# mathlib3documentation

linear_algebra.matrix.special_linear_group

# The Special Linear group $SL(n, R)$ #

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This file defines the elements of the Special Linear group special_linear_group n R, consisting of all square R-matrices with determinant 1 on the fintype n by n. In addition, we define the group structure on special_linear_group n R and the embedding into the general linear group general_linear_group R (n → R).

## Main definitions #

• matrix.special_linear_group is the type of matrices with determinant 1
• matrix.special_linear_group.group gives the group structure (under multiplication)
• matrix.special_linear_group.to_GL is the embedding SLₙ(R) → GLₙ(R)

## Notation #

For m : ℕ, we introduce the notation SL(m,R) for the special linear group on the fintype n = fin m, in the locale matrix_groups.

## Implementation notes #

The inverse operation in the special_linear_group is defined to be the adjugate matrix, so that special_linear_group n R has a group structure for all comm_ring R.

We define the elements of special_linear_group to be matrices, since we need to compute their determinant. This is in contrast with general_linear_group R M, which consists of invertible R-linear maps on M.

We provide matrix.special_linear_group.has_coe_to_fun for convenience, but do not state any lemmas about it, and use matrix.special_linear_group.coe_fn_eq_coe to eliminate it ⇑ in favor of a regular ↑ coercion.

## Tags #

matrix group, group, matrix inverse

def matrix.special_linear_group (n : Type u) [decidable_eq n] [fintype n] (R : Type v) [comm_ring R] :
Type (max u v)

special_linear_group n R is the group of n by n R-matrices with determinant equal to 1.

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Instances for matrix.special_linear_group
@[protected, instance]
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theorem matrix.special_linear_group.ext_iff {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A B : R) :
A = B (i j : n), A i j = B i j
@[ext]
theorem matrix.special_linear_group.ext {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A B : R) :
( (i j : n), A i j = B i j) A = B
@[protected, instance]
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@[protected, instance]
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@[protected, instance]
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@[protected, instance]
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@[protected, instance]
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@[simp]
theorem matrix.special_linear_group.coe_mk {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A : n R) (h : A.det = 1) :
A, h⟩ = A
@[simp]
theorem matrix.special_linear_group.coe_inv {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A : R) :
@[simp]
theorem matrix.special_linear_group.coe_mul {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A B : R) :
(A * B) = A.mul B
@[simp]
theorem matrix.special_linear_group.coe_one {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] :
1 = 1
@[simp]
theorem matrix.special_linear_group.det_coe {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A : R) :
A.det = 1
@[simp]
theorem matrix.special_linear_group.coe_pow {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A : R) (m : ) :
(A ^ m) = A ^ m
theorem matrix.special_linear_group.row_ne_zero {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] [nontrivial R] (g : R) (i : n) :
g i 0
@[protected, instance]
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• matrix.special_linear_group.monoid = matrix.special_linear_group.monoid._proof_1 matrix.special_linear_group.monoid._proof_2 matrix.special_linear_group.monoid._proof_3 matrix.special_linear_group.monoid._proof_4
@[protected, instance]
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A version of matrix.to_lin' A that produces linear equivalences.

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theorem matrix.special_linear_group.to_lin'_apply {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A : R) (v : n R) :
theorem matrix.special_linear_group.to_lin'_symm_apply {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A : R) (v : n R) :

to_GL is the map from the special linear group to the general linear group

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theorem matrix.special_linear_group.coe_to_GL {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A : R) :
@[simp]
theorem matrix.special_linear_group.map_apply_coe {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] {S : Type u_1} [comm_ring S] (f : R →+* S) (g : R) :
def matrix.special_linear_group.map {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] {S : Type u_1} [comm_ring S] (f : R →+* S) :

A ring homomorphism from R to S induces a group homomorphism from special_linear_group n R to special_linear_group n S.

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@[protected, instance]

Coercion of SL n ℤ to SL n R for a commutative ring R.

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@[simp]
@[protected, instance]

Formal operation of negation on special linear group on even cardinality n given by negating each element.

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@[simp]
theorem matrix.special_linear_group.coe_neg {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] [fact (even (fintype.card n))] (g : R) :
@[protected, instance]
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@[simp]
theorem matrix.special_linear_group.SL2_inv_expl_det {R : Type v} [comm_ring R] (A : R) :
matrix.det ![![A.val 1 1, -A.val 0 1], ![-A.val 1 0, A.val 0 0]] = 1
theorem matrix.special_linear_group.SL2_inv_expl {R : Type v} [comm_ring R] (A : R) :
A⁻¹ = ![![A.val 1 1, -A.val 0 1], ![-A.val 1 0, A.val 0 0]], _⟩
theorem matrix.special_linear_group.fin_two_induction {R : Type v} [comm_ring R] (P : Prop) (h : (a b c d : R) (hdet : a * d - b * c = 1), P matrix.of ![![a, b], ![c, d]], _⟩) (g : R) :
P g
theorem matrix.special_linear_group.fin_two_exists_eq_mk_of_apply_zero_one_eq_zero {R : Type u_1} [field R] (g : R) (hg : g 1 0 = 0) :
(a b : R) (h : a 0), g = matrix.of ![![a, b], ![0, a⁻¹]], _⟩
@[protected, instance]
def matrix.special_linear_group.has_coe_to_fun {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] :
(λ (_x : , n n R)

This instance is here for convenience, but is not the simp-normal form.

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@[simp]
theorem matrix.special_linear_group.coe_fn_eq_coe {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (s : R) :

The matrix S = [[0, -1], [1, 0]] as an element of SL(2, ℤ).

This element acts naturally on the Euclidean plane as a rotation about the origin by π / 2.

This element also acts naturally on the hyperbolic plane as rotation about i by π. It represents the Mobiüs transformation z ↦ -1/z and is an involutive elliptic isometry.

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The matrix T = [[1, 1], [0, 1]] as an element of SL(2, ℤ)

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theorem modular_group.coe_T_zpow (n : ) :
= matrix.of ![![1, n], ![0, 1]]
@[simp]
theorem modular_group.T_pow_mul_apply_one (n : ) (g : ) :
* g) 1 = g 1
@[simp]
theorem modular_group.T_mul_apply_one (g : ) :
1 = g 1
@[simp]