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As a $\\mathfrak{t}$-module $\\mathfrak{g}$ decomposes as \\[\\mathfrak{g} = \\mathfrak{s} \\oplus \\big(\\oplus_{\\nu \\in \\mathcal{R}} \\mathfrak{g}^\\nu\\big)\\] where $\\mathfrak{s} \\subset \\mathfrak{g}$ is the reductive part of a parabolic subalgebra of $\\mathfrak{g}$ and $\\mathcal{R}$ is the Kostant root system associated to $\\mathfrak{t}$. When $\\mathfrak{t}$ is a Cartan subalgebra of $\\mathfrak{g}$ the decomposition above is nothing but the root decomposition of $\\ma"},"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":"1612.02851","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-12-08T21:47:48Z","cross_cats_sorted":[],"title_canon_sha256":"27889d862bbead2b4d3219b33442455ba7c0c762b3c832b543cfec6cbd3db2c6","abstract_canon_sha256":"ae80c1f466d4f6844c84be8db9e752d5aca3a1d46016c7be37e495599c3f4a1b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:03.679550Z","signature_b64":"X1pVRNjQ3ui7f82ho6GQuFvvGkRtEXxtx7Yh22vQQq+XTOFZ5l99FAUoGMYfNBwJEZ1nvUV2NL0NDpsAU1iuBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c058a49fb4419a5f556164e7372edb1ab4737039f139b0edbdb4be4197c3253","last_reissued_at":"2026-05-18T00:52:03.678942Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:03.678942Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Positive Systems of Kostant Roots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Ivan Dimitrov, Mike Roth","submitted_at":"2016-12-08T21:47:48Z","abstract_excerpt":"Let $\\mathfrak{g}$ be a simple complex Lie algebra and let $\\mathfrak{t} \\subset \\mathfrak{g}$ be a toral subalgebra of $\\mathfrak{g}$. As a $\\mathfrak{t}$-module $\\mathfrak{g}$ decomposes as \\[\\mathfrak{g} = \\mathfrak{s} \\oplus \\big(\\oplus_{\\nu \\in \\mathcal{R}} \\mathfrak{g}^\\nu\\big)\\] where $\\mathfrak{s} \\subset \\mathfrak{g}$ is the reductive part of a parabolic subalgebra of $\\mathfrak{g}$ and $\\mathcal{R}$ is the Kostant root system associated to $\\mathfrak{t}$. When $\\mathfrak{t}$ is a Cartan subalgebra of $\\mathfrak{g}$ the decomposition above is nothing but the root decomposition of $\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02851","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1612.02851","created_at":"2026-05-18T00:52:03.679034+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.02851v2","created_at":"2026-05-18T00:52:03.679034+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.02851","created_at":"2026-05-18T00:52:03.679034+00:00"},{"alias_kind":"pith_short_12","alias_value":"DQCYUSP3IQM2","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_16","alias_value":"DQCYUSP3IQM2L5KW","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_8","alias_value":"DQCYUSP3","created_at":"2026-05-18T12:30:12.583610+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DQCYUSP3IQM2L5KWCZHHG4XNWG","json":"https://pith.science/pith/DQCYUSP3IQM2L5KWCZHHG4XNWG.json","graph_json":"https://pith.science/api/pith-number/DQCYUSP3IQM2L5KWCZHHG4XNWG/graph.json","events_json":"https://pith.science/api/pith-number/DQCYUSP3IQM2L5KWCZHHG4XNWG/events.json","paper":"https://pith.science/paper/DQCYUSP3"},"agent_actions":{"view_html":"https://pith.science/pith/DQCYUSP3IQM2L5KWCZHHG4XNWG","download_json":"https://pith.science/pith/DQCYUSP3IQM2L5KWCZHHG4XNWG.json","view_paper":"https://pith.science/paper/DQCYUSP3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.02851&json=true","fetch_graph":"https://pith.science/api/pith-number/DQCYUSP3IQM2L5KWCZHHG4XNWG/graph.json","fetch_events":"https://pith.science/api/pith-number/DQCYUSP3IQM2L5KWCZHHG4XNWG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DQCYUSP3IQM2L5KWCZHHG4XNWG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DQCYUSP3IQM2L5KWCZHHG4XNWG/action/storage_attestation","attest_author":"https://pith.science/pith/DQCYUSP3IQM2L5KWCZHHG4XNWG/action/author_attestation","sign_citation":"https://pith.science/pith/DQCYUSP3IQM2L5KWCZHHG4XNWG/action/citation_signature","submit_replication":"https://pith.science/pith/DQCYUSP3IQM2L5KWCZHHG4XNWG/action/replication_record"}},"created_at":"2026-05-18T00:52:03.679034+00:00","updated_at":"2026-05-18T00:52:03.679034+00:00"}