mathlib documentation

algebra.algebra.bilinear

Facts about algebras involving bilinear maps and tensor products #

We move a few basic statements about algebras out of algebra.algebra.basic, in order to avoid importing linear_algebra.bilinear_map and linear_algebra.tensor_product unnecessarily.

def algebra.lmul (R : Type u_1) (A : Type u_2) [comm_semiring R] [semiring A] [algebra R A] :

The multiplication in an algebra is a bilinear map.

A weaker version of this for semirings exists as add_monoid_hom.mul.

Equations
@[simp]
theorem algebra.lmul_apply {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (p q : A) :
((algebra.lmul R A) p) q = p * q
def algebra.lmul_left (R : Type u_1) {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (r : A) :

The multiplication on the left in an algebra is a linear map.

Equations
def algebra.lmul_right (R : Type u_1) {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (r : A) :

The multiplication on the right in an algebra is a linear map.

Equations
def algebra.lmul_left_right (R : Type u_1) {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (vw : A × A) :

Simultaneous multiplication on the left and right is a linear map.

Equations
theorem algebra.commute_lmul_left_right (R : Type u_1) {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (a b : A) :
def algebra.lmul' (R : Type u_1) {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] :
A A →ₗ[R] A

The multiplication map on an algebra, as an R-linear map from A ⊗[R] A to A.

Equations
@[simp]
theorem algebra.lmul'_apply {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] {x y : A} :
(algebra.lmul' R) (x ⊗ₜ[R] y) = x * y
@[simp]
theorem algebra.lmul_left_apply {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (p q : A) :
@[simp]
theorem algebra.lmul_right_apply {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (p q : A) :
@[simp]
theorem algebra.lmul_left_right_apply {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (vw : A × A) (p : A) :
(algebra.lmul_left_right R vw) p = ((vw.fst) * p) * vw.snd
@[simp]
theorem algebra.lmul_left_one {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] :
@[simp]
theorem algebra.lmul_left_mul {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (a b : A) :
@[simp]
theorem algebra.lmul_right_one {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] :
@[simp]
theorem algebra.lmul_right_mul {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (a b : A) :
@[simp]
theorem algebra.lmul_left_zero_eq_zero {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] :
@[simp]
theorem algebra.lmul_right_zero_eq_zero {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] :
@[simp]
theorem algebra.lmul_left_eq_zero_iff {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (a : A) :
@[simp]
theorem algebra.lmul_right_eq_zero_iff {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (a : A) :
@[simp]
theorem algebra.pow_lmul_left {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (a : A) (n : ) :
@[simp]
theorem algebra.pow_lmul_right {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A] [algebra R A] (a : A) (n : ) :
theorem algebra.lmul_left_injective {R : Type u_1} {A : Type u_2} [comm_semiring R] [ring A] [algebra R A] [no_zero_divisors A] {x : A} (hx : x 0) :
theorem algebra.lmul_right_injective {R : Type u_1} {A : Type u_2} [comm_semiring R] [ring A] [algebra R A] [no_zero_divisors A] {x : A} (hx : x 0) :
theorem algebra.lmul_injective {R : Type u_1} {A : Type u_2} [comm_semiring R] [ring A] [algebra R A] [no_zero_divisors A] {x : A} (hx : x 0) :