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We also find a list of thirteen invariants, each of which is the dimension of a space constructed from the relation, such that the 13-vector of multiplicities and the 13-vector of invariants are related by an invertible matrix over $\\mathbb Z$.\n  It turns out to be simpler to do the analysis above for isotropic relations bet"},"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":"1509.04035","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2015-09-14T11:13:21Z","cross_cats_sorted":[],"title_canon_sha256":"457785636a6256bc78005f5004ded70c8733c685e1a138d416fc47fc78df35dc","abstract_canon_sha256":"a8cbb57306932118a3e4428d2e2aae8bc3ba932638066bef0556b24e1669c59b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:35.008503Z","signature_b64":"YyFthHj8o5stdfzy47jKAJg1GTiRLcityXIvoQjTQcCkpyTMamzZBXlw0QB6palF1NZ+HQZWG5RZkWvbbtcaAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f4748babd277c8bd92ad1e0c9ee841fa172caff5222672fa54a5911f86b79469","last_reissued_at":"2026-05-18T00:58:35.007896Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:35.007896Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decomposition of (co)isotropic relations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SG","authors_text":"Alan Weinstein, Jonathan Lorand","submitted_at":"2015-09-14T11:13:21Z","abstract_excerpt":"We identify thirteen isomorphism classes of indecomposable coisotropic relations between Poisson vector spaces and show that every coisotropic relation between finite-dimensional Poisson vector spaces may be decomposed as a direct sum of multiples of these indecomposables. 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