{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:SX4N5M6RFEHO55RI4M4GPNLOLR","short_pith_number":"pith:SX4N5M6R","canonical_record":{"source":{"id":"1707.09384","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2017-07-28T19:30:21Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"84c09cbaf9f68fa911558b6193d9608a563b81edb04ad40d4fa69f7931054c45","abstract_canon_sha256":"54b09af20d8614e48afee57a0e4d8b074870bb7851a5647a757b9cb949b317fb"},"schema_version":"1.0"},"canonical_sha256":"95f8deb3d1290eeef628e33867b56e5c59ad32bd38df1d1ac63004f76aef6012","source":{"kind":"arxiv","id":"1707.09384","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.09384","created_at":"2026-05-18T00:39:11Z"},{"alias_kind":"arxiv_version","alias_value":"1707.09384v1","created_at":"2026-05-18T00:39:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.09384","created_at":"2026-05-18T00:39:11Z"},{"alias_kind":"pith_short_12","alias_value":"SX4N5M6RFEHO","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"SX4N5M6RFEHO55RI","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"SX4N5M6R","created_at":"2026-05-18T12:31:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:SX4N5M6RFEHO55RI4M4GPNLOLR","target":"record","payload":{"canonical_record":{"source":{"id":"1707.09384","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2017-07-28T19:30:21Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"84c09cbaf9f68fa911558b6193d9608a563b81edb04ad40d4fa69f7931054c45","abstract_canon_sha256":"54b09af20d8614e48afee57a0e4d8b074870bb7851a5647a757b9cb949b317fb"},"schema_version":"1.0"},"canonical_sha256":"95f8deb3d1290eeef628e33867b56e5c59ad32bd38df1d1ac63004f76aef6012","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:11.494130Z","signature_b64":"3AoN1sqWLysvysNxwRKY8zlRUXhpW1YpFEqir+dCxuM0ZEySkqkx/Q1e1hAfIHMftrvaEDp+BoUNmHcNGfKiAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95f8deb3d1290eeef628e33867b56e5c59ad32bd38df1d1ac63004f76aef6012","last_reissued_at":"2026-05-18T00:39:11.493466Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:11.493466Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1707.09384","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:39:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wt3uPlqbUFpaCuNxhyIU/nE5xjPeofexPa8wf9lXCIGmznWXspsbB0qZ0VuOCtD18UNSZvgoMA8mxKZNrSz3Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T16:37:22.489236Z"},"content_sha256":"edbafeb7eeb3959f8a839098a7bedc9d67f4250bb6ed00b5bebe4b01a1a94579","schema_version":"1.0","event_id":"sha256:edbafeb7eeb3959f8a839098a7bedc9d67f4250bb6ed00b5bebe4b01a1a94579"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:SX4N5M6RFEHO55RI4M4GPNLOLR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"K-projectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.QA","authors_text":"Alexander Odesskii","submitted_at":"2017-07-28T19:30:21Z","abstract_excerpt":"We study representations of a free associative algebra $T^*(W\\otimes W^*)$ in a vector space $V$ with the property $V\\otimes V\\cong V\\oplus V_0$ where $T^*(W\\otimes W^*)$ acts by zero on $V_0$ and the tensor product $V\\otimes V$ of representations corresponds to the natural homomorphism $W\\otimes W^*\\to W\\otimes W^* \\otimes W\\otimes W^*$. We develop an algebraic theory of such objects and construct a lot of examples."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.09384","kind":"arxiv","version":1},"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":"2026-05-18T00:39:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PXTcP/Z97/A5s4yvTmLH4fDoeBIQ6ZmJF3jNzALdNraiO2w8VCrE7z5ajD8C45VNgUBCb3EBrhECC59+3IvzDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T16:37:22.489841Z"},"content_sha256":"6be7f4a7f25761d2a6fa9d4ae34d60b974702d27ee6193fc7a48696272e3cb3a","schema_version":"1.0","event_id":"sha256:6be7f4a7f25761d2a6fa9d4ae34d60b974702d27ee6193fc7a48696272e3cb3a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SX4N5M6RFEHO55RI4M4GPNLOLR/bundle.json","state_url":"https://pith.science/pith/SX4N5M6RFEHO55RI4M4GPNLOLR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SX4N5M6RFEHO55RI4M4GPNLOLR/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":"2026-05-25T16:37:22Z","links":{"resolver":"https://pith.science/pith/SX4N5M6RFEHO55RI4M4GPNLOLR","bundle":"https://pith.science/pith/SX4N5M6RFEHO55RI4M4GPNLOLR/bundle.json","state":"https://pith.science/pith/SX4N5M6RFEHO55RI4M4GPNLOLR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SX4N5M6RFEHO55RI4M4GPNLOLR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:SX4N5M6RFEHO55RI4M4GPNLOLR","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":"54b09af20d8614e48afee57a0e4d8b074870bb7851a5647a757b9cb949b317fb","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2017-07-28T19:30:21Z","title_canon_sha256":"84c09cbaf9f68fa911558b6193d9608a563b81edb04ad40d4fa69f7931054c45"},"schema_version":"1.0","source":{"id":"1707.09384","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.09384","created_at":"2026-05-18T00:39:11Z"},{"alias_kind":"arxiv_version","alias_value":"1707.09384v1","created_at":"2026-05-18T00:39:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.09384","created_at":"2026-05-18T00:39:11Z"},{"alias_kind":"pith_short_12","alias_value":"SX4N5M6RFEHO","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"SX4N5M6RFEHO55RI","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"SX4N5M6R","created_at":"2026-05-18T12:31:43Z"}],"graph_snapshots":[{"event_id":"sha256:6be7f4a7f25761d2a6fa9d4ae34d60b974702d27ee6193fc7a48696272e3cb3a","target":"graph","created_at":"2026-05-18T00:39:11Z","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 study representations of a free associative algebra $T^*(W\\otimes W^*)$ in a vector space $V$ with the property $V\\otimes V\\cong V\\oplus V_0$ where $T^*(W\\otimes W^*)$ acts by zero on $V_0$ and the tensor product $V\\otimes V$ of representations corresponds to the natural homomorphism $W\\otimes W^*\\to W\\otimes W^* \\otimes W\\otimes W^*$. We develop an algebraic theory of such objects and construct a lot of examples.","authors_text":"Alexander Odesskii","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2017-07-28T19:30:21Z","title":"K-projectors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.09384","kind":"arxiv","version":1},"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:edbafeb7eeb3959f8a839098a7bedc9d67f4250bb6ed00b5bebe4b01a1a94579","target":"record","created_at":"2026-05-18T00:39:11Z","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":"54b09af20d8614e48afee57a0e4d8b074870bb7851a5647a757b9cb949b317fb","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2017-07-28T19:30:21Z","title_canon_sha256":"84c09cbaf9f68fa911558b6193d9608a563b81edb04ad40d4fa69f7931054c45"},"schema_version":"1.0","source":{"id":"1707.09384","kind":"arxiv","version":1}},"canonical_sha256":"95f8deb3d1290eeef628e33867b56e5c59ad32bd38df1d1ac63004f76aef6012","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"95f8deb3d1290eeef628e33867b56e5c59ad32bd38df1d1ac63004f76aef6012","first_computed_at":"2026-05-18T00:39:11.493466Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:39:11.493466Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3AoN1sqWLysvysNxwRKY8zlRUXhpW1YpFEqir+dCxuM0ZEySkqkx/Q1e1hAfIHMftrvaEDp+BoUNmHcNGfKiAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:39:11.494130Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.09384","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:edbafeb7eeb3959f8a839098a7bedc9d67f4250bb6ed00b5bebe4b01a1a94579","sha256:6be7f4a7f25761d2a6fa9d4ae34d60b974702d27ee6193fc7a48696272e3cb3a"],"state_sha256":"a8e29eaf3ea4bbf806636a185962b17c62ddaac6257ee13af2d25e80f1c064ee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"37sB5Jg4Iex3cZ0tAZmODupLUmOVMYRDgLPqogiPCaRQduUeTpZQplgOnOYQ8DnJ+x/Kvauz7vcSNt6jDS6TCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T16:37:22.493702Z","bundle_sha256":"a3220049056f1c54c7873e88f8d55e297b8f1255a2268313b0184e95d98f5155"}}