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We show that a monomial belongs to $\\mathfrak L(V)$ if and only if that this monomial is connected. We obtain the basis for $\\mathfrak L(V)$ of arithmetic root systems and the dimension for $\\mathfrak L(V)$ of finite Cartan type. We give the sufficient and necessary conditions for $\\mathfrak B(V) = F\\oplus \\mathfrak L^-(V)$ and $\\mathfrak L^-(V)= \\mathfrak L(V)$. We obtain an explici"},"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":"1704.06810","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.QA","submitted_at":"2017-04-22T15:19:10Z","cross_cats_sorted":[],"title_canon_sha256":"0486ba259c5ccbcf59044c9100caf473c555999c379e6ecb8e3f27233bf6c3d9","abstract_canon_sha256":"ca88dcc548193d9e0c33e06d469b2b2086d3de2b68a409b328ef0df904fcc162"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:02.391573Z","signature_b64":"hqq84Hl34+T90tfa7/4laqGlG5H2z4gV1YVdyGJc+MdxyBQKk9k7/rZfiskm1Y8RMRDlySm6X6LHPtnVDfnZCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f9195def5e6f7bdaaab0eac0e3f308460455673eadb291607ec79c1433ef6d6","last_reissued_at":"2026-05-18T00:24:02.391064Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:02.391064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Structures of Nichols (braided) Lie algebras of diagonal type","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Jing Wang, Shouchuan Zhang, Weicai Wu, Yao-Zhong Zhang","submitted_at":"2017-04-22T15:19:10Z","abstract_excerpt":"Let $V$ be a braided vector space of diagonal type. Let $\\mathfrak B(V)$, $\\mathfrak L^-(V)$ and $\\mathfrak L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial belongs to $\\mathfrak L(V)$ if and only if that this monomial is connected. We obtain the basis for $\\mathfrak L(V)$ of arithmetic root systems and the dimension for $\\mathfrak L(V)$ of finite Cartan type. We give the sufficient and necessary conditions for $\\mathfrak B(V) = F\\oplus \\mathfrak L^-(V)$ and $\\mathfrak L^-(V)= \\mathfrak L(V)$. 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