{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:44:19Z","timestamp":1760237059718,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,13]],"date-time":"2020-02-13T00:00:00Z","timestamp":1581552000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["2019R1F1A1058962"],"award-info":[{"award-number":["2019R1F1A1058962"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we study the non-degeneracy of 2-forms (skew symmetric     ( 0 , 2 )    -tensor)    \u03b1    along the Pfaffian of    \u03b1   . We consider a symplectic vector space     V     with a non-degenerate skew symmetric     ( 0 , 2 )    -tensor    \u03c9   , and derive various properties of the Pfaffian of    \u03b1   . As an application we show the non-degenerate skew symmetric     ( 0 , 2 )    -tensor    \u03c9    has a property of rigidity that it is determined by its exterior power.<\/jats:p>","DOI":"10.3390\/sym12020280","type":"journal-article","created":{"date-parts":[[2020,2,26]],"date-time":"2020-02-26T04:18:29Z","timestamp":1582690709000},"page":"280","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Non-Degeneracy of 2-Forms and Pfaffian"],"prefix":"10.3390","volume":"12","author":[{"given":"Jae-Hyouk","family":"Lee","sequence":"first","affiliation":[{"name":"Department of Mathematics, Ewha Womans University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1006\/aima.1995.1029","article-title":"A Simple Proof of an Identity Concerning Pfaffians of Skew Symmetric Matrices","volume":"112","author":"Dress","year":"1995","journal-title":"Adv. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1006\/jcta.2000.3117","article-title":"Pfaffian Identities: A Combinatorial Approach","volume":"94","author":"Hamel","year":"2001","journal-title":"J. Comb. Theory Ser. A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1216\/RMJ-1985-15-2-493","article-title":"Properties of Pfaffians","volume":"15","author":"Lomont","year":"1985","journal-title":"Rocky Mt. J. Math."},{"key":"ref_4","first-page":"17","article-title":"Symplectic geometry on symplectic knot spaces","volume":"13","author":"Lee","year":"2007","journal-title":"N. Y. J. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1007\/s00229-011-0448-7","article-title":"Grassmannians of symplectic subspaces","volume":"136","author":"Lee","year":"2011","journal-title":"Manuscripta Math."},{"key":"ref_6","unstructured":"Harvey, F.R. (1990). Spinors and Calibrations, Academic Press, Inc.. Perspectives in Mathematics 9."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1215\/00127094-2008-061","article-title":"E7, Wirtinger inequalites, Cayley 4-form, and homotopy","volume":"146","author":"Bangert","year":"2009","journal-title":"Duke Math. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1090\/conm\/288\/04818","article-title":"Stable forms and special metrics","volume":"288","author":"Hitchin","year":"2001","journal-title":"Contemp. Math."},{"key":"ref_9","unstructured":"Silva, A.C. (2008). Lectures on Symplectic Geometry, Springer. Lecture Notes in Mathematics 1764."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/2\/280\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:57:30Z","timestamp":1760173050000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/2\/280"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,13]]},"references-count":9,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2]]}},"alternative-id":["sym12020280"],"URL":"https:\/\/doi.org\/10.3390\/sym12020280","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,2,13]]}}}