# mathlib3documentation

linear_algebra.clifford_algebra.conjugation

# Conjugations #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This file defines the grade reversal and grade involution functions on multivectors, reverse and involute. Together, these operations compose to form the "Clifford conjugate", hence the name of this file.

https://en.wikipedia.org/wiki/Clifford_algebra#Antiautomorphisms

## Main definitions #

• clifford_algebra.involute: the grade involution, negating each basis vector
• clifford_algebra.reverse: the grade reversion, reversing the order of a product of vectors

## Main statements #

• clifford_algebra.involute_involutive
• clifford_algebra.reverse_involutive
• clifford_algebra.reverse_involute_commute
• clifford_algebra.involute_mem_even_odd_iff
• clifford_algebra.reverse_mem_even_odd_iff
def clifford_algebra.involute {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} :

Grade involution, inverting the sign of each basis vector.

Equations
@[simp]
theorem clifford_algebra.involute_ι {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} (m : M) :
@[simp]
theorem clifford_algebra.involute_comp_involute {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} :
theorem clifford_algebra.involute_involutive {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} :
@[simp]
theorem clifford_algebra.involute_involute {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} (a : clifford_algebra Q) :
@[simp]
theorem clifford_algebra.involute_equiv_symm_apply {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} (ᾰ : clifford_algebra Q) :
@[simp]
theorem clifford_algebra.involute_equiv_apply {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} (ᾰ : clifford_algebra Q) :
def clifford_algebra.involute_equiv {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} :

clifford_algebra.involute as an alg_equiv.

Equations
def clifford_algebra.reverse {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} :

Grade reversion, inverting the multiplication order of basis vectors. Also called transpose in some literature.

Equations
@[simp]
theorem clifford_algebra.reverse_ι {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} (m : M) :
@[simp]
theorem clifford_algebra.reverse.commutes {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} (r : R) :
@[simp]
theorem clifford_algebra.reverse.map_one {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} :
@[simp]
theorem clifford_algebra.reverse.map_mul {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} (a b : clifford_algebra Q) :
@[simp]
theorem clifford_algebra.reverse_comp_reverse {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} :
@[simp]
theorem clifford_algebra.reverse_involutive {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} :
@[simp]
theorem clifford_algebra.reverse_reverse {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} (a : clifford_algebra Q) :
def clifford_algebra.reverse_equiv {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} :

clifford_algebra.reverse as a linear_equiv.

Equations
@[simp]
theorem clifford_algebra.reverse_equiv_symm_apply {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} (ᾰ : clifford_algebra Q) :
@[simp]
theorem clifford_algebra.reverse_equiv_apply {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} (ᾰ : clifford_algebra Q) :
theorem clifford_algebra.reverse_comp_involute {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} :
theorem clifford_algebra.reverse_involute_commute {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} :

clifford_algebra.reverse and clifford_algebra.inverse commute. Note that the composition is sometimes referred to as the "clifford conjugate".

theorem clifford_algebra.reverse_involute {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} (a : clifford_algebra Q) :

### Statements about conjugations of products of lists #

theorem clifford_algebra.reverse_prod_map_ι {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} (l : list M) :

Taking the reverse of the product a list of $n$ vectors lifted via ι is equivalent to taking the product of the reverse of that list.

theorem clifford_algebra.involute_prod_map_ι {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} (l : list M) :
= (-1) ^ l.length l).prod

Taking the involute of the product a list of $n$ vectors lifted via ι is equivalent to premultiplying by ${-1}^n$.

### Statements about submodule.map and submodule.comap#

theorem clifford_algebra.submodule_map_involute_eq_comap {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) (p : ) :
@[simp]
theorem clifford_algebra.ι_range_map_involute {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) :
@[simp]
theorem clifford_algebra.ι_range_comap_involute {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) :
@[simp]
theorem clifford_algebra.even_odd_map_involute {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) (n : zmod 2) :
@[simp]
theorem clifford_algebra.even_odd_comap_involute {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) (n : zmod 2) :
theorem clifford_algebra.submodule_map_reverse_eq_comap {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) (p : ) :
@[simp]
theorem clifford_algebra.ι_range_map_reverse {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) :
@[simp]
theorem clifford_algebra.ι_range_comap_reverse {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) :
theorem clifford_algebra.submodule_map_mul_reverse {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) (p q : ) :

Like submodule.map_mul, but with the multiplication reversed.

theorem clifford_algebra.submodule_comap_mul_reverse {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) (p q : ) :
theorem clifford_algebra.submodule_map_pow_reverse {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) (p : ) (n : ) :

Like submodule.map_pow

theorem clifford_algebra.submodule_comap_pow_reverse {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) (p : ) (n : ) :
@[simp]
theorem clifford_algebra.even_odd_map_reverse {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) (n : zmod 2) :
@[simp]
theorem clifford_algebra.even_odd_comap_reverse {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) (n : zmod 2) :
@[simp]
theorem clifford_algebra.involute_mem_even_odd_iff {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) {x : clifford_algebra Q} {n : zmod 2} :
@[simp]
theorem clifford_algebra.reverse_mem_even_odd_iff {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] (Q : M) {x : clifford_algebra Q} {n : zmod 2} :

TODO: show that these are iffs when invertible (2 : R).

theorem clifford_algebra.involute_eq_of_mem_even {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} {x : clifford_algebra Q} (h : x ) :
theorem clifford_algebra.involute_eq_of_mem_odd {R : Type u_1} [comm_ring R] {M : Type u_2} [ M] {Q : M} {x : clifford_algebra Q} (h : x ) :