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Formal calculus, umbral calculus, and basic axiomatics of vertex algebras
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Formal calculus, umbral calculus, and basic axiomatics of vertex algebras
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ETD_2176
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.000051897
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eng
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theses
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Mathematics
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Calculus
Subject
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(authority = ETD-LCSH)
Topic
Vertex operator algebras
Abstract
The central subject of this thesis is formal calculus together with certain applications to vertex operator algebras and combinatorics. By formal calculus we mean mainly the formal calculus that has been used to describe vertex operator algebras and their modules as well as logarithmic tensor product theory, but we also mean the formal calculus known as umbral calculus. We shall exhibit and develop certain connections between these formal calculi. Among other things we lay out a technique for efficiently proving certain general formal Taylor theorems and we show how to recast much of the classical umbral calculus as stemming from a formal calculus argument that calculates the exponential generating function of the higher derivatives of a composite function. This formal calculus argument is analogous to an important calculation proving the associativity property of lattice vertex operators. We
use some of our results to derive combinatorial identities. Finally, we apply other results to study some basic axiomatics of vertex (operator) algebras. In particular, we enhance well known formal calculus approaches to the axioms by introducing a new axiom, "weak skew-associativity,'' in order to exploit the
$mathcal{S}_{3}$-symmetric nature of the Jacobi identity axiom. In particular, we use this approach to give a simplified proof that the weak associativity and the Jacobi identity axioms for a module for a vertex algebra are equivalent, an important result in the
representation theory of vertex algebras.
use some of our results to derive combinatorial identities. Finally, we apply other results to study some basic axiomatics of vertex (operator) algebras. In particular, we enhance well known formal calculus approaches to the axioms by introducing a new axiom, "weak skew-associativity,'' in order to exploit the
$mathcal{S}_{3}$-symmetric nature of the Jacobi identity axiom. In particular, we use this approach to give a simplified proof that the weak associativity and the Jacobi identity axioms for a module for a vertex algebra are equivalent, an important result in the
representation theory of vertex algebras.
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electronic resource
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viii, 155 p.
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Ph.D.
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Includes bibliographical references (p. 151-154)
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by Thomas J. Robinson
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Robinson
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Thomas J.
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Thomas J. Robinson
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Lepowsky
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James
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James Lepowsky
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Huang
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Yi-Zhi
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Yi-Zhi Huang
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Li
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Haisheng
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Haisheng Li
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Milas
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Antun
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Antun Milas
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Rutgers University
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Graduate School - New Brunswick
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school
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2009
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2009-10
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xx
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Rutgers University Electronic Theses and Dissertations
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ETD
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Graduate School - New Brunswick Electronic Theses and Dissertations
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doi:10.7282/T3G1610G
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ETD doctoral
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The author owns the copyright to this work
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Copyright protected
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Open
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Robinson
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Thomas
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Thomas Robinson
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Rutgers University. Graduate School - New Brunswick
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Author Agreement License
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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