Published paper
Abstract
The article develops the structure theory of reductive algebraic groups over arbitrary fields. Its root-subgroup, parabolic, and unipotent-containment results supply the algebraic-group step in Kazhdan–Margulis Lemma 3.
Role in dependence graphs
Proof-critical source
The Dirichlet spectrum
This paper is included only for the following marked statements:
- Section 2.3 root-subgroup structure · exact Numdam VOR physical PDF pp. 11–12 / printed pp. 64–65Decomposes a torus-normalized subgroup of the positive unipotent radical into its root subgroups.
- Theorem 4.15 and Corollary 4.16 · exact Numdam VOR physical PDF pp. 38–39 / printed pp. 91–92Places a unipotent subgroup in a parabolic and then in a suitable unipotent radical.
- Theorem 8.2 · exact Numdam VOR physical PDF pp. 69–70 / printed pp. 122–123Supplies the rational split-solvable and fixed-point mechanism supporting the positive-root construction.
- Lemma 8.3 and Proposition 8.4 · exact Numdam VOR physical PDF pp. 70–71 / printed pp. 123–124Directly yields containment of the connected unipotent subgroup in the positive-root unipotent group required in Kazhdan–Margulis Lemma 3.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Exact Numdam version of record · Publications Mathématiques de l’IHÉS 27 (1965), 55–151
Armand Borel and Jacques Tits. Groupes réductifs. Publications Mathématiques de l’IHÉS 27 (1965), 55–151.
Open audited source ↗01Statements5 reported findingsNo statement issues found
The exact article supplies the real-characteristic-zero algebraic-group result used in Kazhdan–Margulis Lemma 3: every connected unipotent subgroup is contained in the positive-root unipotent radical associated with a suitable maximal split torus. The supporting conjugacy, parabolic, and root-group statements have the required hypotheses. The separate nilpotent Jordan-decomposition sentence in Kazhdan–Margulis is not assigned to this article without a more precise locator.
Section 2.3No issue found
An ordering of the roots defines the standard unipotent subgroup
Exact Numdam version of record · physical PDF pp. 11–12 (printed pp. 64–65) · Section 2.3
For a maximal torus and an ordering of its roots, Borel–Tits define as the subgroup generated by the positive root groups. They state that is unipotent, that is a Borel subgroup, and that the corresponding negative-root construction gives the opposite subgroup. This is the standard positive-root group appearing as in Kazhdan–Margulis Lemma 3.
Theorem 8.2No issue found
Maximal connected unipotent subgroups are conjugate over a perfect field
Exact Numdam version of record · physical PDF pp. 69–70 (printed pp. 122–123) · Theorem 8.2 and proof
The theorem states conjugacy over of maximal connected -unipotent subgroups when is perfect. For , every algebraic unipotent group is connected, so a given unipotent subgroup can be enlarged to a maximal one and then conjugated to the standard positive-root subgroup. The quantifiers and field hypotheses match the Kazhdan–Margulis application.
Theorem 4.15 and Proposition 8.4No issue found
Maximal unipotent subgroups are precisely the radicals of minimal parabolics
Exact Numdam version of record · physical PDF pp. 38–39 and 71 (printed pp. 91–92 and 124) · Theorem 4.15, Corollary 4.16, and Proposition 8.4
Theorem 4.15 identifies a minimal parabolic, after choosing a maximal split torus and an ordering, with the nonnegative-root subgroup and identifies its unipotent radical with the positive-root subgroup. Proposition 8.4 proves that the maximal connected unipotent subgroups are exactly these radicals. Together they give the precise containment needed in Kazhdan–Margulis.
Kazhdan–Margulis Lemma 3Dependency match verified
The imported Borel–Tits scope is the unipotent-containment step
Kazhdan–Margulis exact MathNet version of record · physical PDF pp. 3–4 (printed pp. 149–150); Borel–Tits exact locators above
Kazhdan–Margulis pass to the algebraic closure of a connected nilpotent subgroup, isolate its unipotent factor , and require a maximal ordered torus for which lies in the positive-root subgroup. Borel–Tits Sections 2.3, 4.15–4.16, and 8.2–8.4 establish exactly that last step over . Their article is therefore a genuine proof parent rather than a retrospective textbook substitute.
Citation boundaryRouted separately
The nilpotent semisimple–unipotent direct product is a separate dependency
Kazhdan–Margulis exact MathNet version of record · physical PDF p. 3 (printed p. 149), paragraph beginning with the Zariski closure of
Kazhdan–Margulis separately state that an algebraic nilpotent subgroup splits as a direct product of its semisimple and unipotent parts. The inspected Borel–Tits results begin with the subsequent containment of the unipotent factor. This report therefore does not silently assign the direct-product assertion to Borel–Tits; it is represented by its own source route in the dependency closure.
02Proofs5 reported findingsNo proof issues found
The selected Borel–Tits argument is coherent in the exact version of record. Its Jacobson–Morozov and Grothendieck inputs terminate at exact book or edited-volume locators; the exact Rosenlicht 1957 source proves the splitness, triangularization, and rational fixed-point inputs; and the exact Chevalley seminar volume proves the positive-root subgroup decomposition. The recursively represented source chain is complete for the selected containment.
Selected internal chainCorrect as an internal reduction
Sections 2.3, 4.15–4.17, and 8.3–8.4 close the containment argument
Exact Numdam version of record · physical PDF pp. 11–12, 38–39, and 69–71 (printed pp. 64–65, 91–92, and 122–124)
Section 2.3 constructs the positive-root unipotent group. Theorem 4.15 and Corollaries 4.16–4.17 identify the relevant minimal parabolics and their unipotent radicals. Lemma 8.3 places any nontrivial unipotent subgroup inside a minimal parabolic, and Proposition 8.4 then puts it inside that radical. In characteristic zero, the connectedness condition is automatic. No logical gap was found in the transitions among these statements.
Jacobson–Morozov inputTerminal published-book dependency
The one-dimensional unipotent reduction terminates at a published book
Exact Numdam version of record · physical PDF p. 70 (printed p. 123) · Lemma 8.3, citation [14, Chapter III, Theorem 17, p. 100]
Lemma 8.3 invokes Nathan Jacobson, Lie Algebras, Interscience Tracts in Pure and Applied Mathematics 10 (1962), Chapter III, Theorem 17, pp. 100–101. The theorem states, in the direction used here, that every nonzero nilpotent element of a completely reducible linear Lie algebra in characteristic zero is contained in a split three-dimensional simple subalgebra. Applied to the adjoint representation of the semisimple Lie algebra, this supplies the required homomorphism.
Grothendieck inputTerminal published-seminar dependency
Existence and density of rational maximal tori have an exact published seminar endpoint
Exact Numdam version of record · physical PDF p. 17 (printed p. 70) · Theorem 2.14, citations [11, Exposé XIV, Theorem 1.1 and §§6.5, 6.7]
The local-section and parabolic conjugacy steps use Theorem 2.14, attributed to Demazure–Grothendieck's Schémas en groupes, Exposé XIV. The same numbered exposé was subsequently published in Schémas en groupes II, Lecture Notes in Mathematics 152 (Springer, 1970). Exposé XIV, Theorem 1.1 gives existence of a maximal torus; Corollaries 6.5 and 6.7 give the unirationality and hence, over an infinite field, density needed for the reductive case.
Rosenlicht 1957Exact source route verified
Split unipotence and the complete-variety fixed-point theorem are verified in the exact source
Exact Numdam version of record · physical PDF pp. 14 and 69–70 (printed pp. 67 and 122–123) · Section 2.7(i), Section 8.1, and proof of Theorem 8.2; citations [23, Propositions 5–7, Corollaries 1–2 on pp. 33–34, and p. 35]
The argument needs the facts that a unipotent group over a perfect field is split and that a split connected solvable group acting on a complete variety with a rational point has a rational fixed point. The exact Springer version of Rosenlicht's article proves Corollaries 1–2, the fixed-point lemma on printed pp. 35–36, and Proposition 7 on p. 36. Their hypotheses and conclusions match the inspected Borel–Tits uses, and the article's own proof-critical parents are represented separately in the graph.
Chevalley seminarExact published-seminar dependency verified
The positive-root subgroup structure closes in the exact published seminar volume
Exact Numdam version of record · physical PDF pp. 11–12 (printed pp. 64–65) · Section 2.3, citation [10, Exposé 13, Theorem 1(d)]
The product decomposition and root-subgroup structure of are cited to the 1956–1958 Séminaire Claude Chevalley, Exposé 13, Theorem 1(d). Exact Numdam scans of Exposé 13 and its same-volume Exposés 9 and 12 inputs were checked. The selected theorem is proved inside that formally published seminar volume plus a Bourbaki book input, so the branch is truthfully terminalized there.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.