mathlib documentation

algebra.lie.direct_sum

Direct sums of Lie algebras and Lie modules #

Direct sums of Lie algebras and Lie modules carry natural algbebra and module structures.

Tags #

lie algebra, lie module, direct sum

The direct sum of Lie modules over a fixed Lie algebra carries a natural Lie module structure.

@[instance]
def direct_sum.lie_ring_module {ι : Type v} {L : Type w₁} {M : ι → Type w} [lie_ring L] [Π (i : ι), add_comm_group (M i)] [Π (i : ι), lie_ring_module L (M i)] :
lie_ring_module L (⨁ (i : ι), M i)
Equations
@[simp]
theorem direct_sum.lie_module_bracket_apply {ι : Type v} {L : Type w₁} {M : ι → Type w} [lie_ring L] [Π (i : ι), add_comm_group (M i)] [Π (i : ι), lie_ring_module L (M i)] (x : L) (m : ⨁ (i : ι), M i) (i : ι) :
@[instance]
def direct_sum.lie_module {R : Type u} {ι : Type v} [comm_ring R] {L : Type w₁} {M : ι → Type w} [lie_ring L] [lie_algebra R L] [Π (i : ι), add_comm_group (M i)] [Π (i : ι), module R (M i)] [Π (i : ι), lie_ring_module L (M i)] [Π (i : ι), lie_module R L (M i)] :
lie_module R L (⨁ (i : ι), M i)
Equations
def direct_sum.lie_module_of (R : Type u) (ι : Type v) [comm_ring R] (L : Type w₁) (M : ι → Type w) [lie_ring L] [lie_algebra R L] [Π (i : ι), add_comm_group (M i)] [Π (i : ι), module R (M i)] [Π (i : ι), lie_ring_module L (M i)] [Π (i : ι), lie_module R L (M i)] [decidable_eq ι] (j : ι) :
M j →ₗ⁅R,L ⨁ (i : ι), M i

The inclusion of each component into a direct sum as a morphism of Lie modules.

Equations
def direct_sum.lie_module_component (R : Type u) (ι : Type v) [comm_ring R] (L : Type w₁) (M : ι → Type w) [lie_ring L] [lie_algebra R L] [Π (i : ι), add_comm_group (M i)] [Π (i : ι), module R (M i)] [Π (i : ι), lie_ring_module L (M i)] [Π (i : ι), lie_module R L (M i)] (j : ι) :
(⨁ (i : ι), M i) →ₗ⁅R,L M j

The projection map onto one component, as a morphism of Lie modules.

Equations

The direct sum of Lie algebras carries a natural Lie algebra structure.

@[instance]
def direct_sum.lie_ring {ι : Type v} {L : ι → Type w} [Π (i : ι), lie_ring (L i)] :
lie_ring (⨁ (i : ι), L i)
Equations
@[simp]
theorem direct_sum.bracket_apply {ι : Type v} {L : ι → Type w} [Π (i : ι), lie_ring (L i)] (x y : ⨁ (i : ι), L i) (i : ι) :
@[instance]
def direct_sum.lie_algebra {R : Type u} {ι : Type v} [comm_ring R] {L : ι → Type w} [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] :
lie_algebra R (⨁ (i : ι), L i)
Equations
def direct_sum.lie_algebra_of (R : Type u) (ι : Type v) [comm_ring R] (L : ι → Type w) [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] [decidable_eq ι] (j : ι) :
L j →ₗ⁅R ⨁ (i : ι), L i

The inclusion of each component into the direct sum as morphism of Lie algebras.

Equations
def direct_sum.lie_algebra_component (R : Type u) (ι : Type v) [comm_ring R] (L : ι → Type w) [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] (j : ι) :
(⨁ (i : ι), L i) →ₗ⁅R L j

The projection map onto one component, as a morphism of Lie algebras.

Equations