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Dechant Pierre-philippe - One of the best experts on this subject based on the ideXlab platform.
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Clifford spinors and root systEm induction: $H_4$ and thE Grand Antiprism
2021Co-Authors: Dechant Pierre-philippeAbstract:REcEnt work has shown that EvEry 3D root systEm allows thE construction of a corrEponding 4D root systEm via an `induction thEorEm'. In this papEr, wE look at thE icosahEdral casE of $H_3\rightarrow H_4$ in dEtail and pErform thE calculations Explicitly. Clifford algEbra is usEd to pErform group thEorEtic calculations basEd on thE vErsor thEorEm and thE Cartan-DiEudonn\'E thEorEm, giving a simplE construction of thE Pin and Spin covErs. Using this connEction with $H_3$ via thE induction thEorEm shEds light on gEomEtric aspEcts of thE $H_4$ root systEm (thE $600$-cEll) as wEll as othEr rElatEd polytopEs and thEir symmEtriEs, such as thE famous Grand Antiprism and thE snub 24-cEll. ThE uniform construction of root systEms from 3D and thE uniform procEdurE of splitting root systEms with rEspEct to subrootsystEms into sEparatE invariant sEts allows furthEr systEmatic insight into thE undErlying gEomEtry. All calculations arE pErformEd in thE EvEn subalgEbra of Cl(3), including thE construction of thE CoxEtEr planE, which is usEd for visualising thE complEmEntary pairs of invariant polytopEs, and arE sharEd as supplEmEntary computational work shEEts. This approach thErEforE constitutEs a morE systEmatic and gEnEral way of pErforming calculations concErning groups, in particular rEflEction groups and root systEms, in a Clifford algEbraic framEwork.CommEnt: 32 pagEs, 11 figurE
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A 3D Spinorial ViEw of 4D ExcEptional PhEnomEna
'Springer Science and Business Media LLC', 2016Co-Authors: Dechant Pierre-philippeAbstract:In this papEr, wE discuss a Clifford algEbra framEwork for discrEtE symmEtriEs -- E.g. rEflEction, CoxEtEr, conformal, modular groups -- that also lEads to a surprising numbEr of nEw rEsults in itsElf. Clifford algEbra affords a particularly simplE dEscription for pErforming rEflEctions (via `sandwiching' with vEctors in thE Clifford algEbra), and sincE via thE Cartan-DiEudonn\'E thEorEm all orthogonal transformations can bE writtEn as products of rEflEctions, all such opErations can bE pErformEd via `sandwiching' with Clifford algEbra multivEctors. WE bEgin by viEwing thE largEst non-crystallographic rEflEction/CoxEtEr group $H_4$ as a group of rotations in two diffErEnt ways -- firstly via a folding from thE largEst ExcEptional group $E_8$, and sEcondly by induction from thE icosahEdral group $H_3$ via Clifford spinors. WE thEn gEnEralisE this lattEr obsErvation and prEsEnt a procEdurE by which starting with any 3D root systEm onE constructs a corrEsponding 4D root systEm. This affords a nEw -- spinorial -- pErspEctivE on 4D phEnomEna, in particular as thE inducEd root systEms arE prEcisEly thE ExcEptional onEs in 4D, and thEir unusual automorphism groups arE Easily ExplainEd in thE spinorial picturE; wE discuss thE widEr contExt of Platonic solids, Arnold's trinitiEs and thE McKay corrEspondEncE. ThE multivEctor groups can bE usEd to pErform concrEtE group thEorEtic calculations, E.g. thosE for $H_3$ and $E_8$, and wE discuss how various rEprEsEntations can also bE constructEd in this Clifford framEwork; in particular, rEprEsEntations of quatErnionic typE arisE vEry naturally
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Clifford algEbra is thE natural framEwork for root systEms and CoxEtEr groups. Group thEory: CoxEtEr, conformal and modular groups
'Springer Science and Business Media LLC', 2016Co-Authors: Dechant Pierre-philippeAbstract:In this papEr, wE makE thE casE that Clifford algEbra is thE natural framEwork for root systEms and rEflEction groups, as wEll as rElatEd groups such as thE conformal and modular groups: ThE mEtric that Exists on thEsE spacEs can always bE usEd to construct thE corrEsponding Clifford algEbra. Via thE Cartan-DiEudonn\'E thEorEm all thE transformations of intErEst can bE writtEn as products of rEflEctions and thus via `sandwiching' with Clifford algEbra multivEctors. ThEsE multivEctor groups can bE usEd to pErform concrEtE calculations in diffErEnt groups, E.g. thE various typEs of polyhEdral groups, and wE trEat thE ExamplE of thE tEtrahEdral group $A_3$ in dEtail. As an asidE, this givEs a constructivE rEsult that inducEs from EvEry 3D root systEm a root systEm in dimEnsion four, which hingEs on thE facts that thE group of spinors providEs a doublE covEr of thE rotations, thE spacE of 3D spinors has a 4D EuclidEan innEr product, and with rEspEct to this innEr product thE group of spinors can bE shown to bE closEd undEr rEflEctions. In particular thE 4D root systEms/CoxEtEr groups inducEd in this way arE prEcisEly thE ExcEptional onEs, with thE 3D spinorial point of viEw also Explaining thEir unusual automorphism groups. This construction simplifiEs Arnold's trinitiEs and puts thE McKay corrEspondEncE into a widEr framEwork. WE finally discuss ExtEnding thE conformal gEomEtric algEbra approach to thE 2D conformal and modular groups, which could havE intErEsting novEl applications in conformal fiEld thEory, string thEory and modular form thEory.CommEnt: 14 pagEs, 1 figurE, 5 tablE
M I Krivoruchenko - One of the best experts on this subject based on the ideXlab platform.
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Hydrostatic Equilibrium of stars without ElEctronEutrality constraint
Physical Review D, 2018Co-Authors: M I Krivoruchenko, D. K. Nadyozhin, A. V. YudinAbstract:ThE gEnEral solution of hydrostatic Equilibrium Equations for a two-componEnt fluid of ions and ElEctrons without a local ElEctronEutrality constraint is found in thE framEwork of NEwtonian gravity thEory. In agrEEmEnt with thE Poincar\'E thEorEm on analyticity and in thE contExt of Dyson's argumEnt, thE gEnEral solution is dEmonstratEd to possEss a fixEd (EssEntial) singularity in thE gravitational constant $G$ at $ G = 0 $. ThE rEgular componEnt of thE gEnEral solution can bE dEtErminEd by pErturbation thEory in $G$ starting from a locally nEutral solution. ThE non-pErturbativE componEnt obtainEd using thE mEthod of WEntzEl, KramErs and Brillouin is ExponEntially small in thE innEr layErs of thE star and grows rapidly in thE outward dirEction. NEar thE surfacE of thE star, both componEnts arE comparablE in magnitudE, and thEir non-linEar intErplay dEtErminEs thE propErtiEs of an ElEctro- or ionosphErE. ThE stEllar chargE variEs within thE limits of $- 0.1 $ to $150$ C pEr solar mass. ThE propErtiEs of ElEctro- and ionosphErEs arE ExponEntially sEnsitivE to variations of thE fluid dEnsitiEs in thE cEntral rEgions of thE star. ThE gEnEral solutions of two Exactly solvablE stEllar modEls without a local ElEctronEutrality constraint arE also prEsEntEd.
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sEmiclassical Expansion of quantum charactEristics for many body potEntial scattEring problEm
arXiv: Nuclear Theory, 2006Co-Authors: M I Krivoruchenko, C Fuchs, Amand FaesslerAbstract:In quantum mEchanics, systEms can bE dEscribEd in phasE spacE in tErms of thE WignEr function and thE star-product opEration. Quantum charactEristics, which appEar in thE HEisEnbErg picturE as thE WEyl's symbols of opErators of canonical coordinatEs and momEnta, can bE usEd to solvE thE Evolution Equations for symbols of othEr opErators acting in thE HilbErt spacE. To any fixEd ordEr in thE Planck's constant, many-body potEntial scattEring problEm simplifiEs to a statistical-mEchanical problEm of computing an EnsEmblE of quantum charactEristics and thEir dErivativEs with rEspEct to thE initial canonical coordinatEs and momEnta. ThE rEduction to a systEm of ordinary diffErEntial Equations pErtains rigorously at any fixEd ordEr in $\hbar$. WE prEsEnt sEmiclassical Expansion of quantum charactEristics for many-body scattEring problEm and providE tools for calculation of avEragE valuEs of timE-dEpEndEnt physical obsErvablEs and cross sEctions. ThE mEthod of quantum charactEristics admits thE consistEnt incorporation of spEcific quantum EffEcts, such as non-locality and cohErEncE in propagation of particlEs, into thE sEmiclassical transport modEls. WE formulatE thE principlE of stationary action for quantum Hamilton's Equations and givE quantum-mEchanical ExtEnsions of thE LiouvillE thEorEm on thE consErvation of phasE-spacE volumE and thE Poincar\'E thEorEm on thE consErvation of $2p$ forms. ThE lowEst ordEr quantum corrEctions to thE KEplEr pEriodic orbits arE constructEd. ThEsE corrEctions show thE rEsonancE bEhavior.
Anotnio Calixto Souza Filho - One of the best experts on this subject based on the ideXlab platform.
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from thE poincar E thEorEm to gEnErators of thE unit group of intEgral group rings of finitE groups
arXiv: Group Theory, 2013Co-Authors: Eric Jespers, S O Juriaans, Ann Kiefer, Antonio De Andrade E Silva, Anotnio Calixto Souza FilhoAbstract:WE givE an algorithm to dEtErminE finitEly many gEnErators for a subgroup of finitE indEx in thE unit group of an intEgral group ring $\mathbb{Z} G$ of a finitE nilpotEnt group $G$, this providEd thE rational group algEbra $\mathbb{Q} G$ doEs not havE simplE componEnts that arE division classical quatErnion algEbras or two-by-two matricEs ovEr a classical quatErnion algEbra with cEntrE $\mathbb{Q}$. ThE main difficulty is to dEal with ordErs in quatErnion algEbras ovEr thE rationals or a quadratic imaginary ExtEnsion of thE rationals. In ordEr to dEal with thEsE wE givE a finitE and Easy implEmEntablE algorithm to computE a fundamEntal domain in thE hypErbolic thrEE spacE $\mathbb{H}^3$ (rEspEctivEly hypErbolic two spacE $\mathbb{H}^2$) for a discrEtE subgroup of ${\rm PSL}_2(\mathbb{C})$ (rEspEctivEly ${\rm PSL}_2(\mathbb{R})$) of finitE covolumE. Our rEsults on group rings arE a continuation of EarliEr work of RittEr and SEhgal, JEspErs and LEal.
Sommen Frank - One of the best experts on this subject based on the ideXlab platform.
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ThE Spin Group in SupErspacE
2019Co-Authors: De Schepper Hennie, Adán, Alí Guzmán, Sommen FrankAbstract:ThErE arE two wEll-known ways of dEscribing ElEmEnts of thE rotation group SO$(m)$. First, according to thE Cartan-DiEudonn\'E thEorEm, EvEry rotation matrix can bE writtEn as an EvEn numbEr of rEflEctions. And sEcond, thEy can also bE ExprEssEd as thE ExponEntial of somE anti-symmEtric matrix. In this papEr, wE study similar dEscriptions of a group of rotations SO${}_0$ in thE supErspacE sEtting. This group can bE sEEn as thE action of thE functor of points of thE orthosymplEctic supErgroup OSp$(m|2n)$ on a Grassmann algEbra. WhilE still bEing connEctEd, thE group SO${}_0$ is thus no longEr compact. As a consEquEncE, it cannot bE fully dEscribEd by just onE action of thE ExponEntial map on its LiE algEbra. InstEad, wE obtain an Iwasawa-typE dEcomposition for this group in tErms of thrEE ExponEntials acting on thrEE dirEct summands of thE corrEsponding LiE algEbra of supErmatricEs. At thE samE timE, SO${}_0$ strictly contains thE group gEnEratEd by supEr-vEctor rEflEctions. ThErEforE, its LiE algEbra is isomorphic to a cErtain ExtEnsion of thE algEbra of supErbivEctors. This mEans that thE Spin group in this sEtting has to bE sEEn as thE group gEnEratEd by thE ExponEntials of thE so-callEd ExtEndEd supErbivEctors in ordEr to covEr SO${}_0$. WE also study thE actions of this Spin group on supErvEctors and providE a propEr subsEt of it that is a doublE covEr of SO${}_0$. Finally, wE show that EvEry fractional FouriEr transform in n bosonic dimEnsions can bE sEEn as an ElEmEnt of this spin group.CommEnt: 28 pagE
Adán, Alí Guzmán - One of the best experts on this subject based on the ideXlab platform.
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ThE Spin Group in SupErspacE
2019Co-Authors: De Schepper Hennie, Adán, Alí Guzmán, Sommen FrankAbstract:ThErE arE two wEll-known ways of dEscribing ElEmEnts of thE rotation group SO$(m)$. First, according to thE Cartan-DiEudonn\'E thEorEm, EvEry rotation matrix can bE writtEn as an EvEn numbEr of rEflEctions. And sEcond, thEy can also bE ExprEssEd as thE ExponEntial of somE anti-symmEtric matrix. In this papEr, wE study similar dEscriptions of a group of rotations SO${}_0$ in thE supErspacE sEtting. This group can bE sEEn as thE action of thE functor of points of thE orthosymplEctic supErgroup OSp$(m|2n)$ on a Grassmann algEbra. WhilE still bEing connEctEd, thE group SO${}_0$ is thus no longEr compact. As a consEquEncE, it cannot bE fully dEscribEd by just onE action of thE ExponEntial map on its LiE algEbra. InstEad, wE obtain an Iwasawa-typE dEcomposition for this group in tErms of thrEE ExponEntials acting on thrEE dirEct summands of thE corrEsponding LiE algEbra of supErmatricEs. At thE samE timE, SO${}_0$ strictly contains thE group gEnEratEd by supEr-vEctor rEflEctions. ThErEforE, its LiE algEbra is isomorphic to a cErtain ExtEnsion of thE algEbra of supErbivEctors. This mEans that thE Spin group in this sEtting has to bE sEEn as thE group gEnEratEd by thE ExponEntials of thE so-callEd ExtEndEd supErbivEctors in ordEr to covEr SO${}_0$. WE also study thE actions of this Spin group on supErvEctors and providE a propEr subsEt of it that is a doublE covEr of SO${}_0$. Finally, wE show that EvEry fractional FouriEr transform in n bosonic dimEnsions can bE sEEn as an ElEmEnt of this spin group.CommEnt: 28 pagE
