feat(linear_algebra/matrix/general_linear_group): GL(n, R) (#8466)

added this file which contains definition of the general linear group as well as the subgroup of matrices with positive determinant.

Author: CBirkbeck

src/linear_algebra/general_linear_group.lean

@@ -0,0 +1,170 @@
+/-
+Copyright (c) 2021 Chris Birkbeck. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Chris Birkbeck
+-/
+import linear_algebra.matrix
+import linear_algebra.matrix.nonsingular_inverse
+import linear_algebra.special_linear_group
+import linear_algebra.determinant
+
+/-!
+# The General Linear group $GL(n, R)$
+This file defines the elements of the General Linear group `general_linear_group n R`,
+consisting of all invertible `n` by `n` `R`-matrices.
+## Main definitions
+* `matrix.general_linear_group` is the type of matrices over R which are units in the matrix ring.
+* `matrix.GL_pos` gives the subgroup of matrices with
+  positive determinant (over a linear ordered ring).
+## Tags
+matrix group, group, matrix inverse
+-/
+
+namespace matrix
+universes u v
+open_locale matrix
+open linear_map
+
+/-- `GL n R` is the group of `n` by `n` `R`-matrices with unit determinant.
+Defined as a subtype of matrices-/
+abbreviation general_linear_group (n : Type u) (R : Type v)
+  [decidable_eq n] [fintype n] [comm_ring R] : Type* := units (matrix n n R)
+
+notation `GL` := general_linear_group
+
+namespace general_linear_group
+
+variables {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R]
+
+/-- The determinant of a unit matrix is itself a unit. -/
+@[simps]
+def det : GL n R →* units R :=
+{ to_fun := λ A,
+  { val := (↑A : matrix n n R).det,
+    inv := (↑(A⁻¹) : matrix n n R).det,
+    val_inv := by rw [←det_mul, ←mul_eq_mul, A.mul_inv, det_one],
+    inv_val := by rw [←det_mul, ←mul_eq_mul, A.inv_mul, det_one]},
+  map_one' := units.ext det_one,
+  map_mul' := λ A B, units.ext $ det_mul _ _ }
+
+/--The `GL n R` and `general_linear_group R n` groups are multiplicatively equivalent-/
+def to_lin : (GL n R) ≃* (linear_map.general_linear_group R (n → R)) :=
+units.map_equiv to_lin_alg_equiv'.to_mul_equiv
+
+/--Given a matrix with invertible determinant we get an element of `GL n R`-/
+def mk' (A : matrix n n R) (h : invertible (matrix.det A)) : GL n R :=
+unit_of_det_invertible A
+
+/--Given a matrix with unit determinant we get an element of `GL n R`-/
+noncomputable def mk'' (A : matrix n n R) (h : is_unit (matrix.det A)) : GL n R :=
+nonsing_inv_unit A h
+
+instance coe_fun : has_coe_to_fun (GL n R) :=
+{ F   := λ _, n → n → R,
+  coe := λ A, A.val }
+
+lemma ext_iff (A B : GL n R) : A = B ↔ (∀ i j, (A : matrix n n R) i j = (B : matrix n n R) i j) :=
+units.ext_iff.trans matrix.ext_iff.symm
+
+/-- Not marked `@[ext]` as the `ext` tactic already solves this. -/
+lemma ext ⦃A B : GL n R⦄ (h : ∀ i j, (A : matrix n n R) i j = (B : matrix n n R) i j) :
+  A = B :=
+units.ext $ matrix.ext h
+
+section coe_lemmas
+
+variables (A B : GL n R)
+
+@[simp] lemma coe_fn_eq_coe : ⇑A = (↑A : matrix n n R) := rfl
+
+@[simp] lemma coe_mul : ↑(A * B) = (↑A : matrix n n R) ⬝ (↑B : matrix n n R) := rfl
+
+@[simp] lemma coe_one : ↑(1 : GL n R) = (1 : matrix n n R) := rfl
+
+lemma coe_inv : ↑(A⁻¹) = (↑A : matrix n n R)⁻¹ :=
+begin
+  letI := A.invertible,
+  exact inv_eq_nonsing_inv_of_invertible (↑A : matrix n n R),
+end
+
+end coe_lemmas
+
+end general_linear_group
+
+namespace special_linear_group
+
+variables {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R]
+
+instance has_coe_to_general_linear_group : has_coe (special_linear_group n R) (GL n R) :=
+⟨λ A, ⟨↑A, ↑(A⁻¹), congr_arg coe (mul_right_inv A), congr_arg coe (mul_left_inv A)⟩⟩
+
+end special_linear_group
+
+section
+
+variables {n : Type u} {R : Type v} [decidable_eq n] [fintype n] [linear_ordered_comm_ring R ]
+
+section
+variables (n R)
+
+/-- This is the subgroup of `nxn` matrices with entries over a
+linear ordered ring and positive determinant. -/
+def GL_pos : subgroup (GL n R) :=
+(units.pos_subgroup R).comap general_linear_group.det
+end
+
+@[simp] lemma mem_GL_pos (A : GL n R) : A ∈ GL_pos n R ↔ 0 < (A.det : R) := iff.rfl
+end
+
+section has_neg
+
+variables {n : Type u} {R : Type v} [decidable_eq n] [fintype n] [linear_ordered_comm_ring R ]
+[fact (even (fintype.card n))]
+
+/-- Formal operation of negation on general linear group on even cardinality `n` given by negating
+each element. -/
+instance : has_neg (GL_pos n R) :=
+⟨λ g,
+   ⟨- g,
+  begin
+    simp only [mem_GL_pos, general_linear_group.coe_det_apply, units.coe_neg],
+    have := det_smul g (-1),
+    simp only [general_linear_group.coe_fn_eq_coe, one_smul, coe_fn_coe_base, neg_smul] at this,
+    rw this,
+    simp [nat.neg_one_pow_of_even (fact.out (even (fintype.card n)))],
+    have gdet := g.property,
+    simp only [mem_GL_pos, general_linear_group.coe_det_apply, subtype.val_eq_coe] at gdet,
+    exact gdet,
+  end⟩⟩
+
+@[simp] lemma GL_pos_coe_neg (g : GL_pos n R) : ↑(- g) = - (↑g : matrix n n R) :=
+rfl
+
+@[simp]lemma GL_pos_neg_elt (g : GL_pos n R): ∀ i j, ( ↑(-g): matrix n n R) i j= - (g i j):=
+begin
+simp,
+end
+
+end has_neg
+
+namespace special_linear_group
+
+variables {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [linear_ordered_comm_ring R]
+
+/-- `special_linear_group n R` embeds into `GL_pos n R` -/
+def to_GL_pos : special_linear_group n R →* GL_pos n R :=
+{ to_fun := λ A, ⟨(A : GL n R), show 0 < (↑A : matrix n n R).det, from A.prop.symm ▸ zero_lt_one⟩,
+  map_one' := subtype.ext $ units.ext $ rfl,
+  map_mul' := λ A₁ A₂, subtype.ext $ units.ext $ rfl }
+
+instance : has_coe (special_linear_group n R) (GL_pos n R) := ⟨to_GL_pos⟩
+
+lemma coe_eq_to_GL_pos : (coe : special_linear_group n R → GL_pos n R) = to_GL_pos := rfl
+
+lemma to_GL_pos_injective :
+  function.injective (to_GL_pos : special_linear_group n R → GL_pos n R) :=
+(show function.injective ((coe : GL_pos n R → matrix n n R) ∘ to_GL_pos),
+ from subtype.coe_injective).of_comp
+
+end special_linear_group
+end matrix

src/ring_theory/subring.lean

@@ -983,3 +983,16 @@ S.to_subsemiring.module
 end subring
 
 end actions
+
+-- while this definition is not about subrings, this is the earliest we have
+-- both ordered ring structures and submonoids available
+
+/-- The subgroup of positive units of a linear ordered commutative ring. -/
+def units.pos_subgroup (R : Type*) [linear_ordered_comm_ring R] [nontrivial R] :
+  subgroup (units R) :=
+{ carrier := {x | (0 : R) < x},
+  inv_mem' := λ x (hx : (0 : R) < x), (zero_lt_mul_left hx).mp $ x.mul_inv.symm ▸ zero_lt_one,
+  ..(pos_submonoid R).comap (units.coe_hom R)}
+
+@[simp] lemma units.mem_pos_subgroup {R : Type*} [linear_ordered_comm_ring R] [nontrivial R]
+  (u : units R) : u ∈ units.pos_subgroup R ↔ (0 : R) < u := iff.rfl

src/ring_theory/subsemiring.lean

@@ -831,3 +831,16 @@ instance [add_comm_monoid α] [module R' α] (S : subsemiring R') : module S α
 end subsemiring
 
 end actions
+
+-- While this definition is not about `subsemiring`s, this is the earliest we have
+-- both `ordered_semiring` and `submonoid` available.
+
+
+/-- Submonoid of positive elements of an ordered semiring. -/
+def pos_submonoid (R : Type*) [ordered_semiring R] [nontrivial R] : submonoid R :=
+{ carrier := {x | 0 < x},
+  one_mem' := show (0 : R) < 1, from zero_lt_one,
+  mul_mem' := λ x y (hx : 0 < x) (hy : 0 < y), mul_pos hx hy }
+
+@[simp] lemma mem_pos_monoid {R : Type*} [ordered_semiring R] [nontrivial R] (u : units R) :
+  ↑u ∈ pos_submonoid R ↔ (0 : R) < u := iff.rfl