feat(algebra/lie/from_cartan_matrix): construction of a Lie algebra from a Cartan matrix (#8206)

Author: ocfnash

docs/references.bib

@@ -964,6 +964,20 @@ @Book{            seligman1967
   mrreviewer    = {R. E. Block}
 }
 
+@book{            serre1965
+  author        = {Serre, Jean-Pierre},
+  title         = {Complex semisimple {L}ie algebras},
+  note          = {Translated from the French by G. A. Jones},
+  publisher     = {Springer-Verlag, New York},
+  year          = {1987},
+  pages         = {x+74},
+  isbn          = {0-387-96569-6},
+  mrclass       = {17-01 (17B20)},
+  mrnumber      = {914496},
+  doi           = {10.1007/978-1-4757-3910-7},
+  url           = {https://doi.org/10.1007/978-1-4757-3910-7},
+}
+
 @Book{            simon2011,
   author        = {Simon, Barry},
   title         = {Convexity: An Analytic Viewpoint},

src/algebra/lie/from_cartan_matrix.lean

@@ -0,0 +1,153 @@
+/-
+Copyright (c) 2021 Oliver Nash. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Oliver Nash
+-/
+import algebra.lie.free
+import algebra.lie.quotient
+
+/-!
+# Lie algebras from Cartan matrices
+
+Split semi-simple Lie algebras are uniquely determined by their Cartan matrix. Indeed, if `A` is
+an `l × l` Cartan matrix, the corresponding Lie algebra may be obtained as the Lie algebra on
+`3l` generators: $H_1, H_2, \ldots H_l, E_1, E_2, \ldots, E_l, F_1, F_2, \ldots, F_l$
+subject to the following relations:
+$$
+\begin{align}
+  [H_i, H_j] &= 0\\
+  [E_i, F_i] &= H_i\\
+  [E_i, F_j] &= 0 \quad\text{if $i \ne j$}\\
+  [H_i, E_j] &= A_{ij}E_j\\
+  [H_i, F_j] &= -A_{ij}F_j\\
+  ad(E_i)^{1 - A_{ij}}(E_j) &= 0 \quad\text{if $i \ne j$}\\
+  ad(F_i)^{1 - A_{ij}}(F_j) &= 0 \quad\text{if $i \ne j$}\\
+\end{align}
+$$
+
+In this file we provide the above construction. It is defined for any square matrix of integers but
+the results for non-Cartan matrices should be regarded as junk.
+
+Recall that a Cartan matrix is a square matrix of integers `A` such that:
+ * For diagonal values we have: `A i i = 2`.
+ * For off-diagonal values (`i ≠ j`) we have: `A i j ∈ {-3, -2, -1, 0}`.
+ * `A i j = 0 ↔ A j i = 0`.
+ * There exists a diagonal matrix `D` over ℝ such that `D ⬝ A ⬝ D⁻¹` is symmetric positive definite.
+
+## Alternative construction
+
+This construction is sometimes performed within the free unital associative algebra
+`free_algebra R X`, rather than within the free Lie algebra `free_lie_algebra R X`, as we do here.
+However the difference is illusory since the construction stays inside the Lie subalgebra of
+`free_algebra R X` generated by `X`, and this is naturally isomorphic to `free_lie_algebra R X`
+(though the proof of this seems to require Poincaré–Birkhoff–Witt).
+
+## References
+
+* [N. Bourbaki, *Lie Groups and Lie Algebras, Chapters 7--9*](bourbaki1975b) chapter VIII, §4.3
+
+* [J.P. Serre, *Complex Semisimple Lie Algebras*](serre1965) chapter VI, appendix
+
+## Main definitions
+
+  * `matrix.to_lie_algebra`
+
+## Tags
+
+lie algebra, semi-simple, cartan matrix
+-/
+
+universes u v w
+
+noncomputable theory
+
+variables (R : Type u) {B : Type v} [comm_ring R] [decidable_eq B] [fintype B]
+variables (A : matrix B B ℤ)
+
+namespace cartan_matrix
+
+variables (B)
+
+/-- The generators of the free Lie algebra from which we construct the Lie algebra of a Cartan
+matrix as a quotient. -/
+inductive generators
+| H : B → generators
+| E : B → generators
+| F : B → generators
+
+instance [inhabited B] : inhabited (generators B) := ⟨generators.H $ default B⟩
+
+variables {B}
+
+namespace relations
+
+open function
+
+local notation `H` := free_lie_algebra.of R ∘ generators.H
+local notation `E` := free_lie_algebra.of R ∘ generators.E
+local notation `F` := free_lie_algebra.of R ∘ generators.F
+local notation `ad` := lie_algebra.ad R (free_lie_algebra R (generators B))
+
+/-- The terms correpsonding to the `⁅H, H⁆`-relations. -/
+def HH : B × B → free_lie_algebra R (generators B) :=
+uncurry $ λ i j, ⁅H i, H j⁆
+
+/-- The terms correpsonding to the `⁅E, F⁆`-relations. -/
+def EF : B × B → free_lie_algebra R (generators B) :=
+uncurry $ λ i j, if i = j then ⁅E i, F i⁆ - H i else ⁅E i, F j⁆
+
+/-- The terms correpsonding to the `⁅H, E⁆`-relations. -/
+def HE : B × B → free_lie_algebra R (generators B) :=
+uncurry $ λ i j, ⁅H i, E j⁆ - (A i j) • E j
+
+/-- The terms correpsonding to the `⁅H, F⁆`-relations. -/
+def HF : B × B → free_lie_algebra R (generators B) :=
+uncurry $ λ i j, ⁅H i, F j⁆ + (A i j) • F j
+
+/-- The terms correpsonding to the `ad E`-relations.
+
+Note that we use `int.to_nat` so that we can take the power and that we do not bother
+restricting to the case `i ≠ j` since these relations are zero anyway. We also defensively
+ensure this with `ad_E_of_eq_eq_zero`. -/
+def ad_E : B × B → free_lie_algebra R (generators B) :=
+uncurry $ λ i j, (ad (E i))^(-A i j).to_nat $ ⁅E i, E j⁆
+
+/-- The terms correpsonding to the `ad F`-relations.
+
+See also `ad_E` docstring. -/
+def ad_F : B × B → free_lie_algebra R (generators B) :=
+uncurry $ λ i j, (ad (F i))^(-A i j).to_nat $ ⁅F i, F j⁆
+
+private lemma ad_E_of_eq_eq_zero (i : B) (h : A i i = 2) : ad_E R A ⟨i, i⟩ = 0 :=
+have h' : (-2 : ℤ).to_nat = 0, { refl, },
+by simp [ad_E, h, h']
+
+private lemma ad_F_of_eq_eq_zero (i : B) (h : A i i = 2) : ad_F R A ⟨i, i⟩ = 0 :=
+have h' : (-2 : ℤ).to_nat = 0, { refl, },
+by simp [ad_F, h, h']
+
+/-- The union of all the relations as a subset of the free Lie algebra. -/
+def to_set : set (free_lie_algebra R (generators B)) :=
+(set.range $ HH R) ∪
+(set.range $ EF R) ∪
+(set.range $ HE R A) ∪
+(set.range $ HF R A) ∪
+(set.range $ ad_E R A) ∪
+(set.range $ ad_F R A)
+
+/-- The ideal of the free Lie algebra generated by the relations. -/
+def to_ideal : lie_ideal R (free_lie_algebra R (generators B)) :=
+lie_submodule.lie_span R _ $ to_set R A
+
+end relations
+
+end cartan_matrix
+
+/-- The Lie algebra corresponding to a Cartan matrix.
+
+Note that it is defined for any matrix of integers. Its value for non-Cartan matrices should be
+regarded as junk. -/
+@[derive [lie_ring, lie_algebra R]]
+def matrix.to_lie_algebra := (cartan_matrix.relations.to_ideal R A).quotient
+
+instance (A : matrix B B ℤ) : inhabited (matrix.to_lie_algebra R A) := ⟨0⟩