{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:KTXOI7MAYGURSSEJ7SFLHV4OI5","short_pith_number":"pith:KTXOI7MA","canonical_record":{"source":{"id":"0911.3073","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-11-16T16:44:05Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"0347089a9207b6252dfedb23fcd49e2de0fd25b3c5431773304d4432de033835","abstract_canon_sha256":"0ea744600616c981e9150b43a4d85eb3c6efe5cf40615c3a756fec07ec84faca"},"schema_version":"1.0"},"canonical_sha256":"54eee47d80c1a9194889fc8ab3d78e475083045239fd3d3c84da9cdba5ef5dc7","source":{"kind":"arxiv","id":"0911.3073","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.3073","created_at":"2026-05-17T23:58:50Z"},{"alias_kind":"arxiv_version","alias_value":"0911.3073v4","created_at":"2026-05-17T23:58:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.3073","created_at":"2026-05-17T23:58:50Z"},{"alias_kind":"pith_short_12","alias_value":"KTXOI7MAYGUR","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"KTXOI7MAYGURSSEJ","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"KTXOI7MA","created_at":"2026-05-18T12:26:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:KTXOI7MAYGURSSEJ7SFLHV4OI5","target":"record","payload":{"canonical_record":{"source":{"id":"0911.3073","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-11-16T16:44:05Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"0347089a9207b6252dfedb23fcd49e2de0fd25b3c5431773304d4432de033835","abstract_canon_sha256":"0ea744600616c981e9150b43a4d85eb3c6efe5cf40615c3a756fec07ec84faca"},"schema_version":"1.0"},"canonical_sha256":"54eee47d80c1a9194889fc8ab3d78e475083045239fd3d3c84da9cdba5ef5dc7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:50.733768Z","signature_b64":"XRYOBZuEj1lvYj88ZVD7yqwy2r39ksmzVB6QTVvLxCnAsaLwFVjhBlSSydVcl04OyXgw9uoCpCJY92DQdE0gCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"54eee47d80c1a9194889fc8ab3d78e475083045239fd3d3c84da9cdba5ef5dc7","last_reissued_at":"2026-05-17T23:58:50.733300Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:50.733300Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0911.3073","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:58:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4JLxjvA0kkuy81Aa3DToveIhvbWnsrQ3n7KWqfQVbFuOO3U8okel4LS266KA2V1Cx5xvJM4XXywiP1R5UN81CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:48:02.688234Z"},"content_sha256":"942c93871cafec132cfafdde5ac816111fa897116d7def57cf2967daabad7db4","schema_version":"1.0","event_id":"sha256:942c93871cafec132cfafdde5ac816111fa897116d7def57cf2967daabad7db4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:KTXOI7MAYGURSSEJ7SFLHV4OI5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The planar algebra of a fixed point subfactor","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.OA","authors_text":"Teodor Banica","submitted_at":"2009-11-16T16:44:05Z","abstract_excerpt":"We consider inclusions of type $(P\\otimes A)^G\\subset(P\\otimes B)^G$, where $G$ is a compact quantum group of Kac type acting on a ${\\rm II}_1$ factor $P$, and on a Markov inclusion of finite dimensional $C^*$-algebras $A\\subset B$. In the case $[A,B]=0$, which basically covers all known examples, we show that the planar algebra of such a subfactor is of the form $P(A\\subset B)^G$, with $G$ acting in some natural sense on the bipartite graph algebra $P(A\\subset B)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.3073","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:58:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vyTV2WqVb76EX7dEACATHwhW0rosFscGxial1Dr0L9CVDJpowXJdXZD5ujxs1q7RX1vJ1db9mTtMktXxX0sLCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:48:02.688903Z"},"content_sha256":"060af4916029ba78afc44540bac07fc46e5f4bfa1104be4f8375f795f8f5ee1d","schema_version":"1.0","event_id":"sha256:060af4916029ba78afc44540bac07fc46e5f4bfa1104be4f8375f795f8f5ee1d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KTXOI7MAYGURSSEJ7SFLHV4OI5/bundle.json","state_url":"https://pith.science/pith/KTXOI7MAYGURSSEJ7SFLHV4OI5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KTXOI7MAYGURSSEJ7SFLHV4OI5/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-08T18:48:02Z","links":{"resolver":"https://pith.science/pith/KTXOI7MAYGURSSEJ7SFLHV4OI5","bundle":"https://pith.science/pith/KTXOI7MAYGURSSEJ7SFLHV4OI5/bundle.json","state":"https://pith.science/pith/KTXOI7MAYGURSSEJ7SFLHV4OI5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KTXOI7MAYGURSSEJ7SFLHV4OI5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:KTXOI7MAYGURSSEJ7SFLHV4OI5","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0ea744600616c981e9150b43a4d85eb3c6efe5cf40615c3a756fec07ec84faca","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-11-16T16:44:05Z","title_canon_sha256":"0347089a9207b6252dfedb23fcd49e2de0fd25b3c5431773304d4432de033835"},"schema_version":"1.0","source":{"id":"0911.3073","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.3073","created_at":"2026-05-17T23:58:50Z"},{"alias_kind":"arxiv_version","alias_value":"0911.3073v4","created_at":"2026-05-17T23:58:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.3073","created_at":"2026-05-17T23:58:50Z"},{"alias_kind":"pith_short_12","alias_value":"KTXOI7MAYGUR","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"KTXOI7MAYGURSSEJ","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"KTXOI7MA","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:060af4916029ba78afc44540bac07fc46e5f4bfa1104be4f8375f795f8f5ee1d","target":"graph","created_at":"2026-05-17T23:58:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We consider inclusions of type $(P\\otimes A)^G\\subset(P\\otimes B)^G$, where $G$ is a compact quantum group of Kac type acting on a ${\\rm II}_1$ factor $P$, and on a Markov inclusion of finite dimensional $C^*$-algebras $A\\subset B$. In the case $[A,B]=0$, which basically covers all known examples, we show that the planar algebra of such a subfactor is of the form $P(A\\subset B)^G$, with $G$ acting in some natural sense on the bipartite graph algebra $P(A\\subset B)$.","authors_text":"Teodor Banica","cross_cats":["math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-11-16T16:44:05Z","title":"The planar algebra of a fixed point subfactor"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.3073","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:942c93871cafec132cfafdde5ac816111fa897116d7def57cf2967daabad7db4","target":"record","created_at":"2026-05-17T23:58:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0ea744600616c981e9150b43a4d85eb3c6efe5cf40615c3a756fec07ec84faca","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-11-16T16:44:05Z","title_canon_sha256":"0347089a9207b6252dfedb23fcd49e2de0fd25b3c5431773304d4432de033835"},"schema_version":"1.0","source":{"id":"0911.3073","kind":"arxiv","version":4}},"canonical_sha256":"54eee47d80c1a9194889fc8ab3d78e475083045239fd3d3c84da9cdba5ef5dc7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"54eee47d80c1a9194889fc8ab3d78e475083045239fd3d3c84da9cdba5ef5dc7","first_computed_at":"2026-05-17T23:58:50.733300Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:58:50.733300Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XRYOBZuEj1lvYj88ZVD7yqwy2r39ksmzVB6QTVvLxCnAsaLwFVjhBlSSydVcl04OyXgw9uoCpCJY92DQdE0gCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:58:50.733768Z","signed_message":"canonical_sha256_bytes"},"source_id":"0911.3073","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:942c93871cafec132cfafdde5ac816111fa897116d7def57cf2967daabad7db4","sha256:060af4916029ba78afc44540bac07fc46e5f4bfa1104be4f8375f795f8f5ee1d"],"state_sha256":"f5b11addaaeb72737a52a481ae3e1c7f45131f3a9fd39bd840889ec50bd192de"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S3GlXPgnWTalywytivxbISdI21lGr2v0dZGV+NhZ9m592RFQFPYvaISBn/ki6PpGsFuc1Ue3Q2p0BIvqKaw+Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T18:48:02.692179Z","bundle_sha256":"e53224e48fb4cea2e31303af4594e06c46aced442596767babc14f36a74b0296"}}