster]], [[Alexander Schenkel]], *Relative Cauchy evolution for linear homotopy AQFTs* ([arXiv:2108.10592](https://arxiv.org/abs/2108.10592))

On [[self-dual higher gauge theory]] on [[Lorentzian spacetimes]] via [[ordinary differential cohomology]]:

* [[Christian Becker]], [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], _Abelian duality on globally hyperbolic spacetimes_, Commun. Math. Phys. **349** (2017) 361-392 [[arXiv:1511.00316](https://arxiv.org/abs/1511.00316), [doi:10.1007/s00220-016-2669-9](https://doi.org/10.1007/s00220-016-2669-9)]

On the [[stack]] of [[Yang-Mills theory|Yang-Mills]] [[gauge fields]]:

* {#BeniniSchenkelSchreiber17} [[Marco Benini]], [[Alexander Schenkel]], [[Urs Schreiber]], _The stack of Yang-Mills fields on Lorentzian manifolds_, Commun. Math. Phys. 359, 765–820 (2018) ([arXiv:1704.01378](https://arxiv.org/abs/1704.01378), [doi:10.1007/s00220-018-3120-1](https://doi.org/10.1007/s00220-018-3120-1))

Review and exposition:

* {#Schenkel17} [[Alexander Schenkel]], _Towards Homotopical Algebraic Quantum Field Theory_, talk at _[Foundational and Structural Aspects of Gauge Theories](https://indico.mitp.uni-mainz.de/event/76/overview)_, Mainz Institute for Theoretical Physics, 29 May – 2 June 2017. ([pdf](http://aschenkel.eu/Mainz17.pdf))

* {#Schenkel14} [[Alexander Schenkel]], _On the problem of gauge theories in locally covariant QFT_, talk at _[Operator and Geometric Analysis on Quantum Theory](http://www.science.unitn.it/~moretti/convegno/convegno.html)_ Trento, 2014 ([[SchenkelTrento2014.pdf:file]]) (with further emphasis on this point in the companion talk [Schreiber 14](field+bundle#Schreiber14))

* {#Schenkel17b} [[Alexander Schenkel]], _From Fredenhagen's universal algebra to homotopy theory and operads_, talk at _Quantum Physics meets Mathematics_, Hamburg, December 2017 ([pdf slides](http://aschenkel.eu/Hamburg17.pdf))

* {#BeniniSchenkel19} [[Marco Benini]], [[Alexander Schenkel]], _Higher Structures in Algebraic Quantum Field Theory_,  Proceedings of LMS/EPSRC Symposium _[[Higher Structures in M-Theory 2018]]_, Fortschritte der Physik 2019  ([arXiv:1903.02878](https://arxiv.org/abs/1903.02878), [doi:10.1002/prop.201910015]( https://doi.org/10.1002/prop.201910015))

An [[operad]] for [[local nets of observables]] in [[AQFT]]

* {#BeniniSchenkelWoike17} [[Marco Benini]], [[Alexander Schenkel]], [[Lukas Woike]], _Operads for algebraic quantum field theory_ ([arXiv:1709.08657](https://arxiv.org/abs/1709.08657))

and its [[model structure on algebras over an operad]] (with respect to the [[model structure on chain complexes]]) is discussed in

* {#BeniniSchenkelWoike18} [[Marco Benini]], [[Alexander Schenkel]], [[Lukas Woike]], _Homotopy theory of algebraic quantum field theories_, Lett. Math. Phys. 109, 1487-1532 (2019) ([arXiv:1805.08795](https://arxiv.org/abs/1805.08795), [doi:10.1007/s11005-018-01151-x](https://doi.org/10.1007/s11005-018-01151-x))

On relating [[homotopy AQFT|homotopy]] [[algebraic quantum field theory]] via [[local nets of observables]] to [[factorization algebras]]:

* [[Marco Benini]], [[Marco Perin]], [[Alexander Schenkel]], _Model-independent comparison between factorization algebras and algebraic quantum field theory on Lorentzian manifolds_, Communications in Mathematical Physics volume 377, pages 971–997 (2020) ([arXiv:1903.03396v2](https://arxiv.org/abs/1903.03396), [doi:10.1007/s00220-019-03561-x](https://doi.org/10.1007/s00220-019-03561-x))

* [[Marco Benini]], Giorgio Musante, [[Alexander Schenkel]], *Quantization of Lorentzian free BV theories: factorization algebra vs algebraic quantum field theory* [[arXiv:2212.02546](https://arxiv.org/abs/2212.02546)]

Discussion of [[orbifold|orbifolding]] via [[categorification]], in [[homotopical algebraic quantum field theory]]:

* [[Marco Benini]], [[Marco Perin]], [[Alexander Schenkel]], [[Lukas Woike]], _Categorification of algebraic quantum field theories_, Lett. Math. Phys. 2021 ([arXiv:2003.13713](https://arxiv.org/abs/2003.13713))

On rigorous [[semi-topological 4d Chern-Simons theory]] via [[homotopical AQFT]]:

* [[Marco Benini]], [[Alexander Schenkel]], Benoit Vicedo, _Homotopical analysis of 4d Chern-Simons theory and integrable field theories_ ([arXiv:2008.01829](https://arxiv.org/abs/2008.01829))

On 1d [[AQFT]] and [[smooth stacks]]:

* [[Marco Benini]], [[Marco Perin]], [[Alexander Schenkel]], *Smooth 1-dimensional algebraic quantum field theories* ([arXiv:2010.13808](https://arxiv.org/abs/2010.13808))

On [[non-perturbative quantum field theory|non-perturbative]] aspects of the [[BV-formalism]]:

* [[Marco Benini]], [[Pavel Safronov]], [[Alexander Schenkel]], *Classical BV formalism for group actions* ([arXiv:2104.14886](https://arxiv.org/abs/2104.14886))

On [[2d CFT]] in terms of [[AQFT on curved spacetimes]]:

* [[Marco Benini]], [[Luca Giorgetti]], [[Alexander Schenkel]], *A skeletal model for 2d conformal AQFTs* ([arXiv:2111.01837](https://arxiv.org/abs/2111.01837))

On the [[time slice axiom]] in [[homotopical AQFT]]:

* [[Marco Benini]], [[Victor Carmona]], [[Alexander Schenkel]], *Strictification theorems for the homotopy time-slice axiom* [[arxiv:2208.04344](https://arxiv.org/abs/2208.04344)]

Exposition and review:

* [[Alexander Schenkel]], *Quantum field theories on Lorentzian manifolds*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#SchenkelMar2023), slides:[[Schenkel-at-QFTAndCobordism2023.pdf:file]], video:[YT](https://www.youtube.com/watch?v=YiQNAgcwZYQ)]

Formulation of the [[CS/WZW correspondence]] in [[homotopical AQFT]]:

* [[Marco Benini]], [[Alastair Grant-Stuart]], [[Alexander Schenkel]], *The linear CS/WZW bulk/boundary system in AQFT*, Annales Henri Poincaré (2023) [[arXiv:2302.06990](https://arxiv.org/abs/2302.06990)]

On the relation between [[functorial quantum field theory]] (axiomatizing the [[Schrödinger picture]] of [[quantum field theory]]) and [[algebraic quantum field theory]] (axiomatizing the [[Heisenberg picture]]):

* {#BunkMacManusSchenkel23} [[Severin Bunk]], [[James MacManus]], [[Alexander Schenkel]], *Lorentzian bordisms in algebraic quantum field theory* [[arXiv:2308.01026](https://arxiv.org/abs/2308.01026)]

## Related entries

* [[AQFT]], [[homotopical AQFT]]

* [[gauge theory]]

* [[ordinary differential cohomology]]

category: people


Alexander Schmeding

Mathematician enjoying higher and infinite-dimensional differential geometry and its applications.

I am an associate professor at NTNU Trondheim.
Before joining NTNU, I was associate professor at Nord University in Levanger and before that at the university in Bergen. See

* [NTNU webpage](https://www.ntnu.edu/employees/alexander.schmeding)

* [Research Gate profile](https://www.researchgate.net/profile/Alexander_Schmeding)

* [Linkedin profile](https://www.linkedin.com/in/alexander-schmeding-248190141/)

## More stuff

On [[differential geometry]] with [[infinite-dimensional manifolds]]:

* [[Alexander Schmeding]],  *An introduction to infinite-dimensional differential geometry*, Cambridge University Press. The open access version is available here: [CUP Open Access version](https://www.cambridge.org/core/books/an-introduction-to-infinitedimensional-differential-geometry/6483795C98EE417C0F3654F6C192C3BC)
Further, here is the [list of Errata](https://github.com/ASchmeding/An-introduction-to-infinite-dimensional-geometry)

* [Youtube](https://www.youtube.com/channel/UCTIB6ykZCrpd0trFhiSETGw) (videos on math in particular a course on infinite-dimensional geometry)

## Related $n$Lab entries

* [[bisection of a Lie groupoid]]

category: people


Alexander Schmidt

* [Web](https://www.mathi.uni-heidelberg.de/~schmidt/)

category: people


Alexander Strohmaier

* [personal page](http://www1.maths.leeds.ac.uk/~pmtast/)

* [Institute page](https://eps.leeds.ac.uk/maths/staff/4080/professor-alexander-strohmaier)

## Selected writings

Generalization of the [[time tube theorem]] to [[AQFT on curved spacetimes|AQFT on curved]] but [[real analytic space|real analytic]] [[spacetimes]]:

* [[Alexander Strohmaier]], *On the Local Structure of the Klein–Gordon Field on Curved Spacetimes*, Letters in Mathematical Physics **54** (2000) 249–261 [[doi:10.1023/A:1010927625112](https://doi.org/10.1023/A:1010927625112), [arXiv:math-ph/0008043](https://arxiv.org/abs/math-ph/0008043)]

* [[Alexander Strohmaier]], [[Edward Witten]], *Analytic states in quantum field theory on curved spacetimes* [[arXiv:2302.02709](https://arxiv.org/abs/2302.02709)]

* [[Alexander Strohmaier]], [[Edward Witten]], *The Timelike Tube Theorem in Curved Spacetime* [[arXiv:2303.16380](https://arxiv.org/abs/2303.16380)]

category: people


Alexander Varchenko

__Alexander Varchenko__ is a mathematician at the University of North Carolina at Chapel Hill; he was a student of [[Vladimir Arnold]]. His work is at the boundary of geometry, special functions, combinatorics, representation theory and mathematical physics.

* [list](http://www.math.unc.edu/Faculty/av/complete.htm)
* [webpage](http://www.math.unc.edu/Faculty/av)

## Selected writings

On the [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of solutions to the [[Knizhnik-Zamolodchikov equation]]:

* [[Vadim Schechtman]], [[Alexander Varchenko]], *Integral representations of N-point conformal correlators in the WZW model*,  Max-Planck-Institut für Mathematik, (1989) Preprint MPI/89- $[$[cds:1044951](http://cds.cern.ch/record/1044951)$]$

* [[Vadim Schechtman]], [[Alexander Varchenko]], *Hypergeometric solutions of Knizhnik-Zamolodchikov equations*, Lett. Math. Phys. **20** (1990) 279–283 $[$[doi:10.1007/BF00626523](https://doi.org/10.1007/BF00626523)$]$

* [[Vadim Schechtman]], [[Alexander Varchenko]], *Arrangements of hyperplanes and Lie algebra homology*, Inventiones mathematicae  **106** 1 (1991) 139-194 $[$[dml:143938](https://eudml.org/doc/143938), [[SchechtmanVarchenko_HyperplaneArrangements.pdf:file]]$]$

On the [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of [[conformal blocks]] for the [[sl(n)|$\mathfrak{sl}(n)$]] [[WZW model]]:

* {#FeiginSchechtmanVarchenko90} [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by correlators in Wess-Zumino-Witten models*,  Lett Math Phys **20** (1990)  291–297  $[$[doi:10.1007/BF00626525](https://doi.org/10.1007/BF00626525)$]$

* {#FeiginSchechtmanVarchenko94} [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by hypergeometric correlators in WZW models. I*, Commun. Math. Phys. **163** (1994) 173–184 $[$[doi:10.1007/BF02101739](https://doi.org/10.1007/BF02101739)$]$

  > (for [[sl(2)|$\mathfrak{sl}(2, \mathbb{C})$]])

* [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by hypergeometric correlators in WZW models. II*, Comm. Math. Phys. **170** 1 (1995) 219-247  $[$[euclid:cmp/1104272957](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-170/issue-1/On-algebraic-equations-satisfied-by-hypergeometric-correlators-in-WZW-models/cmp/1104272957.full)$]$

On [[hypergeometric functions]], the [[Knizhnik-Zamolodchikov equation]] and [[quantum groups]]:

* [[Alexander Varchenko]], *Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base*,  Comm. Math. Phys. **171** 1  (1995) 99-137 $[$[arXiv:hep-th/9403102](https://arxiv.org/abs/hep-th/9403102), [doi:10.1007/BF02103772](https://doi.org/10.1007/BF02103772)$]$

* [[Alexander Varchenko]], _Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups_, Adv. Ser. in Math. Phys. __21__, World Sci. Publ. 1995. x+371 pp. ([doi:10.1142/2467](https://doi.org/10.1142/2467))

* V. Tarasov, [[Alexander Varchenko]], _Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups_, Astérisque __246__ (1997), vi+135 pp. ([arXiv:q-alg/9703044](https://arxiv.org/abs/q-alg/9703044), [numdam:AST_1997__246__R1_0](http://www.numdam.org/item/AST_1997__246__R1_0))

On [[Lie algebra weight systems]] for the [[special linear Lie algebra]] $\mathfrak{sl}(2)$:

* S. Tyurina, [[Alexander Varchenko]], *A remark on $\mathfrak{sl}_2$ approximation of the Kontsevich integral of the unknot*, Journal of Mathematical Sciences **131** (2005) 5270–5274  ([arXiv:math/0111201](https://arxiv.org/abs/math/0111201), [doi:10.1007/s10958-005-0399-1](https://doi.org/10.1007/s10958-005-0399-1))

* [[Sergei Chmutov]], [[Alexander Varchenko]], _Remarks on the Vassiliev knot invariants coming from $\mathfrak{sl}(2)$_, Topology 36 (1), 153-178, 1997 (doi:10.1016/0040-9383(95)00071-2)

Further on 1-[[twisted de Rham cohomology]] related to [[sl(2)|$\mathfrak{sl}(2)$]]-[[Verma modules]]:

* {#SlinkinVarchenko19} [[Alexey Slinkin]], [[Alexander Varchenko]], *Twisted de Rham Complex on Line and Singular Vectors in $\widehat{\mathfrak{sl}}_2$ Verma Modules*, SIGMA **15** (2019) 075 [[arXiv:1812.09791](https://arxiv.org/abs/1812.09791), [doi:10.3842/SIGMA.2019.075](https://doi.org/10.3842/SIGMA.2019.075)]

## Related $n$Lab entries

* [[hypergeometric function]]

* [[quantum group]],

* [[conformal field theory]]

* [[representation theory]]

* [[Kac-Moody Lie algebra]]

* [[singularity theory]]

[[!redirects A. Varchenko]]

category: people


Alexander Verbovetsky

* [webpage](http://diffiety.ac.ru/curvita/ver.htm)

## related $n$Lab entries

* [[diffiety]]

* [[variational bicomplex]]

* [[differential operator]]

* [[jet bundle]]

category: people


Alexander Vilenkin

* [webpage](http://www.phy.tufts.edu/vilenkin.html)

* [Wikipedia entry](http://en.wikipedia.org/wiki/Alexander_Vilenkin)

## Selected writings

On [[cosmic strings]], [[domain walls]], [[magnetic monopoles]], etc.:

* {#VilenkinShellard94} [[Alexander Vilenkin]], E. P. S. Shellard, *Cosmic strings and other topological defects*, Cambridge University Press (1994) [[ISBN:9780521654760](https://www.cambridge.org/ae/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/cosmic-strings-and-other-topological-defects?format=PB&isbn=9780521654760), [spire:1384873](https://inspirehep.net/literature/1384873)]

On [[false vacuum eternal inflation]]:

* {#GarrigaGuthVilenkin06} [[Jaume Garriga]], [[Alan Guth]], [[Alexander Vilenkin]], _Eternal inflation, bubble collisions, and the persistence of memory_, Phys. Rev. D76:123512, 2007 ([arXiv:hep-th/0612242](https://arxiv.org/abs/hep-th/0612242), [doi:10.1103/PhysRevD.76.123512](https://doi.org/10.1103/PhysRevD.76.123512))

## Related entries

* [[cosmic inflation]]

category: people


Alexander Voronov

* [website](http://www.math.umn.edu/~voronov/)

## Selected writings

* [[Murray Gerstenhaber]], [[Alexander Voronov]], _Homotopy G-algebras and moduli space operad_, Internat. Math. Research Notices (1995) 141-153 ([arXiv:hep-th/9409063](https://arxiv.org/abs/hep-th/9409063))

On [[string topology]]:

* [[Ralph Cohen]], [[Alexander Voronov]], _Notes on string topology_, in: [[Ralph Cohen]], [[Kathryn Hess]], [[Alexander Voronov]], _String topology and cyclic homology_, Advanced courses in mathematics CRM Barcelona, Birkhäuser 2006 ([math.GT/05036259](http://arxiv.org/abs/math/0503625), [doi:10.1007/3-7643-7388-1](https://doi.org/10.1007/3-7643-7388-1), [pdf](http://gen.lib.rus.ec/get?md5=adde9464705ede0fea6b435edb58fbe7))

On the [[supergeometry|super]]-[[moduli space]] of [[super Riemann surfaces]] (proving it is generically not "projected" in the presence of [[Ramond punctures]], in that it does not [[retraction|retract]] onto its bosonic [[body]]):

* [[Nadia Ott]], [[Alexander A. Voronov]], *The supermoduli space of genus zero SUSY curves with Ramond punctures* [[arXiv:1910.05655](https://arxiv.org/abs/1910.05655)]

On [[mysterious duality]]/[[U-duality]] understood via [[Hypothesis H]] by [[automorphisms]] of [[iterated loop space|iterated]] [[cyclic loop spaces]] of the [[4-sphere]]:

* [[Alexander Voronov]] (joint with [[Hisham Sati]]), *Mysterious Duality*, talk at Texas Tech 2021 ([abstract pdf](https://www.math.ttu.edu/~lhoang/PureMath/abstract-2021-03-15-Voronov.pdf) [[Voronov_MysteriousAbstract.pdf:file]], [slides pdf](https://www.math.ttu.edu/~lhoang/PureMath/Slides-2021-03-15-Voronov.pdf), [[Voronov_MysteriousSlides.pdf:file]])

* [[Hisham Sati]], [[Alexander Voronov]], *Mysterious Triality and Rational Homotopy Theory* [[arXiv:2111.14810](https://arxiv.org/abs/2111.14810)]

* [[Hisham Sati]], [[Alexander Voronov]], *Mysterious Triality and M-Theory* [[arXiv:2212.13968](https://arxiv.org/abs/2212.13968)

category: people

[[!redirects Alexander A. Voronov]]

[[!redirects Sasha Voronov]]


Alexander Westphal

* [webpage](http://www.desy.de/~westphal/)

## related $n$Lab entries

* [[moduli stabilization]]

category: people


Alexander Zamolodchikov

* [Wikipedia entry](http://en.wikipedia.org/wiki/Alexander_Zamolodchikov)

## Selected writings

Introducing [[2d CFT|2d]] [[conformal field theory]]:

* [[Alexander Belavin]], [[Alexander Polyakov]], [[Alexander Zamolodchikov]], _Infinite conformal symmetry in two–dimensional quantum field theory_, Nuclear Physics B Volume 241, Issue 2, 23 July 1984, Pages 333-380 (doi:10.1016/0550-3213(84)90052-X)

Introducing [[parafermion]] [[2d CFT]]:

* [[Alexander B. Zamolodchikov]], [[Vladimir A. Fateev]] *Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in $Z_N$-symmetric statistical systems*, Sov. Phys. JETP **62** 2 (1985) 215-225 $[$[pdf](http://www.jetp.ras.ru/cgi-bin/dn/e_062_02_0215.pdf), [osti:5929972](https://www.osti.gov/biblio/5929972)$]$

Introducing the [[TT deformation]] in [[2d CFT]]:

* [[Alexander Zamolodchikov]], _Expectation value of composite field $T \bar T$ in two-dimensional quantum field theory_ ([arXiv:hep-th/0401146](https://arxiv.org/abs/hep-th/0401146))

## Related $n$Lab entries

* [[Knizhnik-Zamolodchikov equation]]

* [[conformal bootstrap]]

category: people

[[!redirects Alexander B. Zamolodchikov]]


Alexander's trick

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

#Contents#
* table of contents
{:toc}

## Statement

Consider the [[mapping space]] of those [[homeomorphisms]] from the [[closed ball]] $D^{n+1}$ to itself which restrict to the [[identity]] on the [[boundary]] [[n-sphere|$n$-sphere]].  What is known as *Alexander's trick* is a construction (due to [Alexander 1923](#Alexander23)) of an explicit deformation which witnesses that this space is [[connected topological space|connected]], hence that any two such homemorphisms are connected by an [[isotopy]].

## Related entries

* [[mapping class group]]

* [[braid group]]

## References

Named after:

* {#Alexander23} [[James W. Alexander]], *On the Deformation of an $n$ Cell*, Proc Natl Acad Sci USA. **9** 12 (1923) 406–407 [[doi:10.1073%2Fpnas.9.12.406](https://doi.org/10.1073%2Fpnas.9.12.406) ]

See also:

* Wikipedia, *[Alexander's trick](https://en.wikipedia.org/wiki/Alexander%27s_trick)*

* [[Planet Math]], *[Alexander trick](https://planetmath.org/alexandertrick)*

[[!redirects Alexander trick]]


Alexander-Whitney map

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

#Contents#
* table of contents
{:toc}

## Definition

Let $C : sAb \to Ch_\bullet^+$ be the chains/[[Moore complex]] functor of the [[Dold-Kan correspondence]].

Let $(sAb, \otimes)$ be the standard [[monoidal category]] structure given degreewise by the [[tensor product]] on [[Ab]] and let $(Ch_\bullet^+, \otimes)$ be the standard monoidal structure on the [[category of chain complexes]].

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

For $A,B \in sAb$ two abelian [[simplicial group]]s, the **Alexander-Whitney map** is the [[natural transformation]] on [[chain complex]]es

$$
  \Delta_{A,B} :  C(A \otimes B) \to C(A) \otimes C(B)
$$

defined on two $n$-simplices $a \in A_n$ and $b \in B_n$ by

$$
  \Delta_{A,B} : a \otimes b \mapsto \oplus_{p + q = n}  (\tilde d^p a) \otimes (d^q_0 b)
  \,,
$$

where the _front face map_ $\tilde d^p$ is that induced by

$$
  [p] \to [p+q] : i \mapsto i
$$

and the _back face_ $d^q_0$ map is that induced by

$$
  [q] \to [p+q] : i \mapsto i+p
  \,.
$$

=--

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

This AW map restricts to the normalized chains complex

$$
  \Delta_{A,B} :  N(A \otimes B) \to N(A) \otimes N(B)
  \,.
$$

=--

## Properties

\begin{proposition}
The Alexander-Whitney map is an [[oplax monoidal transformation]] that makes $C$ and $N$ into [[oplax monoidal functors]].
\end{proposition}

Beware that the AW map is _not_ [[symmetric monoidal category|symmetric]].
For details see *[[monoidal Dold-Kan correspondence]]*.

\begin{proposition}\label{EZAWDeformationRetract}
**([[Eilenberg-Zilber/Alexander-Whitney deformation retraction]])**
\linebreak

Let

* $A, B \,\in\, sAb = $ [[SimplicialAbelianGroups]]

and denote

* by $N(A), N(B) \,\in\, Ch^+_\bullet = $ [[ConnectiveChainComplexes]] their [[normalized chain complexes]],

* by $A \otimes B \,\in\, sAb$ the degreewise [[tensor product of abelian groups]],

* by $N(A) \otimes N(B)$ the [[tensor product of chain complexes]].

Then there is a [[deformation retraction]]

\begin{tikzcd}
  N(A) \otimes N(B)
  \ar[
    rr,
    bend right=20,
    "\mathrm{id}"{below},
    "\ "{above, name=t}
  ]
  \ar[
    rr,
    phantom,
    "\ "{name=s, yshift=-6pt}
  ]
  \ar[
    r,
    "\nabla_{A,B}"
  ]
  &
  N( A \otimes B )
  \ar[
    r,
    "\Delta_{A,B}"
  ]
  &
  N(A) \otimes N(B)
  \ar[
    from=s,
    to=t,
    -,
   shift left=1pt
  ]
  \ar[
    from=s,
    to=t,
    -,
   shift right=1pt
  ]
\end{tikzcd}

\begin{tikzcd}
  N( A \otimes B )
  \ar[
    rr,
    bend right=20,
    "\mathrm{id}"{below},
    "\ "{above, name=t}
  ]
  \ar[
    rr,
    phantom,
    "\ "{name=s, yshift=-6pt}
  ]
  \ar[
    r,
    "\Delta_{A,B}"
  ]
  &
  N(A) \otimes N(B)
  \ar[
    r,
    "\nabla_{A,B}"
  ]
  &
  N( A \otimes B )
  \ar[
    from=s,
    to=t,
    Rightarrow
  ]
\end{tikzcd}

where

* $\nabla_{A,B}$ is the [[Eilenberg-Zilber map]];

* $\Delta_{A,B}$ is the [[Alexander-Whitney map]].

\end{proposition}

For unnormalized chain complexes, where we have a [[homotopy equivalence]], this is the original [[Eilenberg-Zilber theorem]] ([Eilenberg & Zilber 1953](#EilenbergZilber53), [Eilenberg & MacLane 1954, Thm. 2.1](#EilenbergMacLane54)). The above [[deformation retraction]] for normalized chain complexes is [Eilenberg & MacLane 1954, Thm. 2.1a](#EilenbergMacLane54). Both are reviewed in [May 1967, Cor. 29.10](#May67). Explicit description of the [[homotopy operator]] is given in [Gonzalez-Diaz & Real 1999](#GonzalezDiazReal99)).

## Related concepts

* **Alexander-Whitney map**

* [[Eilenberg-Zilber map]]

## References

The [[Eilenberg-Zilber theorem]] is due to

* {#EilenbergZilber53} [[Samuel Eilenberg]], [[Joseph Zilber]], _On Products of Complexes_, Amer. Jour. Math. 75 (1): 200–204, (1953) ([jstor:2372629](https://www.jstor.org/stable/2372629), [doi:10.2307/2372629](https://doi.org/10.2307/2372629))

* {#EilenbergMacLane54} [[Samuel Eilenberg]], [[Saunders MacLane]], Section 2 of: *On the Groups $H(\Pi,n)$, II: Methods of Computation*, Annals of Mathematics, Second Series, Vol. 60, No. 1 (Jul., 1954), pp. 49-139 ([jstor:1969702](https://www.jstor.org/stable/1969702))

using the definition of the [[Eilenberg-Zilber map]] in:

* {#EilenbergMacLane53} [[Samuel Eilenberg]], [[Saunders MacLane]], (5.3) of: *On the groups $H(\Pi,n)$*, I*, Ann. of Math. (2) 58, (1953), 55–106. ([jstor:1969820](https://www.jstor.org/stable/1969820))

Review:

* {#May67} [[Peter May]], Section 29 of: _Simplicial objects in algebraic topology_ , Chicago Lectures in Mathematics, University of Chicago Press 1967 ([ISBN:9780226511818](https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html), [djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu), [[May_SimplicialObjectsInAlgebraicTopology.pdf:file]])

* {#GonzalezDiazReal99} Rocio Gonzalez-Diaz, Pedro Real, *A Combinatorial Method for Computing Steenrod Squares*, Journal of Pure and Applied Algebra 139 (1999) 89-108 ([arXiv:math/0110308](https://arxiv.org/abs/math/0110308))

[[!redirects Alexander-Whitney maps]]


Alexander-Čech duality

## Idea

Let $X$ be a [[finite homotopy type|finite]] [[CW complex]]
and $X \subseteq S^{n+1}$. Then there is a map

$$ X \times (S^{n+1}\backslash X) \to S^n,\,\,\,\,\,\, (x,y) \mapsto \frac{x-y}{\|x - y\|}.$$

This element determines an element
$$\delta \in H^n( X \times (S^{n+1}\backslash X)).$$

In [[topology]] there is a [[slant product]] operation, sort of _dividing_,
and in this case one can do slant product with $\delta$.
This way one obtains a map
$$
\delta_{/}  : H_{i}(X)\to H^{n-i}(S^{n+1}\backslash{X}).
$$

This map is an isomorphism and it is called the __Alexander-Čech duality__ (or sometimes simply Alexander duality).
It can be considered for infinite complexes as well, but in that
case one has to change the flavour of (co)homology theories
involved. $H_i$ is then the **Steenrod-Sitnikov homology**
and $H^{n-i}$ has to be cohomology (?).

The [[Spanier-Whitehead duality]] is a generalization, where $S^{n+1}\backslash X$ is
replaced by any space $D_n(X)$, together with an element
$\delta$ such that the corresponding map

$$\delta_/ : H_i(X) \to H^{n-i}(\mathcal{D}_n X)$$

is an isomorphism. It follows that one may replace $\mathcal{D}_n X$ by its
suspension and so on, hence the stable homotopy theory is
a natural setup for this duality.

## References

Named after [[James W. Alexander]] and [[Eduard Čech]].

[[!redirects Alexander duality]]


Alexandr S. Mishchenko

* [personal page](http://mech.math.msu.su/~asmish/)

* [MathNet page](https://www.mathnet.ru/eng/person8599)

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

## Selected writings

On [[vector bundles]]:

* [[Glenys Luke]], [[Alexandr S. Mishchenko]], *Vector bundles and their applications*, Math. and its Appl. **447** Kluwer (1998) [[doi:10.1007/978-1-4757-6923-4](https://doi.org/10.1007/978-1-4757-6923-4), [MR99m:55019](http://www.ams.org/mathscinet-getitem?mr=99m:55019)]

On [[Hermitian K-theory]]:

* {#Mishchenko76} [[Alexandr S. Mishchenko]], *Hermitian K-Theory. The Theory of characteristic classes and methods of functional analysis*, Uspeki Mat. Nauk **31** 2 (1976) 69-134,  Russ. Math. Surv. **31** 71 (1976) [[doi:10.1070/RM1976v031n02ABEH001478](https://iopscience.iop.org/article/10.1070/RM1976v031n02ABEH001478), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/mishch5.pdf)]

[[!redirects Alexandr Mishchenko]]

[[!redirects Alexander S. Mishchenko]]
[[!redirects Alexander Mishchenko]]


Alexandre Deur

* [spire page](https://inspirehep.net/authors/1049232)

## Selected writings

On the [[running coupling constant]] in [[QCD]]:

* [[Alexandre Deur]], [[Stanley Brodsky]], [[Guy de Teramond]], *The QCD Running Coupling*, Prog. Part. Nuc. Phys. 90 1 (2016) ([arXiv:1604.08082](https://arxiv.org/abs/1604.08082), [doi:10.1016/j.ppnp.2016.04.003](https://doi.org/10.1016/j.ppnp.2016.04.003))

category: people


Alexandre Kirillov

**Alexandre Aleksandrovich Kirillov** (Александр Александрович Кириллов) is a professor of mathematics at the University of Pennsylvania. He invented the orbit method in [[representation theory]], developed in other flavours also by Kostant and Souriau. He was a student of [[Israel Gelfand]] and they together lead a famous journal _Functional Analysis and its Applications_.

* [wikipedia](http://en.wikipedia.org/wiki/Alexandre_Kirillov)

## Selected writings

On [[Reshetikhin-Turaev theory]] constructing [[modular functors]] from [[modular tensor categories|modular]] [[tensor categories]]:

* [[Bojko Bakalov]], [[Alexandre Kirillov]], _Lectures on tensor categories and modular functors_, University Lecture Series **21**, Amer. Math. Soc. (2001) ([ISBN:978-1-4704-2168-7](https://bookstore.ams.org/ulect-21), [webpage](http://www.math.sunysb.edu/~kirillov/tensor/tensor.html)).

[[!redirects Александр Кириллов]]

category: people


Alexandre Miquel

* [webpage](http://perso.ens-lyon.fr/alexandre.miquel/)

category: people


Alexandre Vinogradov

[[!redirects Alexander Vinogradov]]
Alexander M. Vinogradov was a Russian mathematician working in the geometric theory of [[differential equations]], the theory of [[diffieties]] with applications to [[variational calculus]].

* [webpage](https://web.archive.org/web/20180914224323/http://diffiety.ac.ru/curvita/amv.htm)

## some publications

* A. M. Vinogradov, _Cohomological analysis of partial differential equations and secondary calculus_, [bookpage](http://diffiety.ac.ru/books/mmono204.htm) Translations of Mathematical Monographs __204__, Amer. Math. Soc. 2001, 247 p.

* A. M. Vinogradov, _Local symmetries and conservation laws_, Acta Appl. Math., Vol. 2, 1984, p. 21, [MR85m:58192](http://www.ams.org/mathscinet-getitem?mr=736872), [doi](http://dx.doi.org/10.1007/BF01405491)

* A. M. Vinogradov, _Symmetries and conservation laws of partial differential equations: basic notions and results_, Acta Appl. Math., Vol. 15, 1989, p. 3. [MR91b:58282](http://www.ams.org/mathscinet-getitem?mr=1007340), [doi](http://dx.doi.org/10.1007/BF00131928)

* A. M. Vinogradov, _Scalar differential invariants, diffieties and characteristic classes_, in: Mechanics, Analysis and Geometry: 200 Years after, 379–414, [MR92e:58244](http://www.ams.org/mathscinet-getitem?mr=1098525)

* [[G. Vezzosi]], A. Vinogradov, _On higher order analogues of De Rham cohomology_, Diff. Geom. and Appl. __19__ (2003), 29-59.

* G. Vezzosi, A.M. Vinogradov, _Infinitesimal Stokes' formula for higher-order de Rham complexes_, Acta Appl. Math. __49__, N. 3 (1997), 311-329.

## related $n$Lab entries

* [[diffiety]]

* [[variational calculus]]

* [[Euler-Lagrange complex]]

category:people
[[!redirects A. Vinogradov]]
[[!redirects Alexander Vinogradov]]
[[!redirects A. M. Vinogradov]]
[[!redirects Aleksander Vinogradov]]
[[!redirects Alexandre M. Vinogradov]]
[[!redirects Aleksander M. Vinogradov]]
[[!redirects Alexander M. Vinogradov]]


Alexandros Anastasiou

* [GoogleScholar page](https://scholar.google.se/citations?user=02U4jccAAAAJ)

## Selected writings

On [[Freudenthal magic square|magic squares]] of [[magic supergravities]]  and [[U-duality]] via the [[classical double copy]]:

* {#ABDHN} [[Alexandros Anastasiou]], [[Leron Borsten]], [[Michael Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], *A magic pyramid of supergravities*, JHEP (2014) 178 [[arXiv:1312.6523](http://arxiv.org/abs/1312.6523)]

category: people


Alexandrov space

# Alexandrov spaces
* table of contents
{: toc}

## Idea

Alexandrov spaces are certain spaces that arise in [[differential geometry]], named after [[Alexander Alexandrov]].  They should not be confused with [[Alexandroff spaces]], which arise in [[general topology]] and are named after [[Paul Alexandroff]].

## References

* Yukio Otsu, _Differential geometric aspects
of Alexandrov spaces_, in: Comparison Geometry, MSRI publications 30, 1997. [pdf](http://library.msri.org/books/Book30/files/otsu.pdf)
* G. Perelman, _Elements of Morse theory on Alexandrov spaces_, SPb Math. J. 1994.
* G. Perelman, DC structure on Alexandrov spaces, [pdf](http://www.math.psu.edu/petrunin/papers/alexandrov/Cstructure.pdf)

* Yu. Burago, M. Gromov, G. Perelman, _Alexandrov spaces with curvature bounded below_, Russ. Math. Surveys 1992.

* MathOverflow [Details of Perelman's example about soul of Alexandrov space](http://mathoverflow.net/questions/22105/details-of-perelmans-example-about-soul-of-alexandrov-space), [Metrically singular Alexandrov space](http://mathoverflow.net/questions/67046/metrically-singular-alexandrov-space), [Is there Domain Invariance for Alexandrov spaces?](http://mathoverflow.net/questions/21512/is-there-domain-invariance-for-alexandrov-spaces), [Rigidity of triangle comparison in Alexandrov spaces](http://mathoverflow.net/questions/44943/rigidity-of-triangle-comparison-in-alexandrov-spaces)

[[!redirects Alexandrov space]]
[[!redirects Alexandrov spaces]]
[[!redirects Alexandrov space (differential geometry)]]
[[!redirects Alexandrov spaces (differential geometry)]]
[[!redirects Alexandrov space (in differential geometry)]]
[[!redirects Alexandrov spaces (in differential geometry)]]
[[!redirects Alexandrov space in differential geometry]]
[[!redirects Alexandrov spaces in differential geometry]]


Alexandru Buium

* [webpage](http://www.math.unm.edu/~buium/)

###Related entries

* [[arithmetic differential geometry]]

category: people


Alexandru Chirvasitu

* [webpage](http://math.berkeley.edu/~calex/)

category: people


Alexandru Dimca

* [personal page](https://math.unice.fr/~dimca/)

* [Wikipedia entry](https://math.unice.fr/~dimca/)

## Selected writings

On ([[abelian sheaf|abelian]]) [[sheaf theory]] in [[topology]], with a focus on [[constructible sheaves]] and [[perverse sheaves]]:

* [[Alexandru Dimca]], *Sheaves in Topology*, Universitext, Springer (2004) $[$[doi:10.1007/978-3-642-18868-8](https://doi.org/10.1007/978-3-642-18868-8)$]$

category: people


Alexandru E. Stanculescu

## Selected writings

On the [[Bousfield-Friedlander theorem]]:

* {#Stanculescu} [[Alexandru E. Stanculescu]], *Note on a theorem of Bousfield and Friedlander*, Topology and its Applications **155** 13 (2008) 1434-1438 [[arxiv:0806.4547](https://arxiv.org/abs/0806.4547), [doi:10.1016/j.topol.2008.05.003](https://doi.org/10.1016/j.topol.2008.05.003)]

On [[bifibrations]], [[factorization systems]] and [[model structures on Grothendieck fibrations]]:

* {#Stanculesu12} [[Alexandru E. Stanculescu]], *Bifibrations and weak factorisation systems*, Applied Categorical Structures **20** 1 (2012) 19–30 [doi:10.1007/s10485-009-9214-3]

On [[model structures on enriched categories]]:

* [[Alexandru Stanculescu]], *Constructing model categories with prescribed fibrant objects*, Theory and Applications of Categories, **29** 23 (2014) 635-653  [[arXiv:1208.6005](http://arxiv.org/abs/1208.6005), [tac:29-23](http://www.tac.mta.ca/tac/volumes/29/23/29-23abs.html)]

category: people

[[!redirects Alexandru Stanculescu]]


Alexandru Solian

## Selected writings

Early discussion of [[2-groups]] as [[Picard 2-groups]]:

* {#Solian72} [[Alexandru Solian]], *Groupe dans une catégorie*, C. R. Acad. Sc. Paris **275** (1972) [[ark:/12148/bpt6k57310477/f7](https://gallica.bnf.fr/ark:/12148/bpt6k57310477/f7.item)]

* {#Solian80} [[Alexandru Solian]], *Coherence in categorical groups*,  Communications in Algebra **9** 10 (1980) 1039-1057 [[doi:10.1080/00927878108822631](https://doi.org/10.1080/00927878108822631)]

category: people


Alexei Abrikosov

* Wikiepedia entry

## Selected writings

On what came to be called [[Abrikosov vortices]] ([[vortex strings]]) in [[type II superconductors]]:

* {#Abrikosov57a} [[Alexei Abrikosov]], *On the Magnetic properties of superconductors of the second group*, Sov. Phys. JETP **5** (1957) 1174-1182; Zh. Eksp. Teor. Fiz. **32** (1957) 1442-1452 [[spire:9138](https://inspirehep.net/literature/9138), [[Abrikosov-MagneticPropertiesOfSuperconductors.pdf:file]]]

* {#Abrikosov57} [[Alexei Abrikosov]], _The magnetic properties of superconducting alloys_, Journal of Physics and Chemistry of Solids Volume 2, Issue 3, 1957, Pages 199-208 (doi:10.1016/0022-3697(57)90083-5)

category: people

[[!redirects Alexei Alexeyevich Abrikosov]]


Alexei Bondal

__Alexei Bondal__ (Алексей Игоревич Бондал, spellings also Aleksei, Alexey etc.) is a mathematician at the Steklov Institute of Russian Academy of Sciences (and at Aberdeen). He has pioneered with [[Mikhail Kapranov]] methods of replacing varieties by [[enhanced triangulated categories]] of [[coherent sheaves]] leading to early developments of [[derived algebraic geometry]], influencing early works of [[Maxim Kontsevich|Kontsevich]] and of [[Carlos Simpson|Simpson]]--[[Bertrand Toen|Toen]] school.

* [data at ras](http://www.mi.ras.ru/staff/bondal_e.html)
* math genealogy [page](http://genealogy.math.ndsu.nodak.edu/id.php?id=85435)

## Selected writings

Introducing [[Serre functors]]:

* {#BondalKapranov90} [[Alexei I. Bondal]], [[Mikhail M. Kapranov]], _Representable functors, Serre functors, and mutations_, Mathematics of the USSR-Izvestiya **35** 3 (1990) 519-541 [[doi:10.1070/IM1990v035n03ABEH000716](https://doi.org/10.1070/IM1990v035n03ABEH000716)]

## Related entries

* [[Bondal-Orlov reconstruction theorem]]

* [[triangulated categories of sheaves]]

category: people

[[!redirects Alexei Bondal]]
[[!redirects Alexei Igorevich Bondal]]
[[!redirects Alexeĭ Bondal]]
[[!redirects Alexej Bondal]]
[[!redirects Alexey Bondal]]
[[!redirects Aleksei Bondal]]
[[!redirects Alekseĭ Bondal]]
[[!redirects Aleksej Bondal]]
[[!redirects Aleksey Bondal]]
[[!redirects Алексей Бондал]]
[[!redirects Алексей Игоревич Бондал]]
[[!redirects A. Bondal]]
[[!redirects Bondal]]
[[!redirects A.I.Bondal]]

[[!redirects Alexei I. Bondal]]


Alexei Davydov

* [website](https://www.ohio.edu/cas/davydov)

## Selected writings
 {#SelectedWritings}

On [[magmoidal categories]]:

* [[Alexei Davydov]], _Nuclei of categories with tensor products_, Theory and Applications of Categories, Vol. 18, 2007, No. 16, pp 440-472 ([tac:18-16](http://www.tac.mta.ca/tac/volumes/18/16/18-16abs.html))

category: people


Alexei Kitaev

* [Wikipedia entry](http://en.wikipedia.org/wiki/Alexei_Kitaev)

## Selected writings

On [[quantum computation]] and [[quantum error correction]] (and stating the [[Solovay-Kitaev theorem]]):

* [[Alexei Kitaev]],  *Quantum computations: algorithms and error correction*, Russian Mathematical Surveys, **52** 6 (1997) [[doi:10.1070/RM1997v052n06ABEH002155](https://iopscience.iop.org/article/10.1070/RM1997v052n06ABEH002155), pdf]

On [[quantum circuits]] with [[mixed quantum states]]/[[density matrices]]:

* [[Dorit Aharonov]], [[Alexei Kitaev]], [[Noam Nisan]], *Quantum Circuits with Mixed States*, *Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computation (STOC)* (1998) 20-30 [[arXiv:quant-ph/9806029](https://arxiv.org/abs/quant-ph/9806029), [doi:10.1145/276698.276708](https://doi.org/10.1145/276698.276708)]

On [[quantum error correcting codes]] associated with planar bulk/boundary systems (precursor to [[holographic tensor networks]]):

* S. B. Bravyi, [[Alexei Kitaev]], *Quantum codes on a lattice with boundary* ([arXiv:quant-ph/9811052](https://arxiv.org/abs/quant-ph/9811052))

On [[computation]] in general and [[quantum computation]] in particular:

* [[Alexei Y. Kitaev]], A. H. Shen, M. N. Vyalyi, *Classical and Quantum Computation*, Graduate Studies in Mathematics **47** (2002) [[doi:10.1090/gsm/047](http://dx.doi.org/10.1090/gsm/047)]

Introducing the idea of [[topological quantum computation]] with [[anyons]]:

* [[Alexei Kitaev]], _Fault-tolerant quantum computation by anyons_, Annals Phys. 303 (2003) 2-30 ([arXiv:quant-ph/9707021](https://arxiv.org/abs/quant-ph/9707021), doi:10.1016/S0003-4916(02)00018-0)

* [[Michael Freedman]], [[Alexei Kitaev]], [[Michael Larsen]], [[Zhenghan Wang]], _Topological quantum computation_,  Bull. Amer. Math. Soc. __40__ (2003), 31-38 ([arXiv:quant-ph/0101025](https://arxiv.org/abs/quant-ph/0101025), [doi:10.1090/S0273-0979-02-00964-3](https://doi.org/10.1090/S0273-0979-02-00964-3), [pdf](http://www.ams.org/journals/bull/2003-40-01/S0273-0979-02-00964-3/S0273-0979-02-00964-3.pdf))

Introducing a model with [[Majorana zero modes]] in quantum wires ([[Kitaev spin chain]]):

* [[Alexei Kitaev]], *Unpaired Majorana fermions in quantum wires*,  Physics-Uspekhi, **44** 10S (2001) 131-136 [[doi:10.1070/1063-7869/44/10S/S29](https://iopscience.iop.org/article/10.1070/1063-7869/44/10S/S29)]

and its [[interaction|interacting]] generalization:

* [[Lukasz Fidkowski]], [[Alexei Kitaev]], *The effects of interactions on the topological classification of free fermion systems*, Phys. Rev. B **81** (2010) 134509  [[arXiv:0904.2197](https://arxiv.org/abs/0904.2197), [doi:10.1103/PhysRevB.81.134509](https://doi.org/10.1103/PhysRevB.81.134509)]

Comprehensive discussion of [[anyons]]:

* {#Kitaev06} [[Alexei Kitaev]], *Anyons in an exactly solved model and beyond*, Annals of Physics **321** 1 (2006) 2-111 $[$[doi:10.1016/j.aop.2005.10.005](https://doi.org/10.1016/j.aop.2005.10.005)$]$

On [[entanglement entropy]] as an indicator of [[topological phases of matter]]:

* [[Alexei Kitaev]], [[John Preskill]], *Topological entanglement entropy*, Phys. Rev. Lett. 96 (2006) 110404 ([arXiv:hep-th/0510092](https://arxiv.org/abs/hep-th/0510092))

On potential [[experiments]] detecting [[uncertainty of fluxes]] in [[quantum electrodynamics|quantum]] [[electromagnetism]]:

* [[Alexei Kitaev]], [[Gregory W. Moore]], [[Kevin Walker]], *Noncommuting Flux Sectors in a Tabletop Experiment* [[arXiv:0706.3410](https://arxiv.org/abs/0706.3410)]

On the classification of [[gapped Hamiltonians]]/[[topological insulators]] by [[topological K-theory]]:

* [[Alexei Kitaev]], _Periodic table for topological insulators and superconductors_, talk at: L.D.Landau Memorial Conference "Advances in Theoretical Physics", June 22-26, 2008, In:AIP Conference Proceedings 1134, 22 (2009) ([arXiv:0901.2686](https://arxiv.org/abs/0901.2686), [doi:10.1063/1.3149495](https://doi.org/10.1063/1.3149495))

Introducing the notion of [[short-range entanglement]] in [[quantum materials]]:

* {#Kitaev11} [[Alexei Kitaev]], *Toward Topological Classification of Phases with Short-range Entanglement*, talk at KITP (2011) $[$[video](https://online.kitp.ucsb.edu/online/topomat11/kitaev/)$]$

* {#Kitaev13} [[Alexei Kitaev]], *On the Classification of Short-Range Entangled States*, talk at Simons Center (2013) $[$[video](https://scgp.stonybrook.edu/archives/7874)$]$

category: people

[[!redirects Alexei Y. Kitaev]]


Alexei Kovalev

* [webpage](https://www.dpmms.cam.ac.uk/~agk22/)

## selected writings

* {#Kovalev03} [[Alexei Kovalev]], _Twisted connected sums and special Riemannian holonomy_, J. Reine Angew. Math. 565 (2003), 125--160 ([arXiv:math/0012189](https://arxiv.org/abs/math/0012189))

  (on [[compact twisted connected sum G2-manifolds]])

## related $n$Lab entries

* [[G2-manifold]]

* [[M-theory on G2-manifolds]]

category: people


Alexei M. Tsvelik

* [GoogleScholar page](https://scholar.google.com/citations?user=Q-A96cUAAAAJ&hl=en)

* [institute page](https://www.bnl.gov/staff/atsvelik)

## Selected writings

An [[integrable model]] for [[cyclic group|$\mathbb{Z}_N$]]-[[parafermion]] [[topological order]]:

* {#Tsvelik14a} [[Alexei M. Tsvelik]], *An integrable model with parafermion zero energy modes*, Phys Rev. Lett. **113** 066401 (2014) $[$[arXiv:1404.2840](https://arxiv.org/abs/1404.2840), [doi:10.1103/PhysRevLett.113.066401](https://doi.org/10.1103/PhysRevLett.113.066401)$]$

* {#Tsvelik14b} [[Alexei M. Tsvelik]], *$\mathbf{Z}_N$ parafermion zero modes without Fractional Quantum Hall effect* $[$[arXiv:1407.4002](https://arxiv.org/abs/1407.4002)$]$

category: people


Alexei Morozov

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

* [iip page](https://www.iip.ufrn.br/eventslecturer?inf===gTEVFe)

* [InSpire page](https://inspirehep.net/authors/996770)

## Selected writings

On [[string theory]]:

* [[Alexei Morozov]], *String theory: what is it?*, Sov. Phys. Usp. **35** (1992) 671-714 [[doi:10.1070/PU1992v035n08ABEH002255](https://iopscience.iop.org/article/10.1070/PU1992v035n08ABEH002255)]

On [[topological quantum computation]] with [[Chern-Simons theory]] (e.g. with [[su(2)-anyons]]):

* [[Dmitry Melnikov]], [[Andrei Mironov]], [[Sergey Mironov]], [[Alexei Morozov]], [[Andrey Morozov]], _Towards topological quantum computer_, Nucl. Phys. B926 (2018) 491-508 ([arXiv:1703.00431](https://arxiv.org/abs/1703.00431), [doi:10.1016/j.nuclphysb.2017.11.016](https://doi.org/10.1016/j.nuclphysb.2017.11.016))

On [[Yangians]]:

* Dmitry Galakhov, [[Alexei Morozov]], Nikita Tselousov, *Towards the theory of Yangians* [[arXiv:2311.00760](https://arxiv.org/abs/2311.00760)]

category: people


Alexei Nurmagambetov

* [inSPIRE page](http://inspirehep.net/author/profile/A.J.Nurmagambetov.3)

## Selected writings

On the [[Green-Schwarz sigma-model]] for the [[M5-brane]]:

* {#BLNPST97} [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _Covariant Action for the Super-Five-Brane of M-Theory_, Phys. Rev. Lett. 78 (1997) 4332-4334 ([arXiv:hep-th/9701149](http://arxiv.org/abs/hep-th/9701149))

* [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _On the equivalence of different formulations of the M Theory five--brane_, Phys. Lett. B408 (1997) 135-141 ([arXiv:hep-th/9703127](http://arxiv.org/abs/hep-th/9703127))

On [[AdS/CFT]] for the [[M5-brane]]:

* [[Alexei Nurmagambetov]], I. Y. Park, _On the M5 and the AdS7/CFT6 Correspondence_ ([arXiv:hep-th/0110192](http://arxiv.org/abs/hep-th/0110192))

On [[type II supergravity]]:

* {#BNS04} [[Igor Bandos]], [[Alexei Nurmagambetov]], [[Dmitri Sorokin]], _Various Faces of Type IIA Supergravity_, Nucl. Phys. B676 (2004) 189-228 ([arXiv:hep-th/0307153](https://arxiv.org/abs/hep-th/0307153))

## Related $n$Lab entries

* [[M5-brane]]

* [[AdS/CFT]]

[[!redirects Aleksei Nurmagambetov]]

[[!redirects Alexei J. Nurmagambetov]]


Alexei Oblomkov

* [webpage](http://people.math.umass.edu/~oblomkov)

## Selected writings

On the [[Gromov-Witten/Donaldson-Thomas correspondence]]:

* [[Davesh Maulik]], [[Alexei Oblomkov]], [[Andrei Okounkov]], [[Rahul Pandharipande]], _Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds_ ([arXiv:0809.3976](https://arxiv.org/abs/0809.3976v1))

category: people


Alexei P. Kopylov

* [personal page](https://www.cs.cornell.edu/people/kopylov/)

## Selected writings

On [[affine logic]]:

* [[Alexei P. Kopylov]], *Decidability of Linear Affine Logic*, Information and Computation **164** 1 (2001) 173-198 [[doi:10.1006/inco.1999.2834](https://doi.org/10.1006/inco.1999.2834)]

category: people

[[!redirects Alexei Kopylov]]


Alexei Pirkovskii

* [Institute page](https://www.hse.ru/en/staff/pirkovskii)

## Selected writings

On [[EFC-algebra|EFC-algebras]]:

* [[Alexei Pirkovskii]], *Holomorphically finitely generated algebras*,  Journal of Noncommutative Geometry 9 (2015), 215–264 ([arXiv:1304.1991](https://arxiv.org/abs/1304.1991), [doi:10.4171/JNCG/192](https://www.ems-ph.org/journals/show_abstract.php?issn=1661-6952&vol=9&iss=1&rank=8)).

category: people


Alexey O. Remizov

* [math-net entry](https://www.mathnet.ru/eng/person12248)

## Selected writings

On basic [[linear algebra]], [[representation theory]] and basic [[geometry]] ([[Euclidean spaces|Euclidean]]-, [[affine spaces|affine]]-, [[projective space|projective]]-, [[hyperbolic spaces]]):

* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), ISBN:9783642309939 [pdf](https://ia801507.us.archive.org/33/items/LinearAlgebraAndGeometryShafarevichIgorR.RemizovAlexey/Linear%20Algebra%20and%20Geometry%20%20%5B%20Shafarevich%2C%20Igor%20R.%2C%20Remizov%2C%20Alexey%20%5D.pdf), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]

[[!redirects Alexey Remizov]]


Alexey Petrov

* [webpage](http://www.physics.wayne.edu/~apetrov/Welcome.html)

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

## Selected writings

On [[D-meson]] [[decays]]:

* Marina Artuso, Brian Meadows, [[Alexey Petrov]], _Charm Meson Decays_, Ann. Rev. Nucl. Part. Sci.58:249-291, 2008 ([arXiv:0802.2934](https://arxiv.org/abs/0802.2934))

On [[effective field theory]]:

* {#PetrovBlechman16} [[Alexey Petrov]], Andrew E Blechman, _Effective Field Theories_, World Scientific 2016 ([doi:10.1142/8619](https://doi.org/10.1142/8619))

On indirect searches for New Physics (i.e. [[phenomenology]] beyond the [[standard model of particle physics]]),

specifically via [[charm quark]] physicsl:

* Alexey Petrov, _Searching for New Physics with Charm_ ([arXiv:1003.0906](https://arxiv.org/abs/1003.0906))

## Related $n$Lab entries

* [[effective field theory]]

category: people


Alexey Slinkin

* [GoogleScholar page](https://scholar.google.ru/citations?user=xeD0KksAAAAJ&hl=ru)

## Selected writings

On 1-[[twisted de Rham cohomology]] related to [[sl(2)|$\mathfrak{sl}(2)$]]-[[Verma modules]]:

* {#SlinkinVarchenko19} [[Alexey Slinkin]], [[Alexander Varchenko]], *Twisted de Rham Complex on Line and Singular Vectors in $\widehat{\mathfrak{sl}}_2$ Verma Modules*, SIGMA **15** (2019) 075 [[arXiv:1812.09791](https://arxiv.org/abs/1812.09791), [doi:10.3842/SIGMA.2019.075](https://doi.org/10.3842/SIGMA.2019.075)]

category: people


Alexis Hazell

Assistant system administrator for the nLab.

[[!redirects AlexisHazell]]


Alexis Toumi

[alexis.toumi.xyz](https://alexis.toumi.xyz)

## Selected writings

On [[DisCoPy]]:

* Giovanni de Felice, [[Alexis Toumi]], [[Bob Coecke]]. _DisCoPy: Monoidal Categories in Python_. Applied Category Theory, 2020. ([arXiv:2005.02975l](https://arxiv.org/abs/2005.02975l))

## Related entries

* [[DisCoPy]]

category: people


Alexis Virelizier

* [personal page](http://math.univ-lille1.fr/~virelizi/)

## Selected writings

On [[Hopf monads]] and [[Hopf adjunctions]]:

* [[Alain Bruguières]], [[Alexis Virelizier]], *Hopf monads*, Advances in Mathematics **215** 2 (2007) 679-733 [[doi:10.1016/j.aim.2007.04.011](https://doi.org/10.1016/j.aim.2007.04.011), [arXiv:math/0604180](https://arxiv.org/abs/math/0604180)]

* [[Alain Bruguières]], [[Steve Lack]], [[Alexis Virelizier]], *Hopf monads on monoidal categories*,  Adv. Math. __227__ 2 (2011) 745-800 [[arXiv:1003.1920](https://arxiv.org/abs/1003.1920), [doi:10.1016/j.aim.2011.02.008](https://doi.org/10.1016/j.aim.2011.02.008)]

category: people


Aleš Pultr

[[!redirects Aleš Pultr]]
Aleš Pultr is a professor at Charles University.

[Website](https://kam.mff.cuni.cz/~pultr/),
[university web page](https://www.mff.cuni.cz/en/faculty/organizational-structure/people?hdl=501).

## Selected writings

A textbook on [[locales]]:

* [[Jorge Picado]], [[Aleš Pultr]], _[[Frames and Locales]]. Topology without points_, Frontiers in Mathematics, Birkhäuser (2012).


Alfred Ewing

* [Wikipedia entry](http://en.wikipedia.org/wiki/A._C._Ewing)
* [Obituary](http://www.anthonyflood.com/blanshardewing.htm), Brand Blanshard

## some writings

* _Idealism: A Critical Survey_, Barnes & Noble (1974) ([PhiPapers](http://philpapers.org/rec/EWIIAC), [Review on JSTOR](http://www.jstor.org/stable/2249864))

## related $n$Lab entries

* [[idealism]]

[[!redirects A. C. Ewing]]
[[!redirects Alfred Cyril Ewing]]

category: people


Alfred Frölicher

* [Wikipedia entry](http://en.wikipedia.org/wiki/Alfred_Fr%C3%B6licher)

## Selected writings

Introducing the [[Frölicher-Nijenhuis bracket]]:

* [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms.  Part I.  Derivations in the graded ring of differential forms_, Indagationes Mathematicae (Proceedings) **59** (1956) 338–350 and 351–359 [doi:10.1016/s1385-7258(56)50046-7, doI;10.1016/s1385-7258(56)50047-9]

and its refinement for [[almost complex structures]]:

* [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms.  Part II.  Almost-complex structures_, Indagationes Mathematicae (Proceedings) **61** (1958) 414–421 and 422-429 [doi:10.1016/S1385-7258(58)50058-4, doi:10.1016/S1385-7258(58)50057-2]

## Related entries

* [[Frölicher space]]

category: people

[[!redirects Alfred Frolicher]]
[[!redirects A. Frölicher]]
[[!redirects Frölicher]]


Alfred Gray

* Wikipedia entry

## Selected writings

* [[Alfred Gray]],  [_A Note on Manifolds Whose Holonomy Group is a Subgroup of Sp(n) $\cdot$ Sp(1)_](https://projecteuclid.org/euclid.mmj/1029000212), Michigan Math. J. Volume 16, Issue 2 (1969), 125-128.

  (on [[Sp(n).Sp(1)]]-[[special holonomy]])

* {#GrayGreen70} [[Alfred Gray]], Paul S. Green, _Sphere transitive structures and the triality automorphism_, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 ([euclid:1102976640](https://projecteuclid.org/euclid.pjm/1102976640))

  (on [[transitive actions]] of [[compact Lie groups]] on [[n-spheres]])

## Related $n$Lab entries

* [[Sp(n).Sp(1)]]

* [[n-sphere]]

* [[Spin(8)]]

* [[G2/SU(3) is the 6-sphere]]

* [[Spin(7)/G2 is the 7-sphere]]

* [[Spin(5)/SU(2) is the 7-sphere]]

* [[Spin(9)/Spin(7) is the 15-sphere]]

category: people


Alfred Rieckers

* [personal page](https://homepages.uni-tuebingen.de/alfred.rieckers/)

* [InSpire page](https://inspirehep.net/authors/2270468)

## Selected writings

On [[continuous field of C-star algebras|continuous fields]] of [[Weyl algebras]] as [[continuous deformation quantizations]] of [[symplectic vector spaces]]:

* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization*, Annales Henri Poincaré **5** (2004) 327–346 [[doi:10.1007/s00023-004-0171-y](https://doi.org/10.1007/s00023-004-0171-y)]

On [[group algebras]] of ([[underlying]] [[discrete group|discrete]]) [[Heisenberg groups]] as [[strict deformation quantizations]] of [[presymplectic manifold|pre-]][[symplectic vector spaces|symplectic]] [[topological vector spaces]] by [[continuous field of C-star algebras|continuous fields of]] [[Weyl algebras]]:

* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Infinite dimensional Heisenberg group algebra and field-theoretic strict deformation quantization*, International Journal of Pure and Applied Mathematics **38** 1 (2007) [[ijpam:2007-38-1/6](https://ijpam.eu/contents/2007-38-1/6/index.html), [pdf](https://www.ijpam.eu/contents/2007-38-1/6/6.pdf)]

* [[Reinhard Honegger]], [[Alfred Rieckers]], *Heisenberg Group Algebra and Strict Weyl Quantization*, Chapter 23 in: *Photons in Fock Space and Beyond, Volume I: From Classical to Quantized Radiation Systems*, World Scientific (2015) [chapter:[doi;10.1142/9789814696586_0023](https://doi.org/10.1142/9789814696586_0023), book:[doi:10.1142/9251-vol1](https://doi.org/10.1142/9251-vol1)]

category: people


Alfred Schild

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

## Selected writings

Introducing [[Kerr-Schild spacetimes]]:

* [[Roy Kerr]], [[Alfred Schild]], _Some algebraically degenerate solutions of Einstein’s gravitational field equations_, Proc. Symp. Appl. Math 17, 199, 1965

category: people


Alfred Tarski

* [SEP entry](https://plato.stanford.edu/entries/tarski/)

* [wikipedia entry](http://en.wikipedia.org/wiki/Alfred_Tarski)

## Selected writings

* [[Alfred Tarski]],  *Introduction to Logic and to the Methodology of Deductive Sciences* Oxford University Press (1994) $[$[doi:10.2307/2180610](https://doi.org/10.2307/2180610), ISBN 978-0-19-504472-0$]$

## Related entries

* [[Tarski universe]]

* [[Jónsson-Tarski algebra]]

* [[Jónsson-Tarski topos]]

* [[Lindenbaum-Tarski algebra]]

* [[Tarski-Seidenberg theorem]]

* [[Knaster-Tarski theorem]]

* [[correspondence theory of truth]]

* [[Euclidean geometry]]

* [[Tarski group]]

category: people

[[!redirects A. Tarski]]
[[!redirects Tarski]]


Alfred Whitehead

* [Wikipedia entry](http://en.wikipedia.org/wiki/Alfred_North_Whitehead)

## Selected writings

* _[[Principia Mathematica]]_ (1910) with [[Bertrand Russell]]

## Related entries

* [[assertion]]

category: people

[[!redirects Alfred North Whitehead]]


Alfred Young

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

## Selected writings

Introducing the [[seminormal representation]] in the [[representation theory of the symmetric group]]:

* [[Alfred Young]], *On quantitative substitutional analysis VI*, Proc. London Math. Soc. (2) **31** (1931), 253-289. Reprinted in: *The Collected Papers of Alfred Young, 1873-1940*, Mathematical Expositions 21, Univ. of Toronto Press, Toronto and Buffalo, 1977

## Related $n$Lab entries

* [[Young diagram]]

* [[Young tableaux]]

* [[semistandard Young tableaux]]

category: people


Alfréd Rényi

* [Wikipedia entry](https://en.wikipedia.org/wiki/Alfr%C3%A9d_R%C3%A9nyi)

## Selected writings

Introducing the notion of [[Rényi entropy]]:

* [[Alfréd Rényi]], *On Measures of Entropy and Information*, Berkeley Symposium on Mathematical Statistics and Probability, 1961: 547-561 (1961) ([euclid](https://projecteuclid.org/ebooks/berkeley-symposium-on-mathematical-statistics-and-probability/Proceedings%20of%20the%20Fourth%20Berkeley%20Symposium%20on%20Mathematical%20Statistics%20and%20Probability,%20Volume%201:%20Contributions%20to%20the%20Theory%20of%20Statistics/chapter/On%20Measures%20of%20Entropy%20and%20Information/bsmsp/1200512181))

[[!redirects Alfred Rényi]]
[[!redirects Alfréd Renyi]]
[[!redirects Alfred Renyi]]

[[!redirects A. Rényi]]
[[!redirects A. Renyi]]

category: people


Alfsen-Shultz theorem

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* table of contents
{:toc}

## Idea

The theorem of ([Alfsen-Shultz 78](#AlfsenShultz78)) says that two [[C*-algebras]] have [[isomorphism|isomorphic]] [[state on an operator algebra|state spaces]] precisely if as [[Jordan algebras]] they are isomorphic.

This is to some extent a [[Jordan algebra]]-analog of the [[Gelfand-Naimark theorem]].

## Related theorems

Other theorems about the foundations and [[interpretation of quantum mechanics]] include:

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

  * [[Kochen-Specker theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

* [[Fell's theorem]]

* [[Gleason's theorem]]

* [[Wigner theorem]]

* [[Bell's theorem]]

* [[Bub-Clifton theorem]]

* [[Kadison-Singer problem]]

## References

* [[Erik Alfsen]], [[Frederic Shultz]], _A Gelfand Neumark theorem for Jordan algebras_, Advances in Math., 28 (1978), 11-56.
  {#AlfsenShultz78}

* [[Erik Alfsen]], H. Hanche-Olsen, [[Frederic Shultz]], _State spaces of $C^\ast$-algebras_, Acta Math., 144 (1980), 267-305.
  {#AlfsenHOShultz80}

See also

* [[Frederic Shultz]], _Dual maps of Jordan homomorphisms and ∗-homomorphisms between $C^\ast$-algebras_,  Pacific J. Math. Volume 93, Number 2 (1981), 435-441 ([Euclid]( http://projecteuclid.org/euclid.pjm/1102736271))
  {#Shultz81}


Alg

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#### Algebra
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{:toc}

## Idea

_Alg_ is the [[category]] with [[algebra|algebras]] as [[object|objects]] and algebra homomorphisms as [[morphism]]s.

More abstractly, we can think of $Alg$ as the [[category|full subcategory]] of $Cat(Vect)$, [[internal category|internal categories]] in [[Vect]], with algebras as objects.

In case of [[Ab]], this gives us Category of Rings, namely, $Alg_Z$.

## Properties

### Relation to algebras with bimodules

Since algebras may be identified with one-object [[category|categories]] [[internalization|internal to]] vector spaces, it is sometimes useful to regard $Alg$ as a strict 2-category, namely as a full sub-2-category of the 2-category $Cat(Vect)$. In this case the 2-morphisms between morphisms of algebras come from "intertwiners": inner endomorphisms of the target algebra.

Precisely analogous statements hold for the category [[Grp]] of groups.

With $Alg$ regarded as a strict 2-category this way there is a canonical 2-functor

$$
  Alg \hookrightarrow Bimod
$$

to the category [[Bimod]], which sends algebra homomorphisms $f : A \to B$ to the $A$-$B$ bimodule ${}_f B$.  This exhibits $Bimod$ as a [[framed bicategory]] in the sense of Shulman.

## Related concepts

* [[Vect]]

* [[Mod]], [[2Mod]], [[nMod]]

* [[dgcAlg]]

* [[Eilenberg-Moore category]]

category: category

[[!redirects Algebras]]

[[!redirects category of algebras]]
[[!redirects categories of algebras]]


Alg(T) > history

< [[Alg(T)]]


algebra

> This page is about **algebra as a theory**. If you are looking for the term **algebra as an object** see [[associative algebra]] or [[algebra over an operad]] or the like. See below for more.

***

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[[geometry]] $\leftarrow$ [[Isbell duality]]  $\rightarrow$ **algebra**

***

#Contents#
* table of contents
{:toc}

## Idea

**Algebra** is the manipulation of symbols without (necessarily) regard for their meaning, especially in a way that can be formalized in [[cartesian logic]].  It is often seen as [[duality|dual]] to [[geometry]]. While modern algebra has ties and applications nearly everywhere in mathematics, traditionally closest ties are with the [[number theory]] and [[algebraic geometry]].

The word 'algebra' is often also used for an algebraic structure:

* often by default an [[associative unital algebra]];
* more generally a [[monoid object]];
* more generally in a different way, a [[nonassociative algebra]];
* an [[algebra over an operad]], of a [[monad]], a [[PROP]], etc;
* an [[algebra for an endofunctor]];
* a model of any [[algebraic theory]] or anything studied in [[universal algebra]];
* higher categorical analogues, many object/family versions of algebras, for example [[algebroid]]s, and [[pseudoalgebra]]s (or [[2-algebra]]s) over [[pseudomonad]]s (or [[2-monad]]s).

Various fields of mathematics or mathematical concepts can be manipulated in an algebraic or symbolic way, and such approaches or formalized subfields have names like [[categorical algebra]], [[homological algebra]], [[homotopical algebra]] and so on. Methods of combinatorics which involve much algebra, and manipulations with [[formal power series]] in particular, are called [[algebraic combinatorics]].

## Related entries

The $n$lab has a number of entries on particular algebraic structures ([[monoid]], [[semigroup]], [[group]], [[ring]], [[noetherian ring]], [[quasigroup]], [[associative algebra]], [[Lie algebra]], [[coalgebra]], [[dg-algebra]], [[bialgebra]], [[graded algebra]], [[Hopf algebra]], [[coring]], [[quasitriangular bialgebra]], [[lattice]], [[rig]], [[near-ring]], $\Omega$-[[$\Omega$-group|group]], [[field]], [[perfect field]], [[skewfield]], [[free field]], [[vector space]], [[vertex operator algebra]], [[crossed module]], [[chain complex]], [[hypermonoid]], [[hyperring]], [[hyperfield]], [[truss]], [[brace]] etc.), entries on their structural features, parts, "envelopes" or localizations ([[ideal]], [[center]], [[centralizer]], [[normal subgroup]], [[normal closure]], [[normalizer]], [[holomorph]], [[Ore set]], [[Ore localization]], [[enveloping algebra]], [[universal enveloping algebra]]) and on algebraic structures internal to other categories ([[topological group]], [[Lie group]], [[Lie groupoid]], [[algebraic group]], [[formal group]], [[dg-algebra]] etc).

There are also few pages on various invariants of algebraic objects or operations on algebraic expressions, e.g. on [[resultants]] of polynomials, [[determinant]] of a matrix, [[quasideterminant]] of a matrix with noncommutative entries.

For many algebraic structures a notion of [[action]] is defined; they embody "symmetry algebras" of some other algebraic objects. An action is expressed via a [[representation]] of one object as a subobject of a full object of another; or as a combination of the object which acts and which is acted upon (e.g. [[action groupoid]]). Objects with action are [[modules]] of the appropriate kind (possibly dualized: [[comodule]], [[contramodule]]; multiple, e.g. [[bimodule]]; or homotopized like $A_\infty$-modules). The possibilities for realizing a given algebra via symmetries of another object are systematically studied in a field called [[representation theory]].

See also

* [[commutative algebra]]

* [[counterexamples in algebra]].

[[!include Isbell duality - table]]

## References

Introductory textbooks:

* [[Carl Faith]], _Algebra: Rings, Modules, Categories I_, Grundlehren der mathematischen Wissenschaften **190**, Springer (1973) [[doi:10.1007/978-3-642-80634-6](https://doi.org/10.1007/978-3-642-80634-6)]

* [[Anthony Knapp]], *Basic Algebra*, Springer (2006) [[doi:10.1007/978-0-8176-4529-8](https://doi.org/10.1007/978-0-8176-4529-8), [pdf](https://www.math.mcgill.ca/darmon/courses/19-20/algebra2/knapp.pdf)]

* {#Aluffi09} [[Paolo Aluffi]], *Algebra: Chapter 0*, Graduate Studies in Mathematics **104**,  AMS (2009) [[ISBN:978-1-4704-1168-8](https://bookstore.ams.org/gsm-104/)]

* [[Joseph A. Gallian]], *Contemporary Abstract Algebra*, Chapman and Hall/CRC (2020)  [[doi:10.1201/9781003142331](https://doi.org/10.1201/9781003142331), [webpage](https://en.wikipedia.org/wiki/Joseph_Gallian), [pdf](https://ict.iitk.ac.in/wp-content/uploads/CS203-Mathematics-for-Computer-Science-III-Gallian.pdf)]

See also:

* Wikipedia, *[Algebra](http://en.wikipedia.org/wiki/Algebra)*

and see the references at *[[ring]]*, *[[module]]*, etc.

[[!redirects algebras]]
[[!redirects algebraist]]


algebra extension

> For extension of [[morphisms]] in the sense dual to [[lift]] see at [[extension]].

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* table of contents
{:toc}

## Idea

Given any kind of object $A$ in [[algebra]], such as an [[associative algebra]] or a [[group]] or a [[Lie algebra]], etc., then an _extension_ of $A$ is an [[epimorphism]]

$$
  \widehat A \overset{p}{\longrightarrow} A
$$

Typically the [[kernel]] of an epimorphism will exist in the given [[category]], leading to a [[short exact sequence]]

$$
  ker(p) \longrightarrow  \widehat A \overset{p}{\longrightarrow} A
  \,.
$$

Then one says that $\widehat A$ is an _extension by $ker(p)$_ of $A$.

## Examples

* [[group extension]]

* [[Lie algebra extension]]

* [[infinitesimal extension]]

* [[central extension]]

## Related concepts

* [[higher extension]]

[[!redirects algebra extensions]]


algebra for a profunctor

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#Contents#
* automatic table of contents goes here
{:toc}

## Idea

The notion of **algebra over an endo-[[profunctor]]** ($C$-$C$-[[bimodule]]) is a joint generalization of the notions [[algebra for an endofunctor]] and [[coalgebra for an endofunctor]].

## Definition

For a category $C$ and a $C$-$C$ [[bimodule]] $H : C^{op} \times C \to Set$, an **algebra** for $H$ is given by a [[functor]] $X \colon D \to C$ and an [[extranatural transformation]] $\ast \to H(X,X)$, where $\ast \colon \mathbf{1} \to Set$ is constant at the [[point]].  $X$ is called the **carrier** of the algebra.  A morphism $(X, \alpha) \to (Y, \beta)$ of $H$-algebras is given by a [[natural transformation]] $\phi \colon X \Rightarrow Y$ such that $H(X,\phi) \circ \alpha = H(\phi,Y) \circ \beta$.

If $D$ is the one-object category, an algebra $(X,\alpha)$ is given by an object $X$ in $C$ and an element $\alpha \in H(X, X)$.  A morphism between two algebras $(X, \alpha)$ and $(Y, \beta)$ is then a morphism $m : X \to Y$ in $C$ such that $H(X, m) (\alpha) = H(m, Y) (\beta)$, these both being elements of $H(X, Y)$.

There is an an obvious [[forgetful functor]] into $C$ from the category of algebras for $H$, which sends each algebra to its carrier and each algebra morphism to its underlying morphism in $C$; among other properties, this functor is always [[faithful]] and [[conservative functor|conservative]].

In fact, the category $Alg(H)$, together with its forgetful functor $U\colon Alg(H)\to C$, has the universal property of an [[Eilenberg-Moore object]], namely that of being the _universal_ $H$-algebra.  Specifically, it is a [[terminal object]] in the category whose objects are functors $G\colon D\to C$ equipped with an [[extranatural transformation]] $\ast \to H(G-,G?)$.  For such an extranatural transformation consists of, for every $d\in D$, an element $\xi_d \in H(G d,G d)$, such that for every morphism $v\colon d\to e$ in $D$, we have $H(id_d,v)(\xi_d) = H(v,id_e)(\xi_e)$.  This is precisely the data of a functor $D\to Alg(H)$ lying over $C$.

## Coalgebras in Prof

One version of [[Yoneda lemma|Yoneda's lemma]] says that for a profunctor $H \colon C ⇸ C$ there is a bijection between extranatural transformations $\ast \to H$ and natural transformations $\hom_C \to H$.  So there are bijections
$$
\array{
  \ast \: {\ddot\to} \: H(X,X) \\
  \hom_D \Rightarrow H(X,X) \\
  C(1,X) \Rightarrow H \circ C(1,X)
}
$$
where the last holds by the usual properties of representable profunctors (see e.g. [[proarrow equipment]]).  This exhibits each $H$-algebra on $X$ in the above sense as a $H$-[[algebra for an endomorphism|coalgebra]] in $Prof$ with carrier $C(1,X)$.

## Examples

* [[algebra for an endofunctor|Algebras]] and [[coalgebra for an endofunctor|coalgebras]] for [[endofunctor|endofunctors]] are special cases of algebras for bimodules; specifically, an algebra for an endofunctor $F$ is an algebra for the bimodule $Hom(F(-), ?)$, while a coalgebra for $F$ is an algebra for the bimodule $Hom(-, F(?))$.

* A [[natural transformation]] between functors $F$ and $G$ from $C$ to $D$ is a [[section]] of the forgetful functor into $C$ from the category of algebras for the $C-C$ bimodule $Hom_D(F(-), G(?))$.  That is, it gives every object of $C$ the structure of an algebra for $Hom_D(F(-), G(?))$ in such a way as that every morphism of $C$ has the property of being an algebra morphism between the algebras on its domain and codomain.

* A [[natural numbers object]] (in the weak, unparametrized sense) in a category $C$ with terminal object $1$ is an initial object in the category of algebras for the bimodule $Hom_C(1, ?) \times Hom_C(-, ?)$.  If $C$ has binary coproducts, then this is of course the same as an initial algebra for the endofunctor $1+(-)$.

## Related concepts

* [[algebra for an endomorphism]]
* [[algebra over a monad]], [[algebra over an endofunctor]], [[coalgebra over an endofunctor]], [[dialgebra]]

[[!redirects algebra for an endo-profunctor]]
[[!redirects algebra for an endoprofunctor]]
[[!redirects algebra for a bimodule]]
[[!redirects algebra for a C-C bimodule]]

[[!redirects algebra over a profunctor]]