Documentation

Mathlib.Algebra.Lie.Matrix

Lie algebras of matrices #

An important class of Lie algebras are those arising from the associative algebra structure on square matrices over a commutative ring. This file provides some very basic definitions whose primary value stems from their utility when constructing the classical Lie algebras using matrices.

Main definitions #

Tags #

lie algebra, matrix

def lieEquivMatrix' {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] :
Module.End R (nR) ≃ₗ⁅R Matrix n n R

The natural equivalence between linear endomorphisms of finite free modules and square matrices is compatible with the Lie algebra structures.

Equations
  • lieEquivMatrix' = let __src := LinearMap.toMatrix'; { toLinearMap := __src, map_lie' := , invFun := __src.invFun, left_inv := , right_inv := }
Instances For
    @[simp]
    theorem lieEquivMatrix'_apply {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (f : Module.End R (nR)) :
    lieEquivMatrix' f = LinearMap.toMatrix' f
    @[simp]
    theorem lieEquivMatrix'_symm_apply {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (A : Matrix n n R) :
    lieEquivMatrix'.symm A = Matrix.toLin' A
    def Matrix.lieConj {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (P : Matrix n n R) (h : Invertible P) :

    An invertible matrix induces a Lie algebra equivalence from the space of matrices to itself.

    Equations
    • P.lieConj h = (lieEquivMatrix'.symm.trans (P.toLinearEquiv' h).lieConj).trans lieEquivMatrix'
    Instances For
      @[simp]
      theorem Matrix.lieConj_apply {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (P : Matrix n n R) (A : Matrix n n R) (h : Invertible P) :
      (P.lieConj h) A = P * A * P⁻¹
      @[simp]
      theorem Matrix.lieConj_symm_apply {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (P : Matrix n n R) (A : Matrix n n R) (h : Invertible P) :
      (P.lieConj h).symm A = P⁻¹ * A * P
      def Matrix.reindexLieEquiv {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] {m : Type w₁} [DecidableEq m] [Fintype m] (e : n m) :

      For square matrices, the natural map that reindexes a matrix's rows and columns with equivalent types, Matrix.reindex, is an equivalence of Lie algebras.

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For
        @[simp]
        theorem Matrix.reindexLieEquiv_apply {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] {m : Type w₁} [DecidableEq m] [Fintype m] (e : n m) (M : Matrix n n R) :
        @[simp]
        theorem Matrix.reindexLieEquiv_symm {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] {m : Type w₁} [DecidableEq m] [Fintype m] (e : n m) :