Module theory MOC

Induced module

Let 𝐴 be a [[K-monoid|𝕂-monoid]], 𝐵 ≤𝐴 be a [[Unital subalgebra|𝕂-submonoid]], and 𝑉 be a 𝐵-module. The 𝐴-module induced by the 𝐵-module 𝑉 is a canonical way of extending 𝑉 to accomodate a representation of 𝐴, as formalized by the Universal property.¹ We have the adjunction

with the Restricted module and more generally we can consider Change of ring along a ring homomorphism.

Universal property

Let 𝐴 be [[K-monoid|𝕂-ring]], 𝐵 ≤𝐴 be a [[Unital subalgebra|𝕂-subring]], and 𝑉 be a 𝐵-module. The 𝐴-module induced by the 𝐵-module 𝑉 is a pair consisting of an 𝐴-module Ind𝐴𝐵⁡𝑉 =𝐴 ⊗𝐵𝑉 and a 𝐵-Module homomorphism 𝜄 :𝑉 →Ind𝐴𝐵⁡𝑉 such that given any 𝐴-module 𝑊 a 𝐵-module homomorphism 𝑓 :𝑉 →𝑊 factorizes uniquely through 𝜄 falg

such that ¯𝑓 :Ind𝐴𝐵⁡𝑉 →𝑊 is an 𝐴-module homomorphism. This admits a unique extension to a functor Ind𝐴𝐵 :𝐵𝖬𝗈𝖽 →𝐴𝖬𝗈𝖽 such that 𝜄 :1 ⇒Ind𝐴𝐵 :𝐵𝖬𝗈𝖽 →𝐵𝖬𝗈𝖽 becomes a natural transformation.

Construction

Let 𝐴 ⊗𝕂𝑉 be the 𝕂-tensor product with the bilinear map ( ⊗) :𝐴 ×𝑉 →𝐴 ⊗𝕂𝑉. Let 𝐾 denote the vector subspace generated by any elements of the form

𝑎𝑏⊗𝑣−𝑎⊗𝑏⋅𝑣

for any 𝑎 ∈𝐴, 𝑏 ∈𝐵, and 𝑣 ∈𝑉. We construct the induced module as the quotient vector space

𝐴⊗𝐵𝑉=𝐴⊗𝕂𝑉𝐾

with its natural projection 𝜋 :𝐴 ⊗𝕂𝑉 ↠𝐴 ⊗𝐵𝑉. The map

(⊗𝐵)=𝜋∘(⊗𝕂)

defines a representation of 𝐴, and the inclusion is given by

𝜄:𝑉↪𝐴⊗𝐵𝑉𝑣↦1⊗𝐵𝑣

Proof of the universal property

Let 𝑊 be an 𝐴-module and 𝑓 :𝑉 →𝑊 be a 𝐵-module homomorphism. Then for the above diagram to commute, we require that ¯𝑓(𝜄(𝑣)) =¯𝑓(1 ⊗𝐵𝑣) =𝑓(𝑣) for 𝑣 ∈𝑉. For ¯𝑓 to be an 𝐴-module homomorphism, it follows ¯𝑓(𝑎 ⊗𝐵𝑣) =𝑎 ⋅𝑓(𝑣) for 𝑎 ∈𝐴 and 𝑣 ∈𝑉. Since elements of this form span 𝐴 ⊗𝐵𝑉, this fully defines ¯𝑓, hence it is unique.

Graded structure

Let 𝐴 be a 𝔄-graded K-monoid, 𝐵 ≤𝐴 be unital graded subalgebra, and 𝑉 be a graded 𝐵-module. Then Ind𝐴𝐵⁡𝑉 has a natural graded structure, where for any 𝑎 ∈𝐴𝛼 and 𝑣 ∈𝑉𝛽, deg⁡(𝑎 ⊗𝐵𝑣) =𝛼 +𝛽.

See also

• Complexification

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §1.5, p. 11 ↩