Bibliographic record
Abstract
The problem posed by Kaplansky [K] of classification of Hopf algebras of a given finite dimension over the complex numbers is a nontrivial one, even for some small dimensions. It has been shown [Z],[Ng], [Ng1], [HN] that for p a prime, Hopf algebras of dimension p, p2,2p and 2p2 are either semisimple, pointed nonsemisimple, or the dual is pointed. It has been conjectured that this is the case also for dimension pq and pn with p, q primes. At this time, the smallest dimension where the classification is incomplete is 24. The classification for dimension 27 has only recently been completed [BG] and in general, although [G], [BG] give many partial results, the classification for dimension p3, p an odd prime, is still open. This talk will give a little survey of what is known for some small dimensions and present some techniques based on [F] using the coradical filtration for dealing with these questions. References [BG] M. Beattie and G. A. Garcia, Techniques for classifying Hopf algebras and applications to dimension p3, arXiv:1108.6037v1 [math.QA] [F] D. Fukuda, Structure of coradical filtration and its application to Hopf algebras of dimension pq, Glasg. Math. J. 50 (2008), no. 2, 183–190. [G] G. A. Garcia, On Hopf algebras of dimension p3, Tsukuba J. Math. 29 (2005), no. 1, 259–284. [HN] M. Hilgemann and S.-H. Ng, Hopf algebras of dimension 2p2, J. London Math. Soc.(2) 80 (2009), 295-310. [K] I. Kaplansky, Bialgebras. Lecture Notes in Mathematics. University of Chicago, Chicago (1975) [Ng] S-H. Ng, Non-semisimple Hopf algebras of dimension p2, J. Algebra 255, no. 1 (2002), 182–197. [Ng1] Hopf algebras of dimension 2p, Proc. Am. Math. Soc. 133(8), 2237-2242. [Z] Y. Zhu, Hopf algebras of prime dimension, Int. Math. Res. Not. 1 (1994), 53–59. Department of Mathematics and Computer Science, Mount Allison University, Sackville, NB E4L 1E6, Canada E-mail address: mbeattie@mta.ca
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".