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A CONJECTURE ON HOPF ALGEBRAS
AND QUIVER (CO)ALGEBRAS
by
Wai-Ting Lam
A Thesis Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF ARTS
(APPLIED MATHEMATICS)
May 2011
Copyright 2011 Wai-Ting Lam
Object Description
| Title | A conjecture on Hopf algebras and Quiver (co)algebras |
| Author | Lam, Wai-Ting |
| Author email | waitingl@usc.edu; waiting212@yahoo.com.hk |
| Degree | Master of Arts |
| Document type | Thesis |
| Degree program | Applied Mathematics |
| School | College of Letters, Arts and Sciences |
| Date defended/completed | 2011-03-28 |
| Date submitted | 2011 |
| Restricted until | Unrestricted |
| Date published | 2011-05-04 |
| Advisor (committee chair) | Montgomery, Susan |
| Advisor (committee member) |
Iovanov, Miodrag C. Guralnick, Robert |
| Abstract | We will give an overview of the theory of coalgebras and Hopf algebras leading to an important conjecture on infinite dimensional Hopf algebras. This conjecture concerns the Hopf algebras which have a nonzero integral. These can be thought of as generalizations of topological compact groups, and are Hopf algebras which are co-Frobenius as coalgebras. This conjecture of Andruskiewitsch and Dascalescu [AD] states that if a Hopf algebra H is co Frobenius, then the coradical filtration of H is finite. The converse of this statement is shown to be true in [AD].; The thesis is organized as follows. Chapter 1 includes a study of semisimple rings and modules, i.e. the development of Wedderburn-Artin theory which was a cornerstone of classical noncommutative ring theory. A review of basic notions and properties of k-algebras and k-coalgebras follows, and we then recall the path and quiver (co)algebras which we use for many examples in this thesis. We introduce bialgebras and Hopf algebras, and we discuss the definition of H-Hopf modules and the Integral theory. We then include some known positive answers for the above stated conjecture; a result of Radford [R] states that it is true if the coradical of H is finite dimensional. We present the ideas of one of the later simplified proofs of this, and some consequences ([AD, C]). Finally, we discuss aspects of this conjecture from the perspective of general co-Frobenius coalgebras and analyze it on several structures coming from graphs and quivers. |
| Keyword | finite coradical filtration; Finiteness conjecture on Hopf algebras; graph algebras; Hopf algebras; quiver algebras; quiver (co)algebras |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m3852 |
| Rights | Lam, Wai-Ting |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | http://www.usc.edu/isd/libraries/services/ask_a_librarian/email/ |
| Filename | etd-Lam-4530 |
| Archival file | uscthesesreloadpub_Volume44/etd-Lam-4530.pdf |
Description
| Title | Page 1 |
| Full text | A CONJECTURE ON HOPF ALGEBRAS AND QUIVER (CO)ALGEBRAS by Wai-Ting Lam A Thesis Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF ARTS (APPLIED MATHEMATICS) May 2011 Copyright 2011 Wai-Ting Lam |
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