Serre — Lie Algebras and Lie Groups (Fast Track 3.13) — Audit + Gap Plan

Book: Jean-Pierre Serre, Lie Algebras and Lie Groups: 1964 Lectures Given at Harvard University (W. A. Benjamin 1965; reprinted as Springer Lecture Notes in Mathematics 1500, 1992; xvi + 168 pp.). Distillation of Serre's 1964 Harvard course. Two parts: Part I — Lie Algebras (Cartan-Killing-style finite-dim semisimple theory, but presented as a companion to the standard machinery and with the Campbell-Hausdorff formula at center) and Part II — Lie Groups (formal groups, $p$ -adic analytic groups via the exponential, the cohomology of Lie algebras / Hochschild-Serre spectral sequence).

Fast Track entry: 3.13. Booklist row tag: "p-adic, homological." This is NOT the same book as Serre's Complex Semisimple Lie Algebras (FT 3.12, booklist tag "Semisimple classification") — that volume is the classification primer (Cartan subalgebras, root systems, Cartan matrix, Serre relations, classification of simple complex Lie algebras). FT 3.13 instead covers the formal-group / $p$ -adic-analytic / Lie-algebra-cohomology arc that Serre developed in parallel and which is conspicuously absent from every other Fast Track Lie slot (Fulton-Harris 3.11 is complex-only and example-driven; Hall is matrix-only; Helgason 3.17 is real symmetric spaces; Knapp is real semisimple structure). FT 3.13 is the only Fast Track entry covering formal groups, the $p$ -adic exponential, and the Chevalley-Eilenberg / Hochschild-Serre cohomology programme.

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol, per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md). Output is a concrete punch-list of new units to write so that Serre's Lie Algebras and Lie Groups (henceforth SLALG to avoid collision with Serre 3.12 "SCSLA") is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

REDUCED audit. No local PDF in reference/textbooks-extra/ (verified — only the cover image reference/fast-track/images/Serre-Lie-Algebras-Lie-Groups-712x1024__dd9fad6ce3.jpg is local). Springer LNM 1500 is gated; Anna's Archive / institutional mirrors were not fetched in this audit window. This audit works from (a) the public TOC structure of LNM 1500 (well-documented: Part I §§I–VI on Lie algebras with Campbell-Hausdorff in §IV; Part II Lie groups split into LG §§1–5 on general / formal groups and LG §§6–8 on analytic / $p$ -adic groups + cohomology), (b) the Bourbaki Groupes et algèbres de Lie Chs. II–III crossreference (which Serre's lectures both predate and helped seed), (c) the Codex's existing shipped units in content/07-representation-theory/06-lie- algebraic/, 03-modern-geometry/03-lie/, 03-modern-geometry/04-differential- forms/03.04.01-lie-algebra.md, and 03-modern-geometry/13-spectral-sequences/, and (d) the originator literature (Campbell 1898, Baker 1905, Hausdorff 1906, Chevalley-Eilenberg 1948, Hochschild-Serre 1953, Lazard 1955/1965). A full line-number audit is deferred until a PDF is acquired. This is consistent with the audit-stub convention used for Brown-Higgins-Sivera (1.05a) and Fulton-Harris (3.11).

§1 What SLALG is for

SLALG is the slim graduate companion that takes the reader from the abstract Lie-algebra definition through the Campbell-Hausdorff formula and out the far side into two specifically Serre-flavoured extensions of the standard theory: formal groups (a category in which one can do Lie theory over any commutative ring, not just fields of characteristic zero) and the $p$ -adic analytic groups that arise when one runs the exponential backwards in a $p$ -adic setting. The third specifically-Serre element is the Chevalley-Eilenberg / Hochschild-Serre development of the cohomology $H^{*} (g, M)$ of a Lie algebra with coefficients in a module $M$ , and the associated spectral sequence for an ideal $\mathfrak{h} \triangleleft \mathfrak{g}$.

Distinctive contributions, in roughly the order SLALG develops them:

Lie-algebra basics done crisply but in Serre's idiosyncratic order. Definitions of Lie algebra, ideal, semidirect product, derivations, the universal enveloping algebra $U (g)$ with the Poincaré-Birkhoff-Witt theorem proved cleanly via filtrations. SLALG Part I §I–§II. Standard content but the PBW proof is the model treatment subsequent texts (e.g. Humphreys, Bourbaki) adopt.
Free Lie algebras and the Magnus expansion. SLALG §III. Constructs the free Lie algebra $L (X)$ on a set $X$ as the subspace of primitive elements in the free associative algebra under the comultiplication $Δ (x) = x \otimes 1 + 1 \otimes x$ . This Hopf-algebraic / primitive-element approach is the clean way to prove Campbell-Hausdorff and is what differentiates SLALG's treatment from textbook expositions that simply state the formula.
The Campbell-Hausdorff formula. SLALG §IV — the centrepiece of Part I. Statement: for $X, Y$ in a topologically complete Lie algebra (or formal Lie algebra), $\log(\exp X \cdot \exp Y) = X + Y + \tfrac{1}{2}[X, Y] + \tfrac{1}{12}([X, [X, Y]] + [Y, [Y, X]]) + \cdots$ is a universal Lie series in $X, Y$ with rational coefficients. Originator citations: J. E. Campbell ("On a law of combination of operators bearing on the theory of continuous transformation groups," Proc. London Math. Soc. 28 (1897), 381–390; ibid. 29 (1898), 14–32); H. F. Baker ("Alternants and continuous groups," Proc. London Math. Soc. (2) 3 (1905), 24–47); F. Hausdorff ("Die symbolische Exponentialformel in der Gruppentheorie," Berichte Sächs. Akad. Wiss. 58 (1906), 19–48). Serre's exposition is the model: prove via the Friedrichs criterion that the BCH series lies in the free Lie algebra, then convergence over $Q$ is automatic; convergence in real / $p$ -adic settings reduces to a single estimate.
Cohomology of Lie algebras $H^{*} (g, M)$ . SLALG Part I §§V–VI (after Campbell-Hausdorff). Chevalley-Eilenberg complex $C^{n} (g, M) = Hom (Λ^{n} g, M)$ with differential $d$ given by the standard formula. Whitehead's lemmas: $H^{1} (g, M) = H^{2} (g, M) = 0$ for $g$ finite-dimensional semisimple and $M$ finite-dimensional — semisimplicity / Weyl's theorem via cohomological vanishing. Originator citation: C. Chevalley, S. Eilenberg, "Cohomology theory of Lie groups and Lie algebras," Trans. Amer. Math. Soc. 63 (1948), 85–124.
Hochschild-Serre spectral sequence for an ideal. SLALG Part I §VI (or sometimes given as an Appendix depending on edition). For $h ◃ g$ , the filtration on the Chevalley-Eilenberg complex by $h$ -degree yields a spectral sequence $E_2^{p, q} = H^p(\mathfrak{g}/\mathfrak{h}, H^q(\mathfrak{h}, M)) \Rightarrow H^{p+q}(\mathfrak{g}, M)$. Originator citation: G. Hochschild, J.-P. Serre, "Cohomology of Lie algebras," Ann. of Math. (2) 57 (1953), 591–603. The Lie-algebra analogue of the Lyndon- Hochschild-Serre spectral sequence for group cohomology and the Serre spectral sequence for a fibration.
Formal groups. SLALG Part II LG §§1–5. A formal group law over a commutative ring $R$ is a power series $F (X, Y) \in R [[X, Y]]$ (or vector-valued $F \in R [[X_{1}, \dots, X_{n}, Y_{1}, \dots, Y_{n}]]^{n}$ in the higher-dimensional case) satisfying $F (X, 0) = X$ , $F (0, Y) = Y$ , $F (F (X, Y), Z) = F (X, F (Y, Z))$ . Bridges the Lie-group / Lie-algebra correspondence: in characteristic 0, every formal group has an associated Lie algebra, and the Lie functor is an equivalence. In characteristic $p > 0$ the picture breaks (formal groups have more structure than their Lie algebras — height, $p$ -typical type, etc.). Originator citation: S. Bochner, "Formal Lie groups," Ann. of Math. (2) 47 (1946), 192–201; M. Lazard, "Sur les groupes de Lie formels à un paramètre," Bull. Soc. Math. France 83 (1955), 251–274; Lois de groupes et analyseurs, Ann. Sci. ENS 72 (1955), 299–400. Comprehensive reformulation in M. Hazewinkel, Formal Groups and Applications (Academic Press 1978, reprinted AMS Chelsea 2012).
$p$ -adic analytic Lie groups. SLALG Part II LG §§6–8. A $p$ -adic Lie group is a group object in the category of $p$ -adic analytic manifolds. Serre constructs the exponential map $exp : g \to G$ as a power-series map convergent on a small neighbourhood of 0 in $g$ (the radius of convergence depends on $p$ via the standard estimate $∣ n! ∣_{p} \to 0$ slowly), and uses the Campbell-Hausdorff formula to define a local group structure on the image. The result: a $p$ -adic Lie group of dimension $n$ is locally isomorphic (as a $p$ -adic analytic manifold-with-group-law) to a neighbourhood of 0 in $Q_{p}^{n}$ with multiplication given by the BCH series — provided one stays inside the radius of convergence. This is the canonical bridge between the algebraic (formal-group) and the analytic ( $p$ -adic-manifold) sides. Originator citation: M. Lazard, "Groupes analytiques $p$ -adiques," Publ. Math. IHES 26 (1965), 5–219 — the definitive treatment, which appeared the same year as Serre's Harvard lectures and which Serre cites repeatedly.
The "Lie-algebra of a $p$ -adic Lie group" functor and its analytic properties. SLALG Part II LG §§7–8. The functor $G \mapsto \mathrm{Lie}(G) $i s f u l l y f ai t h f u l o n t h ec a t e g or y o f$ p$-adic Lie groups, and (in the simply connected case) essentially surjective onto the category of finite-dim $Q_{p}$ -Lie algebras. This is the analogue of Lie's three theorems in the $p$ -adic setting.

SLALG is not a first introduction to Lie theory — it assumes the standard Cartan-Killing theory at the level of SCSLA (Serre 3.12) or Bourbaki GAL Ch. I, and assumes basic homological algebra (derived functors, double complexes) at the level of Cartan-Eilenberg or Weibel. It is not an introduction to algebraic groups (use Borel or Springer); not a treatment of root systems and classification (use SCSLA = Serre 3.12 or Humphreys); and not a treatment of compact / real Lie groups (use Hall, Bump, or Helgason 3.17). It is the entry point if one needs: (a) a textbook derivation of Campbell-Hausdorff in the Hopf-algebraic style, (b) the foundational Chevalley-Eilenberg + Hochschild-Serre cohomology package, (c) an honest introduction to formal groups, or (d) the bridge from Lie algebras to $p$ -adic analytic groups.

Cited peer sources (≥3, per stub spec):

N. Bourbaki, Groupes et algèbres de Lie, Chapitres II–III (Hermann 1972; English translation Springer 1989). Chapter II "Algèbres de Lie libres" is the canonical extended reference for Serre §III (free Lie algebras, Hall bases, Magnus expansion) and the Campbell-Hausdorff formula in §IV. Chapter III "Groupes de Lie" covers the formal-group / $p$ -adic bridge at greater length than Serre. Cross-references SLALG on essentially every page of Chs. II–III.
G. Hochschild, The Structure of Lie Groups (Holden-Day 1965; reprinted Dover 2015). Hochschild's own book, published the same year as SLALG. Covers the Hochschild-Serre cohomology package from Hochschild's perspective; complementary to SLALG's Serre-perspective treatment. Useful for a second pass on §V–§VI.
M. Hazewinkel, Formal Groups and Applications (Academic Press 1978; reprinted AMS Chelsea 2012). The canonical comprehensive reference for the formal-group programme SLALG opens. SLALG Part II §§1–5 is roughly Hazewinkel Chs. 1–3 condensed to lecture form. Hazewinkel goes substantially further: Honda formal groups, the Cartier-Dieudonné classification, applications to algebraic topology (complex cobordism, Lubin-Tate).
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press 1978; reprinted AMS GSM 34, 2001). FT 3.17 sibling. Covers the real / complex Lie group + symmetric space side that SLALG explicitly pushes out of scope. Together SLALG + Helgason = the full Lie-theoretic picture short of structure theory of real semisimple groups (Knapp).
M. Lazard, "Groupes analytiques $p$ -adiques," Publ. Math. IHES 26 (1965). The same-year companion paper Serre cites repeatedly. Definitive for the $p$ -adic analytic side. Goes beyond SLALG by introducing the $p$ -saturation / uniform pro- $p$ group machinery that became the basis for Dixon-du Sautoy-Mann-Segal's modern treatment (Analytic Pro- $p$ Groups, 2nd ed. CUP 1999).

§2 Coverage table (Codex vs SLALG)

Cross-referenced against the current shipped units in:

content/07-representation-theory/06-lie-algebraic/ (9 units: Lie algebra rep, universal enveloping algebra, root system, Weyl group, Dynkin diagram, Verma module, Weyl character formula, Weyl dimension formula, Borel-Weil),
content/03-modern-geometry/03-lie/ (3 units: Lie group, group action, orthogonal group),
content/03-modern-geometry/04-differential-forms/03.04.01-lie-algebra.md,
content/03-modern-geometry/13-spectral-sequences/ (4 files: spectral sequence, Leray-Serre, Leray-Hirsch, exercises).

✓ = covered, △ = partial / different framing, ✗ = not covered.

SLALG topic	Codex unit(s)	Status	Note
Lie algebra (definition, ideals, derivations, semidirect product)	`03.04.01 lie-algebra`	✓	Standard definition shipped.
Solvable / nilpotent Lie algebras; Lie's theorem; Engel's theorem	—	✗	Gap (medium). Touched in `07.04.01 cartan-weyl-classification` but no dedicated unit. SLALG §II prereq for the cohomology vanishing arguments.
Universal enveloping algebra $U (g)$	`07.06.02 universal-enveloping-algebra`	✓
Poincaré-Birkhoff-Witt theorem	△ (inside `07.06.02`)	△	Stated and used in `07.06.02`; SLALG §I gives a model proof. Master-tier deepening candidate.
Free Lie algebra $L (X)$ ; Magnus expansion; primitive-element characterisation	—	✗	Gap (high — SLALG §III centrepiece). No Codex unit on free Lie algebras or the Hopf-algebraic primitive-element picture.
Friedrichs criterion (primitive iff $Δ (x) = x \otimes 1 + 1 \otimes x$ )	—	✗	Gap (medium — depends on free Lie algebra unit). Hopf-algebra machinery.
Campbell-Baker-Hausdorff formula	—	✗	Gap (very high — SLALG §IV centrepiece). Cited in passing in `03.03.01 lie-group` (exponential map) but no derivation, no statement of the universal formula, no convergence discussion. The single largest gap.
Cohomology of Lie algebras $H^{*} (g, M)$	—	✗	Gap (very high — SLALG §V centrepiece). Chevalley-Eilenberg complex absent from the Codex. No dedicated unit.
Whitehead's lemmas ( $H^{1} = H^{2} = 0$ for semisimple)	—	✗	Gap (high). Cohomological proof of Weyl's complete reducibility theorem; SLALG's pedagogical bridge from cohomology to representation theory.
Hochschild-Serre spectral sequence for an ideal	—	✗	Gap (high). Mentioned only at the topological Leray-Serre level in `03.13.02 leray-serre`; the algebraic Lie-algebra version is not its own unit.
Lie group (smooth manifold + group law)	`03.03.01 lie-group`	✓
Lie-algebra of a Lie group ( $T_{e} G$ with bracket)	△ (in `03.04.01 lie-algebra` + `03.03.01 lie-group`)	△	Stated; the functor $G \mapsto Lie (G)$ is sketched but Lie's three theorems are not their own unit.
Exponential map $exp : g \to G$ (real / complex case)	△ (in `03.03.01 lie-group`)	△	Mentioned at intermediate tier; SLALG's universal-power-series viewpoint is not the framing. Gap (low-medium — frame deepening only).
Formal group law $F (X, Y) \in R [[X, Y]]^{n}$	—	✗	Gap (very high — SLALG Part II opening; the entire formal-group programme). No Codex unit.
Lie functor for formal groups (char 0 equivalence)	—	✗	Gap (high). The bridge theorem; depends on formal-group unit.
Lubin-Tate formal groups; height; $p$ -typical formal groups	—	✗	Gap (low — Master-tier deepening; SLALG mentions but does not develop fully). Belongs in a Master extension of the formal-group unit.
$p$ -adic analytic manifold	—	✗	Gap (medium — prereq for $p$ -adic Lie group). $Q_{p}$ and analytic structure on $Q_{p}^{n}$ .
$p$ -adic Lie group	—	✗	Gap (very high — SLALG Part II centrepiece). No Codex unit on $p$ -adic Lie groups.
$p$ -adic exponential and its radius of convergence	—	✗	Gap (high). The technical heart of SLALG Part II §§6–7. Depends on Campbell-Hausdorff and on $p$ -adic analytic manifold.
Lazard's theorem: $p$ -adic Lie group $\leftrightarrow$ $Q_{p}$ -Lie algebra equivalence	—	✗	Gap (high — SLALG's punchline for Part II).
Pro- $p$ groups (Lazard's later development; SLALG sketch)	—	✗	Gap (low — Master-only pointer). SLALG predates the uniform-pro- $p$ machinery; pointer unit at most.
Spectral sequence (general framework)	`03.13.01 spectral-sequence`	✓	General machinery shipped. Lie-algebra specialisation is the gap.
Leray-Serre spectral sequence (fibrations)	`03.13.02 leray-serre`	✓	Topological version. Algebraic Lie-algebra version (= Hochschild-Serre) is the gap.

Aggregate coverage estimate (REDUCED audit basis).

SLALG Part I §§I–II (Lie algebra basics, $U (g)$ , PBW): ~75% covered via 03.04.01, 07.06.02. Gap: dedicated PBW proof unit, dedicated solvable/nilpotent + Lie's/Engel's theorem unit.
SLALG Part I §III (free Lie algebra, Magnus expansion, primitive elements): ~0% covered. Total gap.
SLALG Part I §IV (Campbell-Hausdorff): ~5% covered (one passing mention in 03.03.01). Effectively a total gap and the single largest gap in this audit.
SLALG Part I §§V–VI (cohomology, Hochschild-Serre): ~0% covered. Total gap; only the topological Leray-Serre is in the Codex.
SLALG Part II §§1–5 (formal groups, formal Lie functor): ~0% covered. Total gap.
SLALG Part II §§6–8 ( $p$ -adic analytic groups, $p$ -adic exponential, Lazard's equivalence): ~0% covered. Total gap.

Overall: ~15% of SLALG is covered by the shipped Codex corpus. The gap is substantial and distinct — SLALG covers material that is deliberately absent from the other Lie-strand Fast Track books (Fulton-Harris, Hall, Helgason 3.17, Knapp). Closing this gap is the only way the Fast Track "p-adic, homological" tag for row 3.13 becomes a substantive claim rather than a label.

This is a new-unit-heavy outcome, the opposite of Fulton-Harris's deepening-heavy outcome. SLALG introduces material that has no existing Codex framing to deepen.

§3 Gap punch-list (priority-ordered)

Priority 0 — prereqs from other strands: None blocking — 03.04.01 lie-algebra, 07.06.02 universal-enveloping-algebra, and 03.13.01 spectral-sequence are all shipped. The $p$ -adic units will need a brief $Q_{p}$ / $p$ -adic absolute value prereq, but that can be folded into the relevant unit as a short notation paragraph rather than a dedicated prereq unit (the Codex does not currently have a number-theory strand, and producing one for two units is overkill).

Priority 1 — high-leverage, captures SLALG's distinctive content:

07.06.10 Free Lie algebra and the Magnus expansion. Definition of $L (X)$ on a set $X$ , the universal property, the Hopf-algebraic characterisation as the primitive elements of the free associative algebra $T (X)$ , the Friedrichs criterion. Three-tier; Master tier includes a Hall-basis construction. SLALG §III anchor; Bourbaki GAL Ch. II anchor. ~1800 words. Foundational for the Campbell-Hausdorff unit.
07.06.11 Campbell-Baker-Hausdorff formula. Statement of the universal Lie series, proof via the free-Lie-algebra / primitive-element approach (relies on Priority 1 unit), discussion of convergence over $R$ , $C$ , and $Q_{p}$ . Worked computation of the first several BCH terms with explicit rational coefficients. SLALG §IV anchor; Bourbaki GAL Ch. II §6 anchor. Originator-prose section citing Campbell 1897/1898, Baker 1905, Hausdorff 1906. Three- tier; Master tier includes the Dynkin formula for the BCH coefficients. ~2200 words. The single highest-priority unit in this plan.
07.06.12 Cohomology of Lie algebras $H^{*} (g, M)$ . Chevalley-Eilenberg complex $C^n(\mathfrak{g}, M) = \mathrm{Hom} (\Lambda^n \mathfrak{g}, M)$, the differential, low-degree interpretations ( $H^{0} = M^{g}$ invariants; $H^{1} =$ outer derivations modulo inner; $H^{2} =$ central extensions classifying class). Whitehead's lemmas $H^{1} = H^{2} = 0$ for $g$ finite-dim semisimple, $M$ finite-dim. Cohomological proof of Weyl's complete reducibility theorem. SLALG §V anchor; Chevalley-Eilenberg 1948 originator citation; Weibel An Introduction to Homological Algebra Ch. 7 cross-reference. Three-tier; Master tier handles the derived-functor interpretation $H^n(\mathfrak{g}, M) = \mathrm{Ext}^n_{U(\mathfrak{g})}(k, M)$. ~2200 words. Pulls weight for the cohomology / homological algebra strand and is a prereq for any future BRST / gauge-cohomology material.
07.06.13 Hochschild-Serre spectral sequence for a Lie-algebra ideal. For $h ◃ g$ , the filtration on the Chevalley-Eilenberg complex by $h$ -degree; statement $E_2^{p, q} = H^p(\mathfrak{g}/\mathfrak{h}, H^q(\mathfrak{h}, M)) \Rightarrow H^{p+q}(\mathfrak{g}, M)$; the five-term exact sequence in low degrees. Worked example: the Heisenberg Lie algebra cohomology computed by the spectral sequence with the centre as ideal. SLALG §VI anchor; Hochschild-Serre 1953 originator citation; cross-link to the topological 03.13.02 leray-serre (the parallel statement for fibrations). Three-tier; Master tier includes the derivation of the inflation-restriction sequence. ~1800 words. Closes the Lie-side of the spectral-sequence strand.
03.03.04 Formal group law. Definition over a commutative ring $R$ : one-dim case $F (X, Y) \in R [[X, Y]]$ with axioms; higher-dim vector-valued case. The associated formal Lie algebra $L (F) = (R^{n}, [\cdot, \cdot]_{F})$ from the quadratic part of $F$ . The Lie functor $F \mapsto L (F)$ is an equivalence in characteristic 0 (Cartier; Serre's exposition). Statement-only of the characteristic- $p$ failure (height, $p$ -typical forms). SLALG Part II §§1–5 anchor; Bochner 1946 + Lazard 1955 originator citations; Hazewinkel 1978 secondary anchor. Three-tier; Master tier sketches Lubin-Tate formal groups and their role in local class field theory. ~2000 words. Foundational for 03.03.05 and a long-term hook for future algebraic-topology material (complex cobordism / formal-group law of $MU$ ).

Priority 2 — completes the $p$ -adic Lie group side:

03.03.05 $p$ -adic Lie group and the $p$ -adic exponential. Brief $Q_{p}$ / $p$ -adic absolute value paragraph; $p$ -adic analytic manifold; group object in that category. The $p$ -adic exponential $exp : g \to G$ as a power-series map, radius of convergence $r = p^{- 1/ (p - 1)}$ (the standard bound). The local group structure on $exp (B_{r} (0))$ given by the Campbell-Hausdorff series (depends on Priority 1 unit 2). SLALG Part II §§6–7 anchor; Lazard 1965 originator citation; cross-reference Dixon-du Sautoy-Mann-Segal Analytic Pro- $p$ Groups as the modern follow-up. Three-tier. ~2000 words.
03.03.06 Lazard's equivalence: $p$ -adic Lie groups and $Q_{p}$ -Lie algebras. Statement of the equivalence of categories on the simply-connected side; sketch of the proof using Priority 2 unit 6's $p$ -adic exponential and the Campbell-Hausdorff formula. SLALG Part II §8 anchor; Lazard 1965 anchor. The $p$ -adic analogue of Lie's three theorems. Intermediate + Master tiers. ~1500 words.

Priority 3 — closes Part I solvable/nilpotent prereqs:

07.06.14 Solvable and nilpotent Lie algebras; Engel's theorem and Lie's theorem. Standard content: $g^{(i)}$ derived series, $g_{i}$ lower central series, nilpotent / solvable definitions, Engel ( $g$ nilpotent iff every ad is nilpotent), Lie (over $C$ , solvable acts in upper-triangular form). SLALG §II anchor; Humphreys Lie Algebras Ch. I anchor. Required prereq for the cohomological vanishing arguments in 07.06.12. Intermediate + Master tiers. ~1500 words. Medium priority — load-bearing prereq but currently sufficient at the implicit-reference level in 07.04.01.

Priority 4 — Master-tier deepenings (not strictly required for FT equivalence ≥95% but close the residual gap):

§Master deepening of 07.06.02 universal-enveloping-algebra: PBW theorem with the filtration / associated-graded proof. ~600 words added. SLALG §I anchor.
§Master deepening of 03.03.01 lie-group: Lie's three theorems, rigorously stated. Currently sketched at intermediate; a 700-word master section codifies the functor $G \mapsto Lie (G)$ , the integration theorem (simply connected Lie group from a Lie algebra), and the local-isomorphism theorem. SLALG Part II §1–§2 anchor.
§Master deepening of 07.06.10 (or new pointer unit 07.06.15): Hall basis and Hall sets for the free Lie algebra. ~500 words.
§Master deepening of 03.03.04 formal-group-law (new): Lubin-Tate formal groups, height of a formal group, and pointer to the Cartier-Dieudonné classification. ~700 words. Hazewinkel Chs. 18–25 reference.

Priority 5 — exercise / survey follow-ups (optional):

Exercise pack 07.06.E2 lie-algebra-cohomology-exercises. Worked computations: $H^{*} (sl_{2}, C)$ , $H^{*}$ of the Heisenberg, $H^{*}$ of the abelian Lie algebra (= exterior algebra). ~5 worked exercises.
Pointer paragraph in 07.06.13 cross-linking to BRST / gauge cohomology (when/if a physics/gauge-theory/ BRST unit is shipped).

§4 Implementation sketch (P3 → P4)

Realistic production estimate (mirroring earlier Brown-Higgins-Sivera, Fulton-Harris, and Lawson-Michelsohn batches):

Priority 1 (5 new units, technical and dense): ~4 hours each = ~20 hours. 07.06.10 free-lie-algebra, 07.06.11 campbell-hausdorff, 07.06.12 lie-algebra-cohomology, 07.06.13 hochschild-serre-ss, 03.03.04 formal-group-law. The Campbell-Hausdorff and Chevalley- Eilenberg units skew long because they need both clean statements and worked computations.
Priority 2 (2 new units): ~3.5 hours each = ~7 hours. 03.03.05 p-adic-lie-group, 03.03.06 lazard-equivalence.
Priority 3 (1 new unit): ~3 hours = ~3 hours. 07.06.14 solvable-nilpotent-lie-algebra.
Priority 4 (4 master deepenings): ~45 min each = ~3 hours.
Priority 5 (exercise pack + pointers): ~3 hours.

Total: ~36–38 hours of focused production for full FT-equivalence coverage of SLALG. Fits a focused 5–6 day window. Priority 1 alone (~20 hours, 2.5–3 days) closes ~70% of the SLALG gap and raises effective coverage from ~15% to ~70%. Priority 1+2 (~27 hours, 3.5 days) raises to ~88%. Priority 1+2+3 raises to ~92%. Priority 1+2+3+4 + the exercise pack closes to ≥95%.

Originator-prose targets. Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, originator-prose treatment with primary-source citations should appear in:

Lie 1880s — Sophus Lie, Theorie der Transformationsgruppen I–III (with F. Engel, Teubner 1888–1893). Originates the entire Lie-group / Lie- algebra programme. Cite in 03.03.01 lie-group Master section (and in the 03.04.01 lie-algebra Master section if not already present).
Campbell 1897/1898, Baker 1905, Hausdorff 1906 — see §1 above for full citations. The three originators of the BCH formula. Cite in 07.06.11 campbell-hausdorff Master section.
Chevalley-Eilenberg 1948 — C. Chevalley, S. Eilenberg, "Cohomology theory of Lie groups and Lie algebras," Trans. AMS 63 (1948), 85–124. Originates Lie-algebra cohomology. Cite in 07.06.12 lie-algebra-cohomology Master section.
Hochschild-Serre 1953 — G. Hochschild, J.-P. Serre, "Cohomology of Lie algebras," Ann. of Math. (2) 57 (1953), 591–603. Cite in 07.06.13 hochschild-serre-ss Master section.
Lazard 1955/1965 — M. Lazard, "Sur les groupes de Lie formels à un paramètre," Bull. Soc. Math. France 83 (1955), 251–274; "Groupes analytiques $p$ -adiques," Publ. Math. IHES 26 (1965), 5–219. Cite in 03.03.04 formal-group-law, 03.03.05 p-adic-lie-group, 03.03.06 lazard-equivalence Master sections.
Bourbaki 1972 — Groupes et algèbres de Lie, Ch. II–III (Hermann 1972). Encyclopedic reference; cite as Master-tier anchor in every new unit in this audit.

Notation crosswalk. SLALG uses Bourbaki notation throughout: $U (g)$ for the universal enveloping algebra (already Codex notation); $Lib (X)$ for the free Lie algebra on $X$ (Codex default $L (X)$ is acceptable and is what 07.06.10 should adopt); $H^{*} (g, M)$ for Lie-algebra cohomology (use as-is); $L (G)$ for the Lie algebra of a Lie group $G$ (Codex tends toward $g$ in lowercase fraktur for an abstract Lie algebra and $Lie (G)$ or $g$ for the Lie algebra of $G$ — keep current Codex convention). For $p$ -adic objects: write $Q_{p}$ for the $p$ -adic rationals, $Z_{p}$ for the $p$ -adic integers, $∣ x ∣_{p}$ for the $p$ -adic absolute value, $ord_{p}$ for the $p$ -adic valuation. Record in a §Notation paragraph of 07.06.11 (BCH series ring) and 03.03.05 ( $p$ -adic Lie group).

Cross-strand weaving (Pass-W). New units should link laterally to:

03-modern-geometry/03-lie/ — the existing Lie-group machinery is the global-Lie-group complement to SLALG's local / formal / $p$ -adic emphasis. 07.06.11 campbell-hausdorff directly upgrades 03.03.01 lie-group intermediate-tier exponential discussion.
03-modern-geometry/13-spectral-sequences/ — 07.06.13 hochschild-serre-ss is the algebraic Lie-algebra parallel to 03.13.02 leray-serre (topological). Add a §See-also paragraph in both directions.
07-representation-theory/02-character/, 07-representation-theory/ 04-classification/ — 07.06.12 lie-algebra-cohomology's Whitehead-lemmas section gives the cohomological proof of Weyl's complete reducibility, which is currently a black box in 07.02.01 maschke-theorem (finite-group case) and 07.04.01 cartan-weyl-classification.
A future algebraic-topology/complex-cobordism/ strand (Ravenel 1.06 audit punch-list candidate) — 03.03.04 formal-group-law is the prereq for the formal-group law of $MU$ and chromatic homotopy theory. Add a forward pointer.
A future number-theory/local-fields/ strand (Serre 3.14 A Course in Arithmetic audit candidate) — 03.03.05 p-adic-lie-group shares the $Q_{p}$ prereq paragraph.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem in SLALG (full P1 audit; deferred until a local PDF is acquired). Coverage estimates in §2 are based on the public TOC structure of LNM 1500 plus spot-checks of Codex units. Promote to full P1 audit when PDF is local.
Serre 3.12 Complex Semisimple Lie Algebras — separate book, separate audit. SCSLA covers the classification arc (Cartan subalgebras, root systems, Cartan matrix, Serre relations); SLALG covers Campbell-Hausdorff / cohomology / formal-group / $p$ -adic. The two books are disjoint in content and pedagogically complementary. The 3.12 audit should pick up Cartan-Weyl classification deepenings that 07.04.01 partially covers; this 3.13 audit instead picks up the homological / $p$ -adic gap that no other Codex unit fills.
Serre 3.14 A Course in Arithmetic — separate book, separate audit. Covers Ostrowski, $Q_{p}$ , quadratic forms, modular forms. The $Q_{p}$ paragraph in Priority 2 units 6 and 7 is intentionally thin and is not a substitute for a proper number-theory strand.
Serre 3.15 Linear Representations of Finite Groups — separate book, separate audit. Disjoint content from SLALG.
Modern pro- $p$ group theory (Dixon-du Sautoy-Mann-Segal Analytic Pro- $p$ Groups; uniform pro- $p$ groups; Lazard's " $p$ -saturation" classification of analytic pro- $p$ groups). SLALG predates this material by 30 years and only sketches it. Pointer reference in Priority 2 unit 6 Master section; full coverage deferred.
Algebraic groups / group schemes (Borel, Springer, Milne Algebraic Groups). SLALG is not an algebraic-groups text; deferred to a future algebraic-groups audit (no current Fast Track row).
Chromatic homotopy theory (Lubin-Tate spectra, Morava $E$ - and $K$ -theories, the Adams-Novikov spectral sequence). The formal-group prereq is what 03.03.04 ships; chromatic content is deferred to the Ravenel 1.06 audit + production pass.
Lie-algebroid / Lie-groupoid generalisations (Mackenzie). Adjacent to SLALG only at the conceptual level; deferred to a future Mackenzie audit (Codex has local copies of both Mackenzie texts in reference/textbooks-extra/).

§6 Acceptance criteria for FT equivalence (SLALG)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4, the book is at equivalence-coverage when:

The 5 Priority-1 new units have shipped (07.06.10 free-lie-algebra, 07.06.11 campbell-hausdorff, 07.06.12 lie-algebra-cohomology, 07.06.13 hochschild-serre-ss, 03.03.04 formal-group-law). These close ~70% of the SLALG gap and the entire Part I cohomological half.
≥95% of SLALG's named theorems and major worked computations map to a Codex unit (currently ~15%; after Priority-1 this rises to ~70%; after Priority-1+2 to ~88%; after Priority-1+2+3 to ~92%; full ≥95% requires Priority-4 master deepenings + the exercise pack).
≥90% of SLALG's worked computations have a direct unit or are cross-referenced from a unit that covers them. The Heisenberg / $sl_{2}$ Lie-algebra cohomology computations and the explicit BCH-series terms should be present as worked examples in 07.06.11, 07.06.12, and the 07.06.E2 exercise pack.
Notation crosswalk recorded (see §4): free-Lie-algebra notation, BCH ring, $p$ -adic absolute value, $H^{*} (g, M)$ .
Pass-W weaving connects the new units to 03.13 spectral-sequences/, to the future Ravenel chromatic strand, and to 03-lie/ Lie-group units.
Originator-prose sections (Lie 1880s, Campbell 1897, Baker 1905, Hausdorff 1906, Chevalley-Eilenberg 1948, Hochschild-Serre 1953, Lazard 1955/1965, Bourbaki 1972) appear in the relevant Master tiers.

Equivalence verification protocol (per FASTTRACK_EQUIVALENCE_PLAN.md §4):

4.1 Book-as-input self-check. Sample 5 random theorem statements from SLALG (suggested loci: BCH universality in §IV, Whitehead's first lemma in §V, Hochschild-Serre five-term sequence in §VI, formal-Lie equivalence in Part II §3, $p$ -adic exponential convergence in Part II §6); for each, identify the Codex unit that proves it.
4.2 Exercise reproducibility. Sample 5 random exercises from SLALG; for each, identify the Codex unit(s) sufficient to solve it. SLALG exercises (where present in the LNM 1500 reprint) tend to be brief technical lemmas rather than open-ended; sampling protocol is straightforward.
4.3 Notation comprehension. Sample 5 fragments of SLALG prose (with attention to the Bourbaki conventions and the $p$ -adic exponential's convergence-radius bookkeeping); paraphrase each into Codex notation per the §4 crosswalk.

§7 Sourcing

Local PDF status. Not present in reference/textbooks-extra/ (verified — only the cover image reference/fast-track/images/Serre-Lie-Algebras-Lie-Groups-712x1024__dd9fad6ce3.jpg is local). SLALG is a commercial Springer LNM (originally a Benjamin 1965 hardback; the 1992 Springer LNM 1500 reprint is the canonical available form) and is not author-hosted.
Commercial source. Springer Lecture Notes in Mathematics 1500 (1992 reprint of the 1965 W. A. Benjamin original). Available via link.springer.com/book/10.1007/b13134 and equivalent. ISBN 3-540-55008-9 (softcover). The 1965 Benjamin original is long out of print but appears in university libraries.
Library-mirror sources. Acquire via institutional access (university library Springer subscription) or interlibrary loan. For the local copy, target placement is reference/textbooks-extra/Serre-LieAlgebrasAndLieGroups.pdf to mirror the pattern of other commercial-source FT texts (e.g. Sternberg-LieAlgebras.pdf already in reference/fasttrack-texts/03-modern-geometry/).
Companion / supplementary materials.
- N. Bourbaki, Groupes et algèbres de Lie, Chs. II–III (Hermann 1972; Springer English translation 1989). Encyclopedic companion.
- G. Hochschild, The Structure of Lie Groups (Holden-Day 1965; Dover reprint 2015). Same-year complementary treatment of the cohomology side.
- M. Hazewinkel, Formal Groups and Applications (Academic Press 1978; AMS Chelsea 2012). The canonical extension of SLALG Part II §§1–5.
- M. Lazard, "Groupes analytiques $p$ -adiques," Publ. Math. IHES 26 (1965). Open-access via numdam.org — http://www.numdam.org/item/PMIHES_1965__26__5_0/. The definitive $p$ -adic Lie-group reference; freely available.
- J. D. Dixon, M. P. F. du Sautoy, A. Mann, D. Segal, Analytic Pro- $p$ Groups, 2nd ed. (Cambridge Studies in Advanced Mathematics 61, CUP 1999). The modern pro- $p$ extension.
Open-access alternatives covering portions of SLALG:
- Bourbaki Ch. II §6 (Campbell-Hausdorff section) circulates as PDF via standard mirrors. Covers SLALG §IV at greater length.
- Lazard 1965 (numdam) covers SLALG Part II §§6–8 at much greater length.
- Weibel, An Introduction to Homological Algebra Ch. 7 (CUP 1994) covers SLALG §§V–VI in textbook form. Open Springer eBook on most institutional subscriptions.
Reduced-audit flag. This plan was produced without a local SLALG PDF. Coverage estimates in §2 should be re-verified once a PDF is acquired, particularly for Part II §§7–8 where the convergence estimates and the exact form of Lazard's equivalence statement are load-bearing and where Serre's notation differs subtly from Lazard's. Promote to full P1 audit when PDF is local. Numdam Lazard 1965 may substitute for Part II coverage if a SLALG PDF remains unavailable; in that case rewrite Priority 2 units 6 and 7 with Lazard 1965 as primary anchor and SLALG as secondary.

Serre — *Lie Algebras and Lie Groups* (Fast Track 3.13) — Audit + Gap Plan