Quantum Weyl Group Action on Coroot Lattice via Explicit Abelian Formula — E8 Intelligence Research

{"FINDING:":[0],"Explicit":[1],"abelian":[2],"formula":[3,51],"for":[4],"quantum":[5,13,30],"Weyl":[6,79,119],"group":[7,80,120,124],"action":[8,121,145],"on":[9,43],"coroot":[10,40,74,105],"lattice":[11,41,75,106],"of":[12,36,39,60,63,85,111,160],"loop":[14,31],"algebra":[15],"Uq(Lg),":[16],"given":[17],"in":[18,54,127],"arXiv:2501.02365v2.":[19],"|":[20,96],"MATH:":[21],"Let":[22],"g":[23],"be":[24],"complex":[25],"simple":[26,86,112],"Lie":[27,116],"algebra,":[28,32],"Uq(Lg)":[29],"q":[33,89],"not":[34],"root":[35,71,101],"unity.":[37],"Action":[38],"Q":[42],"finite-dim":[44],"reps":[45],"expressed":[46],"via":[47],"commuting":[48,148],"generators":[49],"—":[50,104,150],"is":[52,107],"exponential":[53],"form":[55],"(likely":[56],"e_x":[57],"=":[58],"exp":[59],"linear":[61],"combination":[62],"Cartan/Drinfeld":[64],"generators,":[65],"with":[66],"q-deformed":[67],"exponents).":[68],"Key":[69],"structures:":[70],"system":[72],"Φ,":[73],"Q∨":[76],"≅":[77],"Z^rank(g),":[78],"W":[81],"acting":[82],"by":[83],"q-analogues":[84],"reflections.":[87],"Constants:":[88],"(deformation":[90],"parameter),":[91],"no":[92],"numeric":[93],"ratios":[94],"given.":[95],"CONNECTION:":[97],"Direct":[98],"link":[99],"to":[100,115,131],"systems":[102],"(crystallographic)":[103],"the":[108,144,161],"integer":[109],"span":[110],"coroots,":[113],"central":[114],"theory.":[117],"Quantum":[118],"encodes":[122],"braid":[123],"symmetries,":[125],"which":[126],"geometric":[128],"terms":[129],"relate":[130],"Coxeter–Dynkin":[132],"diagrams":[133],"(A_n,":[134],"B_n,":[135],"D_n,":[136],"E_6,7,8,":[137],"F_4,":[138],"G_2).":[139],"The":[140],"\\"abelian\\"":[141],"nature":[142],"means":[143],"factors":[146],"through":[147],"tori":[149],"re":[151],"Author:":[152],"Andrew":[153],"Stewart":[154],"Caldin,":[155],"Independent":[156],"Researcher,":[157],"UK.":[158],"Part":[159],"E8":[162],"Intelligence":[163],"Research":[164],"series.":[165],"Platform:":[166],"e8intelligence.com":[167]}

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-17
DOI
https://doi.org/10.5281/zenodo.22805884
Primary Topic
Algebraic structures and combinatorial models
Type
preprint
Controls
|||
ALL TIME
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
preprint

Quantum Weyl Group Action on Coroot Lattice via Explicit Abelian Formula — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Algebraic structures and combinatorial models
preprint

Quantum Weyl Group Action on Coroot Lattice via Explicit Abelian Formula — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: Explicit abelian formula for quantum Weyl group action on coroot lattice of quantum loop algebra Uq(Lg), given in arXiv:2501.02365v2. | MATH: Let g be complex simple Lie algebra, Uq(Lg) quantum loop algebra, q not root of unity. Action of coroot lattice Q on finite-dim reps expressed via commuting generators — formula is exponential in form (likely e_x = exp of linear combination of Cartan/Drinfeld generators, with q-deformed exponents). Key structures: root system Φ, coroot lattice Q∨ ≅ Z^rank(g), Weyl group W acting by q-analogues of simple reflections. Constants: q (deformation parameter), no numeric ratios given. | CONNECTION: Direct link to root systems (crystallographic) — coroot lattice is the integer span of simple coroots, central to Lie theory. Quantum Weyl group action encodes braid group symmetries, which in geometric terms relate to Coxeter–Dynkin diagrams (A_n, B_n, D_n, E_6,7,8, F_4, G_2). The "abelian" nature means the action factors through commuting tori — re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
Algebraic structures and combinatorial models
AI Navigator

Ask Laika to Summarize, Analyze, and Connect papers live on the map.

Summarize Papers & Methodologies

Extract key findings, datasets, and comparative methods across publications.

Benchmark Rankings & Visual Analytics

Rank top research institutions, authors, funders, topics, and journals by Field-Weighted Citation Impact (FWCI) and paper volume with instant charts.

Connect Distant Disciplines

Bridge topological clusters on the map to find hidden collaborative intersections.

Quantum Weyl Group Action on Coroot Lattice via Explicit Abelian Formula — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS