Quasitriangular Hopf algebra




In mathematics, a Hopf algebra, H, is quasitriangular[1] if there exists an invertible element, R, of H⊗H{displaystyle Hotimes H}Hotimes H such that



  • R Δ(x)R−1=(T∘Δ)(x){displaystyle R Delta (x)R^{-1}=(Tcirc Delta )(x)}{displaystyle R Delta (x)R^{-1}=(Tcirc Delta )(x)} for all x∈H{displaystyle xin H}xin H, where Δ{displaystyle Delta }Delta is the coproduct on H, and the linear map T:H⊗H→H⊗H{displaystyle T:Hotimes Hto Hotimes H}T:Hotimes Hto Hotimes H is given by T(x⊗y)=y⊗x{displaystyle T(xotimes y)=yotimes x}T(xotimes y)=yotimes x,


  • (Δ⊗1)(R)=R13 R23{displaystyle (Delta otimes 1)(R)=R_{13} R_{23}}(Delta otimes 1)(R)=R_{{13}} R_{{23}},


  • (1⊗Δ)(R)=R13 R12{displaystyle (1otimes Delta )(R)=R_{13} R_{12}}(1otimes Delta )(R)=R_{{13}} R_{{12}},

where R12=ϕ12(R){displaystyle R_{12}=phi _{12}(R)}R_{12}=phi _{12}(R), R13=ϕ13(R){displaystyle R_{13}=phi _{13}(R)}R_{13}=phi _{13}(R), and R23=ϕ23(R){displaystyle R_{23}=phi _{23}(R)}R_{23}=phi _{23}(R), where ϕ12:H⊗H→H⊗H⊗H{displaystyle phi _{12}:Hotimes Hto Hotimes Hotimes H}phi _{{12}}:Hotimes Hto Hotimes Hotimes H, ϕ13:H⊗H→H⊗H⊗H{displaystyle phi _{13}:Hotimes Hto Hotimes Hotimes H}phi _{{13}}:Hotimes Hto Hotimes Hotimes H, and ϕ23:H⊗H→H⊗H⊗H{displaystyle phi _{23}:Hotimes Hto Hotimes Hotimes H}phi _{{23}}:Hotimes Hto Hotimes Hotimes H, are algebra morphisms determined by


ϕ12(a⊗b)=a⊗b⊗1,{displaystyle phi _{12}(aotimes b)=aotimes botimes 1,}phi _{12}(aotimes b)=aotimes botimes 1,

ϕ13(a⊗b)=a⊗1⊗b,{displaystyle phi _{13}(aotimes b)=aotimes 1otimes b,}phi _{13}(aotimes b)=aotimes 1otimes b,

ϕ23(a⊗b)=1⊗a⊗b.{displaystyle phi _{23}(aotimes b)=1otimes aotimes b.}phi _{23}(aotimes b)=1otimes aotimes b.

R is called the R-matrix.


As a consequence of the properties of quasitriangularity, the R-matrix, R, is a solution of the Yang-Baxter equation (and so a module V of H can be used to determine quasi-invariants of braids, knots and links). Also as a consequence of the properties of quasitriangularity, (ϵ⊗1)R=(1⊗ϵ)R=1∈H{displaystyle (epsilon otimes 1)R=(1otimes epsilon )R=1in H}(epsilon otimes 1)R=(1otimes epsilon )R=1in H; moreover
R−1=(S⊗1)(R){displaystyle R^{-1}=(Sotimes 1)(R)}R^{{-1}}=(Sotimes 1)(R), R=(1⊗S)(R−1){displaystyle R=(1otimes S)(R^{-1})}R=(1otimes S)(R^{{-1}}), and (S⊗S)(R)=R{displaystyle (Sotimes S)(R)=R}(Sotimes S)(R)=R. One may further show that the
antipode S must be a linear isomorphism, and thus S2 is an automorphism. In fact, S2 is given by conjugating by an invertible element: S2(x)=uxu−1{displaystyle S^{2}(x)=uxu^{-1}}S^{2}(x)=uxu^{{-1}} where u:=m(S⊗1)R21{displaystyle u:=m(Sotimes 1)R^{21}}u:=m(Sotimes 1)R^{{21}} (cf. Ribbon Hopf algebras).


It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfeld quantum double construction.


If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding



cU,V(u⊗v)=T(R⋅(u⊗v))=T(R1u⊗R2v){displaystyle c_{U,V}(uotimes v)=Tleft(Rcdot (uotimes v)right)=Tleft(R_{1}uotimes R_{2}vright)}{displaystyle c_{U,V}(uotimes v)=Tleft(Rcdot (uotimes v)right)=Tleft(R_{1}uotimes R_{2}vright)}.



Contents






  • 1 Twisting


  • 2 See also


  • 3 Notes


  • 4 References





Twisting


The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element F=∑ifi⊗fi∈A⊗A{displaystyle F=sum _{i}f^{i}otimes f_{i}in {mathcal {Aotimes A}}}F=sum _{i}f^{i}otimes f_{i}in {mathcal {Aotimes A}} such that (ε⊗id)F=(id⊗ε)F=1{displaystyle (varepsilon otimes id)F=(idotimes varepsilon )F=1}(varepsilon otimes id)F=(idotimes varepsilon )F=1 and satisfying the cocycle condition


(F⊗1)∘(Δ⊗id)F=(1⊗F)∘(id⊗Δ)F{displaystyle (Fotimes 1)circ (Delta otimes id)F=(1otimes F)circ (idotimes Delta )F}(Fotimes 1)circ (Delta otimes id)F=(1otimes F)circ (idotimes Delta )F

Furthermore, u=∑ifiS(fi){displaystyle u=sum _{i}f^{i}S(f_{i})}u=sum _{i}f^{i}S(f_{i}) is invertible and the twisted antipode is given by S′(a)=uS(a)u−1{displaystyle S'(a)=uS(a)u^{-1}}S'(a)=uS(a)u^{{-1}}, with the twisted comultiplication, R-matrix and co-unit change according to those defined for the quasi-triangular quasi-Hopf algebra. Such a twist is known as an admissible (or Drinfeld) twist.



See also



  • Quasi-triangular quasi-Hopf algebra

  • Ribbon Hopf algebra



Notes




  1. ^ Montgomery & Schneider (2002), p. 72.



References




  • Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  • Montgomery, Susan; Schneider, Hans-Jürgen (2002). New directions in Hopf algebras. Mathematical Sciences Research Institute Publications. 43. Cambridge University Press. ISBN 978-0-521-81512-3. Zbl 0990.00022.




-

Popular posts from this blog

Schooner

Image
For other uses, see Schooner (disambiguation). A traditional gaff topsail schooner Schooner rigging: 1 ) Bowsprit 2 ) Jib & fore staysail 3 ) Fore gaffsail & topsail 4 ) Main staysail 5 ) Main gaff topsail 6 ) Mainsail 7 ) End of boom A schooner / ˈ s k uː n ər / is a type of sailing vessel with fore-and-aft sails on two or more masts. The most common type has two masts, the foremast being shorter than the main. While the schooner was originally gaff-rigged, modern schooners typically carry a Bermuda rig. Contents 1 Etymology 2 History 3 Usage 4 Schooner sail plan 4.1 Schooner rationale 5 Multi-masted schooners 6 Famous schooners 7 Gallery 8 See also 9 References 10 External links Etymology Traditional square topsail schooner Shenandoah , sailing in Nantucket Sound. The first detailed definition of a schooner, describing the vessel as two-masted vessel with fore and aft gaff-rigged sails appeared in ...
Read more

巴黎地鐵5號線

Image
巴黎地铁5號線 啟用年代 1906年 运营商 RATP 车型 MF 01 （ 法语 ： MF 01 ） (50辆) 车辆存放所 博比尼 路線站數 22 14.6公里 平均站距 697公尺 营运日 每日 年均載客量 86.1 巴黎地鐵5號線 （法语： Ligne 5 du métro de Paris ）為巴黎地鐵路網之一，於1906年通車。本線略呈"厂"字型，自巴黎東北方的博比尼起，向西南经巴黎北站、巴黎东站後轉南，经共和广场和巴士底，穿越塞纳河，最终訖於5、6、7三條地铁線交會的意大利广场。 目录 1 歷史 2 线路走向与車站列表 2.1 车站设施 2.2 车站曾用名 2.3 车站特色 3 使用車型 4 使用状况 5 未来计划 5.1 线路延伸 5.1.1 南延段 5.1.2 北环转乘站 6 周边主要旅游景点 7 外部連結 歷史 开通之初的奥斯特里茨车站 5号线穿越塞纳河的路径 关于巴黎地铁建设的想法源于1845年，铁路公司偏好利用连接郊区铁路网构建地铁；而市政府希望在市内新建不同系统的中小型铁路，双方争执数十年。直到巴黎交通状况持续恶化暨1900年世博会临近，最终市政府的计划被采用，而5号线即为巴黎地铁网络一期工程的一部分。 5号线通车的同时，4号线尚未南北贯通，故5号线实际上是巴黎第一条南北向穿越塞纳河的地铁。 1903年11月，用于5号线穿越塞纳河的奥斯特里茨铁桥开工建设，铁桥的设计长度仅有140米，以此避免在河中加装支撑柱来给过往船只带来不便。 1906年6月2日，義大利廣場站和奥斯特里茨车站（当时叫做奥尔良车站）之间的路段通車。同年7月13日，铁桥完工，线路从桥上穿越塞纳河，通过一个极大地弯角下陡坡，即 拉佩河边街铁桥 （ 法语 ： Viaduc du quai de la Rapée ） ，向北延伸至 拉佩河边街站 （ 法语 ： Quai de la Rapée (métro de Paris) ） （当时叫做马萨广场）。然而由于北部路段尚未完工，...
Read more

Y

Image
This article is about the letter of the alphabet. For other uses, see Y (disambiguation). See also: Wye (disambiguation) Y Y y (See below) Usage Writing system Latin script Type Alphabetic and Logographic Language of origin Latin language Phonetic usage [ y ] [ ɨ ] [ j ] [ iː ] [ ɪ ] [ ɘ ] [ ə ] [ ɯ ] [ ɛː ] [ j ] [ ɥ ] [ ɣ̟ ] / w aɪ / / aɪ / Unicode value U+0059, U+0079 Alphabetical position 25 History Development Υ υ 𐌖 Y y Time period 54 to present Descendants  • U  • V  • W  • Ỿ  • ¥  • Ꮙ  • Ꮍ  • Ꭹ Sisters F Ѵ У Ў Ұ Ү ו و ܘ וּ וֹ ࠅ 𐎆 𐡅 ወ વ ૂ ુ उ Variations (See below) Other Other letters commonly used with y(x), ly, ny This article contains IPA phonetic symbols. Without proper rendering support, you may see question marks, boxes, or other symbols instead of Unicode characters. For an introductory guide on IPA symbols, see Help:IPA. ISO basic Latin alphabet Aa Bb Cc D...
Read more