Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules

Communications of the American Mathematical Society Pub Date : 2023-05-01 DOI:10.1090/cams/19

Cain EDIE-MICHELL

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In this work we classify the braided auto-equivalences of the categories of local modules for all known type <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I\">\n <mml:semantics>\n <mml:mi>I</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">I</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> quantum subgroups of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper C left-parenthesis German s German l Subscript r plus 1 Baseline comma k right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"fraktur\">s</mml:mi>\n <mml:mi mathvariant=\"fraktur\">l</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>r</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:mi>k</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {C}(\\mathfrak {sl}_{r+1}, k)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We find that the symmetries are all non-exceptional except for four cases (up to level-rank duality). 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stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>2</mml:mn>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:mn>0</mml:mn>\n </mml:msubsup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {C}(\\mathfrak {sl}_{2}, 16)^0_{\\operatorname {Rep}(\\mathbb {Z}_{2})}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper C left-parenthesis German s German l Subscript 3 Baseline comma 9 right-parenthesis Subscript upper R e p left-parenthesis double-struck upper Z 3 right-parenthesis Superscript 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi>\n 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</mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper C left-parenthesis German s German l Subscript 4 Baseline comma 8 right-parenthesis Subscript upper R e p left-parenthesis double-struck upper Z 4 right-parenthesis Superscript 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"fraktur\">s</mml:mi>\n <mml:mi mathvariant=\"fraktur\">l</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>4</mml:mn>\n </mml:mrow>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:mn>8</mml:mn>\n <mml:msubsup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>Rep</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>4</mml:mn>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:mn>0</mml:mn>\n </mml:msubsup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {C}(\\mathfrak {sl}_{4}, 8)^0_{\\operatorname {Rep}(\\mathbb {Z}_{4})}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper C left-parenthesis German s German l Subscript 5 Baseline comma 5 right-parenthesis Subscript upper R e p left-parenthesis double-struck upper Z 5 right-parenthesis Superscript 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi>\n 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</mml:semantics>\n</mml:math>\n</inline-formula>.</p>\n\n<p>We develop several technical tools in this work. 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引用次数: 0

Abstract

This paper is the first of a pair that aims to classify a large number of the type I I II quantum subgroups of the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . In this work we classify the braided auto-equivalences of the categories of local modules for all known type I I quantum subgroups of C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . We find that the symmetries are all non-exceptional except for four cases (up to level-rank duality). These exceptional cases are the orbifolds C ( s l 2 , 16 ) Rep ⁡ ( Z 2 ) 0 \mathcal {C}(\mathfrak {sl}_{2}, 16)^0_{\operatorname {Rep}(\mathbb {Z}_{2})} , C ( s l 3 , 9 ) Rep ⁡ ( Z 3 ) 0 \mathcal {C}(\mathfrak {sl}_{3}, 9)^0_{\operatorname {Rep}(\mathbb {Z}_{3})} , C ( s l 4 , 8 ) Rep ⁡ ( Z 4 ) 0 \mathcal {C}(\mathfrak {sl}_{4}, 8)^0_{\operatorname {Rep}(\mathbb {Z}_{4})} , and C ( s l 5 , 5 ) Rep ⁡ ( Z 5 ) 0 \mathcal {C}(\mathfrak {sl}_{5}, 5)^0_{\operatorname {Rep}(\mathbb {Z}_{5})} .

We develop several technical tools in this work. We give a skein theoretic description of the orbifold quantum subgroups of

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输入𝐼𝐼的量子子群:𝔩_{𝔑}。局部模块的对称性

本文是对范畴C (sl r+1, k) \mathcal {C}(\mathfrak {sl}_{r+1}， k)的大量II II型量子子群进行分类的第一个对。本文对C (sl r+1, k) \mathcal {C}(\mathfrak {sl}_{r+1}， k)的所有已知I I型量子子群的局部模范畴的编织自等价进行了分类。我们发现除了四种情况(直到水平-秩对偶)外，对称性都是非例外的。这些例外情况是轨道C (s 1,2, 16) Rep (Z 2) 0 \mathcal {C}(\mathfrak {sl}_{2}， 16)^0_{\operatorname {Rep}(\mathbb {Z}_{2})}，C (sl 3,9) Rep (z3) 0 \mathcal {C}(\mathfrak {sl}_{3}， 9)^0_{\operatorname {Rep}(\mathbb {Z}_{3})}，C (sl 1,8) Rep (Z 4) 0 \mathcal {C}(\mathfrak {sl}_{4}， 8)^0_{\operatorname {Rep}(\mathbb {Z}_{4})}，和C (s 1,5,5) Rep (z5) 0 \mathcal {C}(\mathfrak {sl}_{5}， 5)^0_{\operatorname {Rep}(\mathbb {Z}_{5})}我们在这项工作中开发了几个技术工具。本文给出了

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Communications of the American Mathematical Society

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