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Group-Graded Twisted Calabi-Yau Alge...

Baki, Yasmeen Sophia,

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Group-Graded Twisted Calabi-Yau Algebras /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Group-Graded Twisted Calabi-Yau Algebras // Yasmeen Sophia Baki.
Author:	Baki, Yasmeen Sophia,
Description:	1 electronic resource (67 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 87-01, Section: B.
Contained By:	Dissertations Abstracts International87-01B.
Subject:	Mathematics. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=32045326
ISBN:	9798288801327

Group-Graded Twisted Calabi-Yau Algebras /
Baki, Yasmeen Sophia,

Group-Graded Twisted Calabi-Yau Algebras /Yasmeen Sophia Baki. - 1 electronic resource (67 pages)

Source: Dissertations Abstracts International, Volume: 87-01, Section: B.

Historically, the study of graded (twisted or otherwise) Calabi-Yau algebras has meant the study of such algebras under an N-grading. In this dissertation, we propose a suitable definition for a twisted G-graded Calabi-Yau algebra, for G an arbitrary abelian group. Building on the work of Reyes and Rogalski, we show that a G-graded algebra is twisted Calabi-Yau if and only if it is G-graded twisted Calabi-Yau. We then prove that localizations of twisted Calabi-Yau algebras at elements which form both left and right denominator sets remain twisted Calabi-Yau. As such, we obtain a large class of Z-graded twisted Calabi- Yau algebras arising as localizations of Artin-Schelter regular algebras. Throughout the dissertation, we survey a number of concrete examples of G-graded twisted Calabi-Yau algebras, including the Weyl algebras, families of generalized Weyl algebras, and universal enveloping algebras of finite dimensional Lie algebras.In the second half of this dissertation, we begin the work required to extend our previous results to the case where G is not necessarily abelian. Towards this end, we develop a theory of G-graded bimodules which is robust enough to perform all of our usual computations. We further provide an example of an algebra A which is both graded by the binary dihedral group of order 12 and twisted Calabi-Yau of dimension 2. We prove that A is G-graded homologically smooth, and discuss current obstructions towards defining what it means to be G-graded twisted Calabi-Yau in the non-abelian case. We conclude by describing how certain results of Marcus can be used to resolve these issues in the case where G is finite and the grading is strong.

English

ISBN: 9798288801327Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Lie algebras

Group-Graded Twisted Calabi-Yau Algebras /
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245 1 0 $a Group-Graded Twisted Calabi-Yau Algebras / $c Yasmeen Sophia Baki.
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500 $a Source: Dissertations Abstracts International, Volume: 87-01, Section: B.
500 $a Advisors: Reyes, Manuel Committee members: Baranovsky, Vladimir; Ascher, Kenneth.
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520 $a Historically, the study of graded (twisted or otherwise) Calabi-Yau algebras has meant the study of such algebras under an N-grading. In this dissertation, we propose a suitable definition for a twisted G-graded Calabi-Yau algebra, for G an arbitrary abelian group. Building on the work of Reyes and Rogalski, we show that a G-graded algebra is twisted Calabi-Yau if and only if it is G-graded twisted Calabi-Yau. We then prove that localizations of twisted Calabi-Yau algebras at elements which form both left and right denominator sets remain twisted Calabi-Yau. As such, we obtain a large class of Z-graded twisted Calabi- Yau algebras arising as localizations of Artin-Schelter regular algebras. Throughout the dissertation, we survey a number of concrete examples of G-graded twisted Calabi-Yau algebras, including the Weyl algebras, families of generalized Weyl algebras, and universal enveloping algebras of finite dimensional Lie algebras.In the second half of this dissertation, we begin the work required to extend our previous results to the case where G is not necessarily abelian. Towards this end, we develop a theory of G-graded bimodules which is robust enough to perform all of our usual computations. We further provide an example of an algebra A which is both graded by the binary dihedral group of order 12 and twisted Calabi-Yau of dimension 2. We prove that A is G-graded homologically smooth, and discuss current obstructions towards defining what it means to be G-graded twisted Calabi-Yau in the non-abelian case. We conclude by describing how certain results of Marcus can be used to resolve these issues in the case where G is finite and the grading is strong.
546 $a English
590 $a School code: 0030
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Applied mathematics. $3 2122814
653 $a Lie algebras
653 $a Binary dihedral group
653 $a Calabi-Yau algebra
653 $a Finite dimensional
690 $a 0405
690 $a 0642
690 $a 0364
710 2 $a University of California, Irvine. $b Mathematics. $e degree granting institution. $3 3823182
720 1 $a Reyes, Manuel $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 87-01B.
790 $a 0030
791 $a Ph.D.
792 $a 2025
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=32045326