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Title: Classification of rank 5 premodular categories

Abstract

We survey a number of classification tools developed in recent years and employ them to classify pseudounitary rank 5 premodular categories up to Grothendieck equivalence.

Authors:
ORCiD logo [1];  [1]
  1. Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
Publication Date:
Research Org.:
Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1511176
Alternate Identifier(s):
OSTI ID: 1421110
Report Number(s):
PNNL-SA-120942
Journal ID: ISSN 0022-2488
Grant/Contract Number:  
AC05-76RL01830; PNNL-SA-120942
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 59; Journal Issue: 1; Journal ID: ISSN 0022-2488
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING

Citation Formats

Bruillard, Paul, and Ortiz-Marrero, Carlos M. Classification of rank 5 premodular categories. United States: N. p., 2018. Web. doi:10.1063/1.5020256.
Bruillard, Paul, & Ortiz-Marrero, Carlos M. Classification of rank 5 premodular categories. United States. https://doi.org/10.1063/1.5020256
Bruillard, Paul, and Ortiz-Marrero, Carlos M. Mon . "Classification of rank 5 premodular categories". United States. https://doi.org/10.1063/1.5020256. https://www.osti.gov/servlets/purl/1511176.
@article{osti_1511176,
title = {Classification of rank 5 premodular categories},
author = {Bruillard, Paul and Ortiz-Marrero, Carlos M.},
abstractNote = {We survey a number of classification tools developed in recent years and employ them to classify pseudounitary rank 5 premodular categories up to Grothendieck equivalence.},
doi = {10.1063/1.5020256},
journal = {Journal of Mathematical Physics},
number = 1,
volume = 59,
place = {United States},
year = {Mon Jan 08 00:00:00 EST 2018},
month = {Mon Jan 08 00:00:00 EST 2018}
}

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Cited by: 6 works
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Works referenced in this record:

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font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2012-04-01">April 2012</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Bruillard, Paul; Rowell, Eric C.</span> </li> <li> Proceedings of the American Mathematical Society, Vol. 140, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1090/s0002-9939-2011-11476-x" class="text-muted" target="_blank" rel="noopener noreferrer">10.1090/s0002-9939-2011-11476-x<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1155/s1073792804131206" target="_blank" rel="noopener noreferrer" class="name"><title/><span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; 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font-size:0.75rem;"><br/> <span class="type">preprint</span>, <span class="date" data-date="2008-01-01">January 2008</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Etingof, Pavel; Nikshych, Dmitri; Ostrik, Victor</span> </li> <li> arXiv</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.48550/arxiv.0809.3031" class="text-muted" target="_blank" rel="noopener noreferrer">10.48550/arxiv.0809.3031<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.48550/arxiv.1507.05139" target="_blank" rel="noopener noreferrer" class="name">On classification of modular categories by rank<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">preprint</span>, <span class="date" data-date="2015-01-01">January 2015</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Bruillard, Paul; Ng, Siu-Hung; Rowell, Eric C.</span> </li> <li> arXiv</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.48550/arxiv.1507.05139" class="text-muted" target="_blank" rel="noopener noreferrer">10.48550/arxiv.1507.05139<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.48550/arxiv.hep-th/9409028" target="_blank" rel="noopener noreferrer" class="name">Quantum Invariants of Knots and 3-Manifolds<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">preprint</span>, <span class="date" data-date="1994-01-01">January 1994</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Turaev, V.</span> </li> <li> arXiv</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.48550/arxiv.hep-th/9409028" class="text-muted" target="_blank" rel="noopener noreferrer">10.48550/arxiv.hep-th/9409028<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> </div> <div class="pagination-container small"> <a class="pure-button prev page" href="#" rel="prev"><span class="sr-only">Previous Page</span><span class="fa fa-angle-left"></span></a> <ul class="pagination d-inline-block" style="padding-left:.2em;"></ul> <a class="pure-button next page" href="#" rel="next"><span class="sr-only">Next Page</span><span class="fa fa-angle-right"></span></a> </div> </div> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a href="" class="reference-type-filter tab-nav" data-tab="biblio-references" data-filter="type" data-pattern="*"><span class="fa fa-angle-right"></span> All References</a></li> <li class="small" style="margin-left:.75em; 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margin-top:0px;">Similar Records in DOE PAGES and OSTI.GOV collections:</p> <aside> <ul class="item-list" itemscope itemtype="http://schema.org/ItemList" style="padding-left:0; list-style-type: none;"> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="1" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/1340755-rank-premodular-categories" itemprop="url">Rank 4 Premodular Categories</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Bruillard, Paul J.</span> ; <span class="author">Galindo, Cesar</span> ; <span class="author">Ng, Siu Hung</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - New York Journal of Mathematics, 22:775-800</span> </span> </div> <div class="abstract">We consider the classification problem for rank 4 premodular categories. We uncover a formula for the 2nd Frobenius-Schur indicator of a premodular category is determined and the classification of rank 4 premodular categories (up to Grothendieck equivalence) is completed. In the appendix we show rank finiteness for premodular categories.</div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="2" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/1340753-classification-modular-categories-rank-table" itemprop="url">On Classification of Modular Categories by Rank: Table A.1</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Bruillard, Paul</span> ; <span class="author">Ng, Siu-Hung</span> ; <span class="author">Rowell, Eric C.</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - International Mathematics Research Notices</span> </span> </div> <div class="abstract">The feasibility of a classification-by-rank program for modular categories follows from the Rank-Finiteness Theorem. We develop arithmetic, representation theoretic and algebraic methods for classifying modular categories by rank. As an application, we determine all possible fusion rules for all rank=5 modular categories and describe the corresponding monoidal equivalence classes.</div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.1093/imrn/rnw020" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1340753" data-product-type="Journal Article" data-product-subtype="AC" >https://doi.org/10.1093/imrn/rnw020</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="3" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/22661347-galah-survey-classification-diagnostics-sne-reduction-spectral-information" itemprop="url">The Galah Survey: Classification and Diagnostics with t-SNE Reduction of Spectral Information</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Traven, G.</span> ; <span class="author">Zwitter, T.</span> ; <span class="author">Žerjal, M.</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Astrophysical Journal, Supplement Series</span> </span> </div> <div class="abstract">Galah is an ongoing high-resolution spectroscopic survey with the goal of disentangling the formation history of the Milky Way using the fossil remnants of disrupted star formation sites that are now dispersed around the Galaxy. It is targeting a randomly selected magnitude-limited ( V ≤ 14) sample of stars, with the goal of observing one million objects. To date, 300,000 spectra have been obtained. Not all of them are correctly processed by parameter estimation pipelines, and we need to know about them. We present a semi-automated classification scheme that identifies different types of peculiar spectral morphologies in an effort to<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> discover and flag potentially problematic spectra and thus help to preserve the integrity of the survey results. To this end, we employ the recently developed dimensionality reduction technique t-SNE ( t -distributed stochastic neighbor embedding), which enables us to represent the complex spectral morphology in a two-dimensional projection map while still preserving the properties of the local neighborhoods of spectra. We find that the majority (178,483) of the 209,533 Galah spectra considered in this study represents normal single stars, whereas 31,050 peculiar and problematic spectra with very diverse spectral features pertaining to 28,579 stars are distributed into 10 classification categories: hot stars, cool metal-poor giants, molecular absorption bands, binary stars, H α /H β emission, H α /H β emission superimposed on absorption, H α /H β P-Cygni, H α /H β inverted P-Cygni, lithium absorption, and problematic. Classified spectra with supplementary information are presented in the catalog, indicating candidates for follow-up observations and population studies of the short-lived phases of stellar evolution.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.3847/1538-4365/228/2/24" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="22661347" data-product-type="Journal Article" data-product-subtype="AC" >https://doi.org/10.3847/1538-4365/228/2/24</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="4" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/1340754-classification-weakly-integral-modular-categories" itemprop="url">On the classification of weakly integral modular categories</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Bruillard, Paul</span> ; <span class="author">Galindo, César</span> ; <span class="author">Ng, Siu-Hung</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Journal of Pure and Applied Algebra</span> </span> </div> <div class="abstract">In this paper we classify all modular categories of dimension 4m, where m is an odd square-free integer, and all rank 6 and rank 7 weakly integral modular categories. This completes the classification of weakly integral modular categories through rank 7. In particular, our results imply that all integral modular categories of rank at most 7 are pointed (that is, every simple object has dimension 1). All the non-integral (but weakly integral) modular categories of ranks 6 and 7 have dimension 4m, with m an odd square free integer, so their classification is an application of our main result. The<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> classification of rank 7 integral modular categories is facilitated by an analysis of the two group actions on modular categories: the Galois group of the field generated by the entries of the S-matrix and the group of invertible isomorphism classes of objects. We derive some valuable arithmetic consequences from these actions.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.1016/j.jpaa.2015.11.010" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1340754" data-product-type="Journal Article" data-product-subtype="AC" >https://doi.org/10.1016/j.jpaa.2015.11.010</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="6" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/1828164-phasor-measurement-unit-based-data-analytics-using-digital-twin-phasoranalytics-software" itemprop="url">Phasor-Measurement-Unit-Based Data Analytics Using Digital Twin and PhasorAnalytics Software</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Technical Report</small><span class="authors"> <span class="author">Hart, Philip</span> ; <span class="author">Yan, Weizhong</span> ; <span class="author">Wang, Tianyi</span> ; <span class="author">...</span> <span class="text-muted pubdata"></span> </span> </div> <div class="abstract">A major objective of this project was to apply GE’s commercial machine learning and data analytics toolsets to large-scale, real-world, anonymized Phasor Measurement Unit (PMU) datasets in order to extract signatures, correlated and/or causal factors, and precursor patterns associated with significant power system phenomena. The project had a particular emphasis on extraction of insights relevant to asset health monitoring, real-time load modeling and cybersecurity monitoring. Additionally, the team was directed to undertake a comprehensive data quality analysis for the provided datasets and encouraged to estimate the ‘machine-learning readiness’ of the datasets by documenting any major obstacles to the application of<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> commercial machine learning algorithms. To accomplish the aforementioned objectives, the project team’s work centered around the identification of key event signatures and application of the identified event signatures for event detection and event classification. The industry-validated, semi-supervised machine learning strategy employed for event signature identification involved several major tasks, including data-preprocessing, generation of an overabundance of features, normal data identification, normality modeling, and event signature identification through a methodical, quantitative ranking of features in order of relevance to each studied event type. Throughout the project, data quality issues and mitigation techniques were investigated. In this report, insights are provided regarding the readiness of the provided synchrophasor datasets for application of machine learning and data analytics. The methodologies employed for this technical strategy are summarized in this report. With regards to data preprocessing and feature generation, the provided Training and Test Datasets were ingested into GE’s big data environment. Subsequently, the team applied bad data cleansing and data imputation scripts, event detection scripts, and application programming interfaces (APIs) to the datasets for convenient data access. The project team completed development and validation of dozens of physics-based, statistics-based and transformation-based feature functions used for the extraction of over 60 synchrophasor features. Using a new parallel feature generation technology developed on this project, over 60 features have been rapidly generated for the full two years’ worth of Training and Test Dataset data associated with both the Eastern and Western interconnects. Even accommodating for temporal down-sampling inherent to the feature extraction procedure, this parallel feature generation activity resulted in a massive feature set with a storage requirement approximately equal to that of the raw training dataset itself. With regards to normal data identification and normality modeling, a normality model was built using the feature data extracted from the Training Dataset and iteratively refined subsequent to incremental adjustments and expansions of the Training Dataset feature data. With respect to event characterization and signature identification, an event signature identification pipeline was developed and used in conjunction with the normality model to identify over 15 event signatures for key event categories within the Training Dataset. The identified event signatures were used to characterize hundreds of key events in terms of relative severity, duration, and location of the event. An investigation was undertaken to identify correlated and causal factors involved in transformer events. A separate investigation into temporal trends in ring-down analysis results was undertaken to determine possible associations between system dynamics and various other factors such as loading, season or year. To validate the identified event signatures, additional work was undertaken to develop signature-based anomaly detection and classification tools suitable for convenient application to the synchrophasor datasets. The anomaly detection and classification tools, suitable for online application, were then applied to the entirety of the Eastern Interconnect Training and Test Datasets. Performance of the event detection and classification tools was evaluated upon receipt of the Test Dataset event logs (i.e., the labels for events contained in the Test Dataset), and promising results were obtained despite several challenges (documented herein) associated with application of supervised or semi-supervised machine learning methods to large-scale, anonymized datasets. Finally, the detection and classification tools were used to detect, classify, and characterize thousands of new events not included in the original event logs provided by the DOE within both the Training and Test Datasets.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.2172/1828164" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1828164" data-product-type="Technical Report" data-product-subtype="" >https://doi.org/10.2172/1828164</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/servlets/purl/1828164" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1828164" data-product-type="Technical Report" data-product-subtype="" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> </ul> </aside> </div> </section> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a class="tab-nav disabled" data-tab="related" style="color: #636c72 !important; 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