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University of Colorado

1. McGregor-Dorsey, Zachary Strider. Some properties of full heaps.

Degree: PhD, Mathematics, 2013, University of Colorado

URL: https://scholar.colorado.edu/math_gradetds/28

A full heap is a labeled infinite partially ordered set with labeling taken from the vertices of an underlying Dynkin diagram, satisfying certain conditions intended to capture the structure of that diagram. The notion of full heaps was introduced by R. Green as an affine extension of the minuscule heaps of J. Stembridge. Both authors applied these constructions to make observations of the Lie algebras associated to the underlying Dynkin diagrams. The main result of this thesis, Theorem 4.7.1, is a complete classification of all full heaps over Dynkin diagrams with a finite number of vertices, using only the general notion of Dynkin diagrams and entirely elementary methods that rely very little on the associated Lie theory. The second main result of the thesis, Theorem 5.1.7, is an extension of the Fundamental Theorem of Finite Distributive Lattices to locally finite posets, using a novel analogue of order ideal posets. We apply this construction in an analysis of full heaps to find our third main result, Theorem 5.5.1, an ADE classification of the full heaps over simply laced affine Dynkin diagrams.
*Advisors/Committee Members: Richard M. Green, Nathaniel Thiem, Martin E. Walter, J. M. Douglas, Stephen R. Doty.*

Subjects/Keywords: ADE Classification; Combinatorial Algebra; Dynkin Diagram; Full Heap; Lie Algebra; Minuscule Representation

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

McGregor-Dorsey, Z. S. (2013). Some properties of full heaps. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/28

Chicago Manual of Style (16^{th} Edition):

McGregor-Dorsey, Zachary Strider. “Some properties of full heaps.” 2013. Doctoral Dissertation, University of Colorado. Accessed October 26, 2020. https://scholar.colorado.edu/math_gradetds/28.

MLA Handbook (7^{th} Edition):

McGregor-Dorsey, Zachary Strider. “Some properties of full heaps.” 2013. Web. 26 Oct 2020.

Vancouver:

McGregor-Dorsey ZS. Some properties of full heaps. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2020 Oct 26]. Available from: https://scholar.colorado.edu/math_gradetds/28.

Council of Science Editors:

McGregor-Dorsey ZS. Some properties of full heaps. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/math_gradetds/28

2.
Edie-Michell, Cain.
The *classification* of categories generated by an object of small dimension
.

Degree: 2018, Australian National University

URL: http://hdl.handle.net/1885/146634

The goal of this thesis is to attempt the classification of
unitary fusion categories generated by a normal object (\refi{an
object comuting with its dual}{1}) of dimension less than 2. This
classification has recently become accessible due to a result of
Morrison and Snyder, which shows that any such category must be a
cyclic extension of an adjoint subcategory of one of the ADE
fusion categories. Our main tool is the classification of graded
categories from , which classifies graded
extensions of a fusion category in terms of the Brauer-Picard
group, and Drinfeld centre of that category.
We compute the Drinfeld centres, and Brauer-Picard groups of the
adjoint subcategories of the ADE fusion categories. Using this
information we apply the machinery of graded extensions to
classify the cyclic extensions that are generated by a normal
object of dimension less than 2, of the adjoint subcategories of
the ADE fusion categories. Unfortunately, our classification
has a gap when the dimension of the object is √{2+√{2}}
corresponding to the possible existence of an interesting new
fusion category. Interestingly we prove the existence of a new
category, generated by a normal object of dimension
2\cos(\frac{π}{18}), which we call the DEE fusion category.
We include the fusion rules for the DEE fusion categories in an
appendix to this thesis.

Subjects/Keywords: Unitary fusion categories; classification; ADE

…the *ADE* fusion categories.
An important piece of data in the *classification* of graded… …subcategories of the *ADE* fusion categories. The recipe for such
a *classification*, as laid out in [… …The Brauer-Picard groups of the adjoint subcategories of the
*ADE* fusion categories
3.1 The… …Drinfeld Centres of the *ADE* fusion categories . . . . . . . . .
3.2 Planar algebra automorphisms… …3.3 The Brauer-Picard groups of the *ADE* fusion categories . . . . . .
3.4 Explicit…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Edie-Michell, C. (2018). The classification of categories generated by an object of small dimension . (Thesis). Australian National University. Retrieved from http://hdl.handle.net/1885/146634

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Edie-Michell, Cain. “The classification of categories generated by an object of small dimension .” 2018. Thesis, Australian National University. Accessed October 26, 2020. http://hdl.handle.net/1885/146634.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Edie-Michell, Cain. “The classification of categories generated by an object of small dimension .” 2018. Web. 26 Oct 2020.

Vancouver:

Edie-Michell C. The classification of categories generated by an object of small dimension . [Internet] [Thesis]. Australian National University; 2018. [cited 2020 Oct 26]. Available from: http://hdl.handle.net/1885/146634.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Edie-Michell C. The classification of categories generated by an object of small dimension . [Thesis]. Australian National University; 2018. Available from: http://hdl.handle.net/1885/146634

Not specified: Masters Thesis or Doctoral Dissertation