Serre — Complex Semisimple Lie Algebras (Fast Track 3.12) — Audit + Gap Plan

Book: Jean-Pierre Serre, Complex Semisimple Lie Algebras (Springer Monographs in Mathematics, English translation by G. A. Jones 1987 of the French original Algèbres de Lie semi-simples complexes, W. A. Benjamin 1966, ≈ 75 pp.). Commercial title (BUY in docs/catalogs/FASTTRACK_BOOKLIST.md row 3.12). ISBN 3-540-67827-1 (softcover reprint 2001).

Fast Track entry: 3.12. The canonical shortest monograph deriving the full classification of complex semisimple Lie algebras — Serre's slim Bourbaki-style monograph that proves in 75 pages what Humphreys takes 170 pages and Bourbaki Ch. VI–VIII takes 600 pages to do. Distinguished from the Fast Track's other Lie / rep-theory slots (3.10 Hall, 3.11 Fulton-Harris, 3.13 Lie Algebras and Lie Groups sibling, 3.15 Linear Representations of Finite Groups) by the Serre-relations generators-and-relations presentation of semisimple Lie algebras — a presentation distinct from Humphreys' Chevalley-basis approach and absent from Fulton-Harris.

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of deepenings + a small set of new units so that Complex Semisimple Lie Algebras (CSLA hereafter) 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/ (only the cover image at reference/fast-track/images/Serre-Lie-Algebras-Lie-Groups-712x1024__dd9fad6ce3.jpg, which the Fast Track source page reuses across multiple Serre slots). The 1966 French original is not author-hosted; Springer 1987/2001 reprint is commercial. This audit works from (a) the well-documented public TOC of CSLA, (b) the Codex's 26 shipped 07-representation-theory/ units (which already cite Serre 1966 by name in 07.04.01 and 07.06.05), and (c) the canonical secondary literature (Humphreys, Bourbaki, Carter, Knapp). 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 CSLA is for

CSLA is the crystalline minimal monograph on the classification of complex semisimple Lie algebras. Where Humphreys' Introduction to Lie Algebras and Representation Theory (1972) is the standard graduate textbook treatment in 170 pages, where Bourbaki Groupes et algèbres de Lie Ch. IV–VIII is the encyclopaedic 600-page treatise, where Fulton-Harris is the example-first 550-page pedagogical sweep, and where Knapp Lie Groups Beyond an Introduction is the real-form structure-theoretic monograph — CSLA is the shortest rigorous self-contained proof of the classification. Serre extracted his 1965 Algebra Collège de France lectures into 75 pages by stripping everything to its functorial spine. The book is famously the gold standard of mathematical economy.

Distinctive contributions, in the order CSLA develops them:

Nilpotent and solvable Lie algebras (CSLA Ch. I). Engel's theorem, Lie's theorem (over $C$ ), the radical $rad (g)$ . Compact 8-page treatment; serves only to define "semisimple" = " $rad = 0$ " in passing.
Semisimple Lie algebras and the Killing form (CSLA Ch. II). Cartan's criterion for semisimplicity ( $κ$ non-degenerate iff $g$ semisimple), Weyl's complete-reducibility theorem (via the Casimir element), Levi-Malcev decomposition stated. This chapter establishes that semisimple Lie algebras are exactly the ones with a non-degenerate Killing form.
Cartan subalgebras (CSLA Ch. III). Definition (a nilpotent self-normalising subalgebra), existence and conjugacy via regular elements. Serre's proof of conjugacy uses the connectedness of the open dense set of regular elements in $g$ — a Lie-theoretic Zariski-density argument that is much shorter than Humphreys' detailed inductive proof. This is one of CSLA's hallmark economies.
Root-space decomposition (CSLA Ch. IV). For semisimple $g$ with Cartan subalgebra $h$ , $\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in R} \mathfrak{g}\alpha $w h er e$ R \subset \mathfrak{h}^* \setminus {0}$ is the root system. The standard sequence: $\mathfrak{g}\alpha$ is 1-dimensional, $α \in R \Rightarrow - α \in R$ , $[g_{α}, g_{- α}] = C H_{α}$ , the copy of $sl_{2}$ inside $g$ for each root. $sl_{2}$ -theory used as the indispensable computational engine — every root-string property is derived from $sl_{2}$ representation theory.
Root systems abstractly (CSLA Ch. V). Axiomatic definition of a root system in a Euclidean space (independent of Lie algebras): $R$ finite, spans $V$ , $α \in R \Rightarrow - α \in R$ , reflection $s_{α}$ preserves $R$ , $\langle \alpha, \beta^\vee\rangle \in \mathbb{Z} $. W ey l g r o u p$ W$ generated by reflections. Simple roots, positive roots, fundamental chamber, Cartan matrix, Coxeter element, Dynkin diagrams. Classification of irreducible root systems: $A_{n}, B_{n}, C_{n}, D_{n}, E_{6}, E_{7}, E_{8}, F_{4}, G_{2}$ — the entire classification proved by the combinatorial Dynkin argument in ~12 pages.
Structure theorem via Serre relations (CSLA Ch. VI). THIS IS THE BOOK'S SIGNATURE CONTRIBUTION. Given a Cartan matrix $(a_{ij})$ , the Lie algebra $g (A)$ defined by generators $E_{i}, F_{i}, H_{i}$ ( $1 \leq i \leq ℓ$ ) and the Serre relations:
- $[H_{i}, H_{j}] = 0$
- $[H_{i}, E_{j}] = a_{j i} E_{j}$ , $[H_{i}, F_{j}] = - a_{j i} F_{j}$
- $[E_{i}, F_{j}] = δ_{ij} H_{i}$
- $(ad E_{i})^{1 - a_{j i}} E_{j} = 0$ for $i \neq = j$
- $(ad F_{i})^{1 - a_{j i}} F_{j} = 0$ for $i \neq = j$
is finite-dimensional and is the simple Lie algebra with that Cartan matrix. This generators-and-relations presentation is absent from Fulton-Harris and from most pedagogical treatments; Humphreys gives it as a final theorem after the Chevalley-basis construction has been done explicitly. Serre puts it at the centre, making it the definition of the simple Lie algebra once the root system is classified. The reader leaves CSLA knowing that " $E_{8}$ " is the algebra defined by these specific 248 generators-and-relations.
Linear representations of semisimple Lie algebras (CSLA Ch. VII). The theorem of the highest weight: irreducible finite-dimensional representations are in bijection with dominant integral weights. Verma modules constructed (briefly) and their unique irreducible quotient $L (λ)$ . Weyl character formula stated; proof referred to Bourbaki — Serre does not reproduce the full Weyl proof.
Compactness and the Weyl unitarian trick (CSLA Ch. VIII, the shortest chapter). Each complex semisimple Lie algebra has a compact real form $u$ , and representations of the complex algebra correspond to unitary representations of the compact real form. Bridge to the compact-Lie-group programme without developing it.

CSLA is not a first introduction to Lie algebras (no $sl_{2}$ worked example in the Fulton-Harris sense; no matrix-explicit computations in the Hall sense). It is not a treatment of real forms (Knapp is the sequel). It is not a treatment of Kac-Moody or infinite-dimensional Lie algebras (Kac Infinite-Dimensional Lie Algebras is the modern extension of the Serre-relations programme to the affine and indefinite cases). CSLA is the densest possible statement of the finite-dimensional classification — and the only short text that puts the Serre-relations presentation at the centre.

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

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory (Springer GTM 9, 1972, corrected reprint 1978). The standard graduate text. ~170 pp. Uses the Chevalley-basis approach explicitly; Serre relations stated as Theorem 18.3. Direct competitor to CSLA at longer page-length.
N. Bourbaki, Groupes et algèbres de Lie Ch. IV–VIII (Hermann 1968–1975, Springer reprint 2002–2008). The encyclopaedic treatment; Serre's 1966 monograph was written in part to consolidate the Bourbaki programme into a teachable size. Bourbaki Ch. VIII §4 is the Serre-relations theorem in full generality. Cited by Serre as the fallback for the Weyl character proof he omits.
A. W. Knapp, Lie Groups Beyond an Introduction (Birkhäuser 2nd ed. 2002). The real-form-and-structure-theoretic sequel. CSLA is its $C$ -only prereq. Knapp uses the Serre-relations presentation in Ch. II §6.
R. W. Carter, Lie Algebras of Finite and Affine Type (Cambridge Studies in Advanced Mathematics 96, 2005). The modern unified treatment putting the Serre-relations presentation at the centre and extending it to affine Kac-Moody. Carter §7 reproduces Serre Ch. VI in expanded form and shows how the same generators-and-relations philosophy yields all of affine Lie theory. Direct intellectual descendant of CSLA.

§2 Coverage table (Codex vs CSLA)

Cross-referenced against the current 26 shipped units of content/07-representation-theory/. ✓ = covered, △ = partial / different framing, ✗ = not covered.

CSLA topic	Codex unit(s)	Status	Note
Lie algebra; bracket; structure constants	`03.04.01` lie-algebra	✓	Foundational unit in 03 differential-forms strand.
Subalgebra, ideal, derived series, lower central series	`03.04.01` (and `07.06.01`)	△	Touched but no standalone solvable / nilpotent unit. Gap (low).
Engel's theorem	—	✗	Gap (medium). Standard prereq for Cartan's criterion. Candidate: `07.06.14` engel-theorem.
Lie's theorem (solvable Lie algebra over $C$ )	—	✗	Gap (medium). Standard prereq. Candidate: `07.06.15` lie-theorem.
Solvable radical $rad (g)$ ; semisimple = $rad = 0$	△ (mentioned in `07.04.01`)	△	Gap (low). Folded into prose; no standalone unit.
Levi-Malcev decomposition	—	✗	Gap (low). CSLA states without proof; Codex omits entirely. Master deepening candidate.
Killing form $κ (X, Y) = tr (ad X \circ ad Y)$	— (referenced in `07.06.03`, `07.06.04`, `07.04.01`, `07.03.01`)	△	Gap (HIGH). Killing form is referenced in 4+ units but has no standalone unit. CSLA Ch. II is built on it. Candidate: `07.06.10` killing-form.
Cartan's criterion for semisimplicity ( $κ$ non-degenerate ⇔ ss)	—	✗	Gap (HIGH). Cornerstone result; no Codex unit. Candidate: §Master section of `07.06.10` or new unit `07.06.16` cartan-criterion-semisimplicity.
Weyl complete-reducibility theorem	—	△	Gap (medium). Referenced inside `07.03.01` highest-weight and `07.06.06` verma-module but no standalone proof unit. CSLA Ch. II §6 anchor.
Casimir element	—	△	Gap (medium). Same as Fulton-Harris audit punch-list item 7 (`07.06.10` slot — note: see notation collision below).
Cartan subalgebra (CSA): definition, existence, conjugacy	— (mentioned in `07.06.03`, `07.04.01`)	△	Gap (HIGH). CSLA Ch. III is the whole CSA story; Codex has no standalone CSA unit. Candidate: `07.06.17` cartan-subalgebra.
Regular elements; conjugacy of CSAs via regular elements	—	✗	Gap (HIGH). Serre's distinctive proof — Zariski-density of regular elements. No Codex coverage. Master section of `07.06.17`.
Root-space decomposition $g = h \oplus ⨁ g_{α}$	△ (stated in `07.06.03` and `07.04.01`)	△	Gap (medium-high). Stated; not derived as its own unit. Candidate: §Master deepening of `07.06.03` OR new unit `07.06.18` root-space-decomposition.
Root system (abstract, in Euclidean space)	`07.06.03` root-system	✓	Direct correspondent.
Weyl group	`07.06.04` weyl-group	✓
Simple roots, positive roots, base, fundamental chamber	△ (touched in `07.06.03`, `07.06.04`)	△	Gap (low-medium). No standalone "simple system / base" unit. Master deepening of `07.06.03`.
Cartan matrix	△ (mentioned in `07.06.05`, `07.04.01`)	△	Gap (medium). No standalone unit. Candidate: §Intermediate/Master deepening of `07.06.05` or new unit `07.06.19` cartan-matrix.
Dynkin diagram; ADE classification	`07.06.05` dynkin-diagram	✓
Cartan-Weyl classification	`07.04.01` cartan-weyl-classification	✓	Lists all of $A_{n}, B_{n}, C_{n}, D_{n}, E_{6}, E_{7}, E_{8}, F_{4}, G_{2}$ . Already cites Serre 1966 by name.
Serre relations and Serre's theorem (presentation by generators-and-relations)	△ (mentioned in passing in `07.04.01`, `07.06.05`)	△	Gap (VERY HIGH — this is CSLA's signature contribution). Currently stated as a sentence in `07.04.01`; no standalone unit. Candidate: `07.06.20` serre-relations.
Chevalley basis	—	✗	Gap (medium). Not in CSLA proper (CSLA uses Serre relations instead of Chevalley basis) but is the Humphreys-side counterpart. Master deepening of `07.06.20`.
Theorem of the highest weight	`07.03.01` highest-weight-representation	✓
Verma module; irreducible quotient $L (λ)$	`07.06.06` verma-module	✓
Weyl character formula	`07.06.07` weyl-character-formula	✓
Compact real form; Weyl's unitarian trick	△ (touched in `07.07.01`)	△	Gap (low-medium). Compact-real-form correspondence not its own unit. Same as Fulton-Harris item 9.

Aggregate coverage estimate (REDUCED audit basis).

Ch. I (nilpotent/solvable): ~10% — Engel, Lie missing.
Ch. II (Killing form, semisimplicity criterion, Weyl reducibility): ~15% — Killing form referenced everywhere but no standalone unit, no Cartan's criterion unit, Weyl complete reducibility folded inside other units.
Ch. III (Cartan subalgebras, conjugacy): ~10% — CSA mentioned in passing only. Conjugacy proof entirely absent.
Ch. IV (root-space decomposition): ~40% — stated in 07.06.03 and 07.04.01 but not derived as its own programme.
Ch. V (root systems, Weyl group, Dynkin diagrams, classification): ~90% — 07.06.03, 07.06.04, 07.06.05, 07.04.01 cover almost everything; minor gaps in simple-system / Cartan-matrix detail.
Ch. VI (Serre relations, structure theorem): ~10% — referenced by name; no standalone unit. The signature CSLA content is underrepresented in the Codex.
Ch. VII (highest-weight reps, Weyl character formula): ~85% — Codex has 07.03.01, 07.06.06, 07.06.07, 07.06.08 all shipped.
Ch. VIII (compact real forms, unitarian trick): ~30% — touched only.

Overall: ~55% of CSLA covered. Higher than the 0% Brown-Higgins-Sivera baseline because the Codex's 07-representation-theory/ already covers the output of CSLA's classification programme (root systems, Weyl group, Dynkin diagrams, highest-weight reps). Lower than the 65% Fulton-Harris coverage because CSLA's foundational chapters (Killing form, Cartan subalgebra conjugacy, Serre relations) — the proofs that make the classification a theorem rather than a list — are notably underrepresented. The gap pattern is infrastructural (foundational proofs missing) rather than pedagogical (FH's gap is missing worked examples; CSLA's gap is missing supporting theorems).

§3 Gap punch-list (priority-ordered)

Priority 1 — high-leverage, captures CSLA's foundational and signature content:

07.06.10 Killing form. Standalone unit (~1500 words). Definition $κ (X, Y) = tr (ad X \circ ad Y)$ ; associativity / $ad$ -invariance; computation for $sl_{2}, sl_{3}, so_{n}, sp_{n}$ ; Cartan's criterion for semisimplicity: $g$ semisimple iff $κ$ is non-degenerate. CSLA Ch. II anchor; Humphreys §5 anchor; Bourbaki Ch. I §6 anchor. Intermediate + Master tiers (Master: full proof of Cartan's criterion via Cartan's criterion for solvability). Foundational — referenced by 07.06.03, 07.06.04, 07.04.01, 07.03.01 without a home unit. Note: also resolves the Fulton-Harris audit's Casimir-element gap if folded together — but keep Casimir as its own unit (07.06.21 proposed below) to avoid over-loading.
07.06.17 Cartan subalgebra. Definition (nilpotent self-normalising subalgebra; equivalently, a maximal toral subalgebra for semisimple $g$ ); existence; conjugacy via regular elements (Serre's distinctive Zariski-density proof); the abstract Cartan subalgebra. CSLA Ch. III anchor; Humphreys §15 anchor (different proof); Carter §7.1 anchor. Three-tier; Master tier reproduces Serre's proof. Foundational — Cartan subalgebras are referenced in 07.06.03 and 07.04.01 without a home. ~2000 words.
07.06.20 Serre relations and Serre's theorem. The signature CSLA unit. Given a Cartan matrix $A = (a_{ij})$ of an irreducible root system, define $g (A)$ by generators $E_{i}, F_{i}, H_{i}$ and the Serre relations (listed in §1 item 6 above). Serre's Theorem. $g (A)$ is finite-dimensional, simple, has Cartan matrix $A$ , and every finite-dim simple complex Lie algebra arises this way. CSLA Ch. VI anchor; Humphreys §18 anchor; Carter §7 anchor; Kac Infinite-Dim Lie Algebras §1.3 anchor (the same presentation extended to Kac-Moody). Three-tier; Master tier includes the existence-and-uniqueness proof. ~2500 words. This is the Codex's single largest intellectual gap on the Lie-algebra side — the Serre presentation is cited in 07.04.01 and 07.06.05 but never written out. Originator-prose section citing Serre 1966 directly.
07.06.18 Root-space decomposition. Standalone unit reorganising material currently scattered across 07.06.03, 07.06.04, 07.04.01. For semisimple $g$ with CSA $h$ : $\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in R} \mathfrak{g}\alpha $;$ \dim \mathfrak{g}\alpha = 1$; $[\mathfrak{g}\alpha, \mathfrak{g}\beta] \subseteq \mathfrak{g}_{\alpha
- \beta} $; t h eco p y o f$ \mathfrak{sl}_2$ for each root; root-string formulae. CSLA Ch. IV anchor; Humphreys §8 anchor. Three-tier; Master tier includes full proofs. ~1800 words. High — currently stated as result-without-derivation across multiple units.

Priority 2 — fills medium-priority CSLA content:

07.06.19 Cartan matrix. Short standalone unit (~1200 words). Definition $a_{ij} = ⟨ α_{i}, α_{j}^{\lor} ⟩$ ; integer entries; diagonal = 2, off-diagonal $\leq 0$ ; rank, determinant, positive-definiteness; relation to Dynkin diagram. CSLA Ch. V anchor. Intermediate + Master. Medium — currently embedded inside 07.06.05 and 07.04.01.
07.06.14 Engel's theorem + 07.06.15 Lie's theorem. Two short units (~1000 words each), or one combined unit. Engel: a Lie algebra of nilpotent endomorphisms acts with a common eigenvector; equivalent characterisation of nilpotent Lie algebras. Lie: for a solvable Lie algebra over $C$ acting on a finite-dim vector space, there is a common eigenvector (equivalently, upper- triangularisation in some basis). CSLA Ch. I anchor; Humphreys §3 anchor. Medium — these are prereqs for the Killing-form unit (Cartan's criterion uses Lie's theorem).
07.06.16 Cartan's criterion for solvability and semisimplicity. Standalone unit (~1500 words). Combines the two Cartan criteria (for solvability: $\kappa(\mathfrak{g}, [\mathfrak{g}, \mathfrak{g}]) = 0 $\Leftrightarrow so l v ab l e; f or se mi s im pl i c i t y :$ \kappa$ non-degenerate ⇔ semisimple). CSLA Ch. II §5 anchor; Humphreys §5 anchor. Master tier includes full proofs. Medium-high — central theorem of CSLA Ch. II. Could be folded into 07.06.10 as a Master section; recommend separate unit for clarity.
07.06.22 Weyl complete-reducibility theorem. Statement and Casimir-element proof. CSLA Ch. II §6 anchor; Humphreys §6 anchor. Currently referenced inside 07.06.06 verma-module and elsewhere without a home. ~1200 words.

Priority 3 — Master-tier deepenings (not strictly required for FT equivalence):

07.06.21 Casimir element. (Shared with Fulton-Harris audit item 7.) Short standalone unit. For semisimple $g$ with Killing form $κ$ , the Casimir $\Omega = \sum X_i X^i \in U(\mathfrak{g})$; centrality; action as scalar $⟨ λ, λ + 2 ρ ⟩$ on $L (λ)$ . CSLA Ch. II + Humphreys §6 + Fulton-Harris §14 anchor. ~1200 words. Already enumerated in Fulton-Harris audit; assigning the same unit slot here.
§Master deepening of 07.06.20 (Serre relations): Chevalley basis. ~700 words added. Cross-references Humphreys §25.2 for the Chevalley-basis construction (the alternative integral form, used for Chevalley groups and modular reps).
§Master deepening of 07.06.10 (Killing form): Levi-Malcev decomposition. ~500 words added. $\mathfrak{g} = \mathfrak{r} \rtimes \mathfrak{s} $w h er e$ \mathfrak{r} = \mathrm{rad}(\mathfrak{g})$ and $s$ is semisimple. Statement + reference to Bourbaki for proof.
§Master deepening of 07.07.01: compact real form and Weyl's unitarian trick. (Shared with Fulton-Harris audit item 9.) ~500 words added. Weyl's unitarian trick: complex reps of $g_{C}$ ↔ unitary reps of compact real form $u$ . CSLA Ch. VIII anchor.
§Master deepening of 07.06.18 (root-space decomposition): root strings and the $sl_{2}$ subalgebra for each root. ~600 words. Already implicit in 07.06.03; promote to explicit master section here.

Priority 4 — survey pointers (optional, Master-only):

§Pointer in 07.06.20: extension to Kac-Moody and affine Lie algebras via the same generators-and-relations philosophy. ~300 words. Statement only; full development deferred to a hypothetical Kac Infinite-Dimensional Lie Algebras audit.

§4 Implementation sketch (P3 → P4)

Realistic production estimate (mirroring earlier Brown / Fulton-Harris audit batches):

Priority 1 (4 new units): ~3 hours each = ~12 hours. 07.06.10 killing-form, 07.06.17 cartan-subalgebra, 07.06.20 serre-relations, 07.06.18 root-space-decomposition.
Priority 2 (4 new units, some short): ~2.5 hours each = ~10 hours.
Priority 3 (5 master deepenings + 1 unit): ~45 min each = ~4 hours.
Priority 4 (pointer): ~30 min.

Total: ~26–27 hours of focused production for full FT-equivalence coverage of CSLA. Fits a 4-day window. Priority 1 alone (~12 hours, 1.5 days) closes the foundational gap and raises effective coverage from ~55% to ~85%.

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

Killing 1888–1890 — Wilhelm Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen, I–IV," Math. Ann. 31, 33, 34, 36 (1888–1890). Originated the classification programme (with errors that Cartan corrected). Originated the Killing form itself. Cite in 07.06.10 and 07.04.01 Master sections.
Cartan 1894 — Élie Cartan, Sur la structure des groupes de transformations finis et continus, thèse, Paris 1894. Corrected and completed Killing's classification; introduced Cartan subalgebras and the rigorous semisimplicity criterion. Cite in 07.06.10, 07.06.17, 07.06.16, 07.04.01.
Weyl 1925/1926 — Hermann Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen, I–IV," Math. Z. 23–24 (1925–1926). Compact-form integration; complete reducibility; character formula. Cite in 07.06.22 and 07.06.07 Master sections.
Chevalley 1955 — C. Chevalley, "Sur certains groupes simples," Tôhoku Math. J. 7 (1955) 14–66. The Chevalley basis; integral forms; Chevalley groups. Cite in 07.06.20 Master deepening (Chevalley-basis section).
Serre 1966 — J.-P. Serre, Algèbres de Lie semi-simples complexes, W. A. Benjamin, New York 1966. Cited as the originator of the Serre-relations presentation. Cite in 07.06.20 (this is the originator citation for the unit) and as the book-anchor for every Priority-1 unit.

Notation crosswalk. CSLA uses $g$ for the Lie algebra, $h$ for the Cartan subalgebra, $R$ for the root system, $R^{+}$ for positive roots, $Π = {α_{1}, \dots, α_{ℓ}}$ for the simple roots, $W$ for the Weyl group, $H_{i}, E_{i}, F_{i}$ for the Serre generators, $\langle \beta, \alpha^\vee\rangle = 2(\beta, \alpha)/(\alpha, \alpha) $f or t h e C a r t an p ai r in g (w i t h$ \alpha^\vee$ the coroot). Codex matches CSLA on $g, h, R, W$ (per 07.06.03, 07.06.04); the simple-root notation in 07.06.05 is $α_{i}$ matching CSLA. The principal notation concern is the Serre vs Fulton-Harris vs Humphreys split for the Cartan-Weyl generators:

CSLA / Serre: $E_{i}, F_{i}, H_{i}$ for the Chevalley-Serre generators attached to simple roots only; for each root $α$ , the $sl_{2}$ -triple is $(e_{α}, f_{α}, h_{α})$ .
Fulton-Harris: $X_{α}, Y_{α}, H_{α}$ for the $sl_{2}$ -triple attached to root $α$ ; no special notation for simple-root generators (because FH does not centre the Serre presentation).
Humphreys: $x_{α}, y_{α}, h_{α}$ for the Chevalley basis; $e_{i}, f_{i}, h_{i}$ for the Serre generators in §18.
Carter: $e_{i}, f_{i}, h_{i}$ for Serre generators throughout (lowercase) to match the Kac-Moody literature.

Codex notation decision. Adopt the Serre / CSLA / Carter convention: $e_{i}, f_{i}, h_{i}$ for the Serre generators attached to simple roots $α_{i}$ (lowercase to match Kac-Moody convention and to disambiguate from group elements). For the $sl_{2}$ -triple attached to a general root $α$ , write $(e_{α}, f_{α}, h_{α})$ . Record this in a §Notation paragraph of 07.06.20 and cross-reference in 07.06.18, 07.06.10, 07.06.17. This is a deliberate departure from Fulton-Harris notation (where $X_{α}, Y_{α}$ are used) — the Codex follows the modern Carter / Kac convention which is the de facto standard for current Lie-theory research.

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

03.04.01 lie-algebra (basic Lie-algebra definitions) — strict prereq for 07.06.10, 07.06.14, 07.06.15, 07.06.16, 07.06.17.
Fulton-Harris audit punch-list items (07.06.11 sl2, 07.06.12 sl3): both depend on 07.06.10 killing-form and 07.06.17 cartan-subalgebra, so ordering matters — ship CSLA Priority-1 units before FH Priority-1 worked-example units.
physics/quantum-mechanics/ angular momentum strand — 07.06.10 Killing form on $su (2)$ is the inner product on physical angular momentum.
02.02 quantum-theory-groups-representations audit (Woit, FT 2.02) — Woit Ch. 8–10 uses the Serre presentation implicitly for $su (3)$ ; cross-link to 07.06.20.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem in CSLA (full P1 audit; deferred until a local PDF is acquired). Coverage estimates in §2 are based on CSLA's well-known chapter structure (the 8-chapter organisation is canonical and reproduced in every secondary source) plus spot-checks of the 26 Codex units.
Serre's Lie Algebras and Lie Groups (FT 3.13 sibling). Per the stub spec, deferred to its own dedicated audit. Note that 3.13 covers p-adic Lie groups and Campbell-Hausdorff homological algebra — topics orthogonal to 3.12; the sibling audit is therefore independent rather than overlapping.
Serre's Linear Representations of Finite Groups (FT 3.15). Different book; covered by its own audit. CSLA does not overlap with finite-group representation theory.
Serre's A Course in Arithmetic (FT 3.14). Different book; its rep-theory fragment is a fraction of FT 3.15 and is anchored there.
Infinite-dimensional Kac-Moody Lie algebras (Kac, Infinite-Dim Lie Algebras, 3rd ed. 1990). CSLA is the finite-dim progenitor of the Kac programme; the affine and indefinite extensions are deferred to a hypothetical future Kac audit. Pointer only in 07.06.20 Priority-4.
Real forms and structure theory (Knapp Beyond an Introduction). Knapp is the real-form sequel to CSLA's complex-form-only treatment. Deferred to a Knapp audit if/when promoted.
Quantum groups and Hopf-algebraic deformations (Drinfeld 1985, Jimbo 1985, Lusztig). The Serre relations admit a $q$ -deformed form $U_{q} (g)$ ; this is the modern descendant of CSLA's presentation but is deferred to a category-theory / quantum-group audit.
Algebraic groups and Chevalley groups in positive characteristic (Steinberg lectures; Jantzen Representations of Algebraic Groups). Chevalley basis briefly addressed as a Master deepening in 07.06.20; the full positive-characteristic theory is out of scope.
Exercise-pack production. CSLA has very few exercises (Serre's monographs are uniformly exercise-light). The exercise pack for 07-representation-theory/ will mostly draw from Humphreys and Fulton-Harris rather than CSLA.

§6 Acceptance criteria for FT equivalence (CSLA)

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

The 4 Priority-1 new units have shipped (07.06.10 killing-form, 07.06.17 cartan-subalgebra, 07.06.20 serre-relations, 07.06.18 root-space-decomposition). These close the foundational gap.
≥95% of CSLA's named theorems in Ch. I–VIII map to a Codex unit (currently ~55%; after Priority-1 this rises to ~85%; after Priority-1+2 to ~92%; full ≥95% requires Priority-3 master deepenings).
≥90% of CSLA's structure proofs (Cartan criteria, Weyl reducibility, CSA conjugacy via regular elements, Serre's theorem) have a direct unit covering them.
Notation crosswalk recorded — in particular, the $e_{i}, f_{i}, h_{i}$ Serre-generator convention is adopted across the Lie-algebraic strand. See §4.
Pass-W weaving connects the new units to 03.04 lie-algebra, to the Fulton-Harris audit punch-list, and to the Woit FT 2.02 audit.
Originator-prose sections (Killing 1888, Cartan 1894, Weyl 1925, Chevalley 1955, Serre 1966) 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 CSLA (e.g., Engel's theorem, Cartan's criterion for semisimplicity, CSA conjugacy, Serre's theorem, Weyl character formula); for each, identify the Codex unit that proves it.
4.2 Exercise reproducibility. CSLA is exercise-light — sample exercises should be drawn from the Humphreys / Fulton-Harris exercise corpus on the same topics. For each, identify the Codex unit(s) sufficient to solve it.
4.3 Notation comprehension. Sample 5 fragments of CSLA prose; paraphrase each into the Codex notation system (per §4 crosswalk). Particular attention to the Serre-generator notation in CSLA Ch. VI.

§7 Sourcing

Local PDF status. Not present in reference/textbooks-extra/ (verified — only the cover image at reference/fast-track/images/Serre-Lie-Algebras-Lie-Groups-712x1024__dd9fad6ce3.jpg is local, and that image is shared with the FT 3.13 Lie Algebras and Lie Groups slot). CSLA is a commercial Springer Monographs in Mathematics title and is not author-hosted.
Commercial source. Springer Monographs in Mathematics. Print + Springer eBook available at link.springer.com/book/10.1007/978-3-642-56884-8. ISBN 3-540-67827-1 (softcover 2001 reprint), 0-8053-8633-5 (1966 Benjamin original).
Library-mirror sources. Acquire via institutional access (university library Springer subscription) or interlibrary loan. The 1966 Benjamin original is out of print but available via inter-library loan in major research libraries; the 1987 Jones translation is the Springer text in circulation. For the local copy, target placement is reference/textbooks-extra/Serre-Complex-Semisimple-Lie-Algebras.pdf to mirror the pattern of other commercial-source FT texts.
Companion / supplementary materials.
- J. E. Humphreys, Introduction to Lie Algebras and Representation Theory (Springer GTM 9, 1972). The standard graduate textbook; longer treatment of the same material. Often easier to acquire than CSLA.
- N. Bourbaki, Groupes et algèbres de Lie Ch. IV–VIII (Hermann 1968–1975; Springer reprint 2002–2008). Encyclopaedic.
- R. W. Carter, Lie Algebras of Finite and Affine Type (Cambridge 2005). Modern unified treatment in 600 pages; reproduces CSLA Ch. VI in expanded form.
- A. W. Knapp, Lie Groups Beyond an Introduction (Birkhäuser 2nd ed. 2002). The real-form sequel.
Open-access alternatives covering ~80% of CSLA:
- B. C. Hall, Lie Groups, Lie Algebras, and Representations (Springer GTM 222) — substantial preview on Google Books; covers root systems, Weyl group, classification at a gentler pace.
- Various lecture notes — Erdmann-Wildon Introduction to Lie Algebras (Springer SUMS 2006, ~250 pp.) is the canonical undergraduate companion; Etingof et al. Introduction to Lie Algebras (free MIT 18.745 notes) covers similar ground.
Reduced-audit flag. This plan was produced without a local CSLA PDF. Coverage estimates in §2 should be re-verified once a PDF is acquired, particularly for Ch. VI (Serre relations) where the proof detail is the load-bearing content. Promote to full P1 audit when PDF is local.

Serre — *Complex Semisimple Lie Algebras* (Fast Track 3.12) — Audit + Gap Plan