{"title":"Kauffman托架串代数的维数与迹","authors":"C. Frohman, J. Kania-Bartoszyńska, Thang T. Q. Lê","doi":"10.1090/BTRAN/69","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\n <mml:semantics>\n <mml:mi>F</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a finite type surface and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"zeta\">\n <mml:semantics>\n <mml:mi>ζ<!-- ζ --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\zeta</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> a complex root of unity. The Kauffman bracket skein algebra <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K Subscript zeta Baseline left-parenthesis upper F right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>K</mml:mi>\n <mml:mi>ζ<!-- ζ --></mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>F</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">K_\\zeta (F)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K Subscript zeta Baseline left-parenthesis upper F right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>K</mml:mi>\n <mml:mi>ζ<!-- ζ --></mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>F</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">K_\\zeta (F)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> over its center, and we extend a theorem of the first and second authors in [Math. Z. 289 (2018), pp. 889–920] which says the skein algebra has a splitting coming from two pants decompositions of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\n <mml:semantics>\n <mml:mi>F</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"34 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"16","resultStr":"{\"title\":\"Dimension and Trace of the Kauffman Bracket Skein Algebra\",\"authors\":\"C. Frohman, J. Kania-Bartoszyńska, Thang T. Q. Lê\",\"doi\":\"10.1090/BTRAN/69\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper F\\\">\\n <mml:semantics>\\n <mml:mi>F</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">F</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> be a finite type surface and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"zeta\\\">\\n <mml:semantics>\\n <mml:mi>ζ<!-- ζ --></mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\zeta</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> a complex root of unity. The Kauffman bracket skein algebra <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper K Subscript zeta Baseline left-parenthesis upper F right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>K</mml:mi>\\n <mml:mi>ζ<!-- ζ --></mml:mi>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>F</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">K_\\\\zeta (F)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper K Subscript zeta Baseline left-parenthesis upper F right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>K</mml:mi>\\n <mml:mi>ζ<!-- ζ --></mml:mi>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>F</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">K_\\\\zeta (F)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> over its center, and we extend a theorem of the first and second authors in [Math. Z. 289 (2018), pp. 889–920] which says the skein algebra has a splitting coming from two pants decompositions of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper F\\\">\\n <mml:semantics>\\n <mml:mi>F</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">F</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>.</p>\",\"PeriodicalId\":377306,\"journal\":{\"name\":\"Transactions of the American Mathematical Society, Series B\",\"volume\":\"34 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-02-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"16\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the American Mathematical Society, Series B\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/BTRAN/69\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/BTRAN/69","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Dimension and Trace of the Kauffman Bracket Skein Algebra
Let FF be a finite type surface and ζ\zeta a complex root of unity. The Kauffman bracket skein algebra Kζ(F)K_\zeta (F) is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of Kζ(F)K_\zeta (F) over its center, and we extend a theorem of the first and second authors in [Math. Z. 289 (2018), pp. 889–920] which says the skein algebra has a splitting coming from two pants decompositions of FF.
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