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Categorification of Braid Group Repr...

Adair, Andrew,

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Categorification of Braid Group Representations /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Categorification of Braid Group Representations // Andrew Adair.
作者:	Adair, Andrew,
面頁冊數:	1 electronic resource (102 pages)
附註:	Source: Dissertations Abstracts International, Volume: 85-11, Section: B.
Contained By:	Dissertations Abstracts International85-11B.
標題:	Mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31235119
ISBN:	9798382631585

Categorification of Braid Group Representations /
Adair, Andrew,

Categorification of Braid Group Representations /Andrew Adair. - 1 electronic resource (102 pages)

Source: Dissertations Abstracts International, Volume: 85-11, Section: B.

In this dissertation, we define categorifications of two braid group representations: the Burau representation and a specialization of the Lawrence-Krammer-Bigelow representation. We first prove a modestly novel realization of each representation in terms of a basis of Temperley-Lieb diagrams, relying on definitions of these representations in terms of representations of the quantum group Uq(sl2). We then categorify the basis diagrams using Bar-Natan's categories BN of tangles and cobordisms from [BN] and the categorified Jones-Wenzl projectors from work of Cooper-Krushkal in [CK] and Rozansky in [Roz]. Finally, we generate certain subcategories of the dg category of complexes over BN that are closed under a dg-categorical braid group action. In an appropriate sense, these subcategories decategorify to representations of the braid group isomorphic to Burau and (a specialization of) LKB. As far as we know, this is the first known categorification of any version of the LKB representation. In addition, our categorification of Burau is a slight generalization of prior work due to Khovanov-Seidel in [KS]: specifically, our categorification is equivalent to one in terms of a dg category of dg modules over a dg algebra; and up to a small modification, its zeroth homology is equivalent to the algebraic construction in [KS].

English

ISBN: 9798382631585Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Quantum group

Categorification of Braid Group Representations /
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520 $a In this dissertation, we define categorifications of two braid group representations: the Burau representation and a specialization of the Lawrence-Krammer-Bigelow representation. We first prove a modestly novel realization of each representation in terms of a basis of Temperley-Lieb diagrams, relying on definitions of these representations in terms of representations of the quantum group Uq(sl2). We then categorify the basis diagrams using Bar-Natan's categories BN of tangles and cobordisms from [BN] and the categorified Jones-Wenzl projectors from work of Cooper-Krushkal in [CK] and Rozansky in [Roz]. Finally, we generate certain subcategories of the dg category of complexes over BN that are closed under a dg-categorical braid group action. In an appropriate sense, these subcategories decategorify to representations of the braid group isomorphic to Burau and (a specialization of) LKB. As far as we know, this is the first known categorification of any version of the LKB representation. In addition, our categorification of Burau is a slight generalization of prior work due to Khovanov-Seidel in [KS]: specifically, our categorification is equivalent to one in terms of a dg category of dg modules over a dg algebra; and up to a small modification, its zeroth homology is equivalent to the algebraic construction in [KS].
546 $a English
590 $a School code: 0153
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Applied mathematics. $3 2122814
650 4 $a Theoretical physics. $3 2144760
653 $a Quantum group
653 $a Burau representation
653 $a Categorification
653 $a Zeroth homology
690 $a 0405
690 $a 0642
690 $a 0753
690 $a 0364
710 2 $a The University of North Carolina at Chapel Hill. $b Mathematics. $e degree granting institution. $3 3820964
720 1 $a Rose, David $e degree supervisor.
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