The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization

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• 作者：Hyun Kyu Kim ^{hkim@kias.re.kr" class="auth_mail" title="E-mail the corresponding author hyunkyu87@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Quantum Teichmü ; ller theory ; Thompson group T ; Ptolemy&ndash ; Thompson group ; Kashaev quantization ; Braided Ptolemy&ndash ; Thompson group ; Universal Teichmü ; ller space ; Stable braid group ; Infinite braid group
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：30 April 2016
• 年：2016
• 卷：293
• 期：Complete
• 页码：529-588
• 全文大小：1265 K

文摘

Quantization of universal Teichmüller space provides projective representations of the Ptolemy&ndash;Thompson group, which is isomorphic to the Thompson group T. This yields certain central extensions of T   by an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000694&_mathId=si1.gif&_user=111111111&_pii=S0001870816000694&_rdoc=1&_issn=00018708&md5=f7fe284fa8b8f7f1c2f8fe7cc19b1654" title="Click to view the MathML source">Zan>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">athvariant="double-struck">Zath>an>an>an>, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension an id="mmlsi2" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000694&_mathId=si2.gif&_user=111111111&_pii=S0001870816000694&_rdoc=1&_issn=00018708&md5=8475264c941d20afcc72a55852bab6bd">ass="imgLazyJSB inlineImage" height="17" width="42" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816000694-si2.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">accent="true">Tˆathvariant="normal">Kashath>an>an>an> of T   resulting from the Kashaev quantization, and show that it corresponds to 6 times the Euler class in an id="mmlsi3" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000694&_mathId=si3.gif&_user=111111111&_pii=S0001870816000694&_rdoc=1&_issn=00018708&md5=8d26ba876a2176f73bed7abf0742623d" title="Click to view the MathML source">H²(T;Z)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">H2alse">(T;athvariant="double-struck">Zalse">)ath>an>an>an>. Meanwhile, the braided Ptolemy&ndash;Thompson groups an id="mmlsi4" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000694&_mathId=si4.gif&_user=111111111&_pii=S0001870816000694&_rdoc=1&_issn=00018708&md5=18ff5bcaeb012260eedd9b596c50c508" title="Click to view the MathML source">T^⁎an>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">T⁎ath>an>an>an>, an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000694&_mathId=si5.gif&_user=111111111&_pii=S0001870816000694&_rdoc=1&_issn=00018708&md5=32cee6dea8dadf989e191d72cf4bcf71" title="Click to view the MathML source">T^♯an>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">T♯ath>an>an>an> of Funar&ndash;Kapoudjian are extensions of T   by the infinite braid group an id="mmlsi6" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000694&_mathId=si6.gif&_user=111111111&_pii=S0001870816000694&_rdoc=1&_issn=00018708&md5=e466b45b14806216a1811bef72c0d900" title="Click to view the MathML source">B_∞an>an class="mathContainer hidden">an class="mathCode">ath altimg="si6.gif" overflow="scroll">B∞ath>an>an>an>, and by abelianizing the kernel an id="mmlsi6" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000694&_mathId=si6.gif&_user=111111111&_pii=S0001870816000694&_rdoc=1&_issn=00018708&md5=e466b45b14806216a1811bef72c0d900" title="Click to view the MathML source">B_∞an>an class="mathContainer hidden">an class="mathCode">ath altimg="si6.gif" overflow="scroll">B∞ath>an>an>an> one constructs central extensions an id="mmlsi114" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000694&_mathId=si114.gif&_user=111111111&_pii=S0001870816000694&_rdoc=1&_issn=00018708&md5=490f7db8916453559b4f94093c37649d">ass="imgLazyJSB inlineImage" height="16" width="25" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816000694-si114.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si114.gif" overflow="scroll">Tathvariant="normal">ab⁎ath>an>an>an>, an id="mmlsi8" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000694&_mathId=si8.gif&_user=111111111&_pii=S0001870816000694&_rdoc=1&_issn=00018708&md5=c532cd7d5234b8b304f5966b569137a3">ass="imgLazyJSB inlineImage" height="20" width="25" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816000694-si8.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si8.gif" overflow="scroll">Tathvariant="normal">ab♯ath>an>an>an> of T   by an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000694&_mathId=si1.gif&_user=111111111&_pii=S0001870816000694&_rdoc=1&_issn=00018708&md5=f7fe284fa8b8f7f1c2f8fe7cc19b1654" title="Click to view the MathML source">Zan>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">athvariant="double-struck">Zath>an>an>an>, which are of topological nature. We show an id="mmlsi760" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000694&_mathId=si760.gif&_user=111111111&_pii=S0001870816000694&_rdoc=1&_issn=00018708&md5=a12c9405280846b6e8a234228680e4b4">ass="imgLazyJSB inlineImage" height="22" width="88" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816000694-si760.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si760.gif" overflow="scroll">accent="true">Tˆathvariant="normal">Kash≅Tathvariant="normal">ab♯ath>an>an>an>. Our result is analogous to that of Funar and Sergiescu, who computed a presentation of another dilogarithmic central extension an id="mmlsi10" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000694&_mathId=si10.gif&_user=111111111&_pii=S0001870816000694&_rdoc=1&_issn=00018708&md5=d4b65ba3e0735afcc759ed9daea54f93">ass="imgLazyJSB inlineImage" height="17" width="31" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816000694-si10.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si10.gif" overflow="scroll">accent="true">Tˆathvariant="normal">CFath>an>an>an> of T   resulting from the Chekhov&ndash;Fock(&ndash;Goncharov) quantization and thus showed that it corresponds to 12 times the Euler class and that an id="mmlsi11" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000694&_mathId=si11.gif&_user=111111111&_pii=S0001870816000694&_rdoc=1&_issn=00018708&md5=9993f0ca98b16bcd592306e9627b5225">ass="imgLazyJSB inlineImage" height="21" width="77" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816000694-si11.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si11.gif" overflow="scroll">accent="true">Tˆathvariant="normal">CF≅Tathvariant="normal">ab⁎ath>an>an>an>. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations.