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

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

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      @[simp]
      theorem lieEquivMatrix'_apply {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (f : Module.End R (n → R)) :
      lieEquivMatrix' f = LinearMap.toMatrix' f
      source
      @[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
      source
      def Matrix.lieConj {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (P : Matrix n n R) (h : Invertible P) :
      Matrix n n R ≃ₗ⁅R⁆ Matrix n n R

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

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          @[simp]
          theorem Matrix.lieConj_apply {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (P A : Matrix n n R) (h : Invertible P) :
          (P.lieConj h) A = P * A * P⁻¹
          source
          @[simp]
          theorem Matrix.lieConj_symm_apply {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (P A : Matrix n n R) (h : Invertible P) :
          (P.lieConj h).symm A = P⁻¹ * A * P
          source
          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) :
          Matrix n n R ≃ₗ⁅R⁆ Matrix m m R

          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.

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              @[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) :
              (reindexLieEquiv e) M = (reindex e e) M
              source
              @[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) :
              (reindexLieEquiv e).symm = reindexLieEquiv e.symm
              source
              instance Matrix.instLieRingModuleForall {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] :
              LieRingModule (Matrix n n R) (n → R)
              Equations
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                instance Matrix.instLieModuleForall {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] :
                LieModule R (Matrix n n R) (n → R)
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                @[simp]
                theorem Matrix.lie_apply {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (A : Matrix n n R) (v : n → R) :
                ⁅A, v⁆ = A.mulVec v
                source
                @[simp]
                theorem LieModule.toEnd_matrix {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] :
                toEnd R (Matrix n n R) (n → R) = lieEquivMatrix'.symm.toLieHom
                source
                instance LieModule.instIsFaithfulMatrixForall {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] :
                IsFaithful R (Matrix n n R) (n → R)