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Schneider, this algebra is isomorphic to the Nichols algebra associated to the Yetter-Drinfeld module $V$, over the symmetric group $\\mathbb{S}_3$, corresponding to the conjugacy class of all transpositions and the sign representation. Exploiting this identification, we compute the cohomology ring $Ext_{FK_3}^*(\\Bbbk,\\Bbbk)$, showing that it is a polynomial ring $S[X]$ "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1404.5101","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2014-04-21T03:51:56Z","cross_cats_sorted":["math.KT","math.RA"],"title_canon_sha256":"5d3dd4e0b2cd8d92009c2c3eddf689cd90a83076c9c4746bb4f31d9c8622ecb5","abstract_canon_sha256":"7d8a2972528fb6c535c45e74e3a79669c58c6dddee5e86f9c1062a7c3d2fda50"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:16.297089Z","signature_b64":"leP5jcducUT+xN+yDt45jozMnqMak5oLj73ljc3FXXUYrInTjnM07uDwYKnczgo+ZMX3W3ZdqbsMsyVB3Yk0CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"78bf13ec489416d31df54a097fa0f087cfd714769c397a2628c998948c8e53ed","last_reissued_at":"2026-05-18T01:21:16.296216Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:16.296216Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The cohomology ring of the 12-dimensional Fomin-Kirillov algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT","math.RA"],"primary_cat":"math.QA","authors_text":"Cristian Vay, Dragos Stefan","submitted_at":"2014-04-21T03:51:56Z","abstract_excerpt":"The $12$-dimensional Fomin-Kirillov algebra $FK_3$ is defined as the quadratic algebra with generators $a$, $b$ and $c$ which satisfy the relations $a^2=b^2=c^2=0$ and $ab+bc+ca=0=ba+cb+ac$. 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