Module over a $𝕂$ -monoid

Let $𝐴$ be a $𝕂$ -monoid over $𝕂$ . A (left) $𝐴$ -module is a $𝕂$ -vector space $𝑉$ equipped with a bilinear map

𝐴 \times 𝑉 \to 𝑉 (𝑎, 𝑣) \mapsto 𝑎 \cdot 𝑣

such that

$1 \cdot 𝑣 = 𝑣$ for $𝑣 \in 𝑉$
$(𝑎 𝑏) \cdot 𝑣 = 𝑎 \cdot (𝑏 \cdot 𝑣)$ for $𝑎, 𝑏 \in 𝐴$ , $𝑣 \in 𝑉$

which is a curried version of a unital algebra homomorphism

𝐴 \to E n d 𝕂 (𝑉) .

We also call this a representation of $𝐴$ carried by $𝑉$ .

Properties and further terminology

$𝑉$ automatically carries a Lie algebra representation of the commutator algebra of $𝐴$ and any Lie subalgebra.
A Submodule of $𝑉$ is an invariant subspace under the action of $𝐴$ .
A module is irreducible iff it has no proper nontrivial submodules.
A module is indecomposable iff it cannot be decomposed into the direct sum of two nonzero submodules.
A module isomorphism is an Equivalence of group representations.
The Regular representation shows that $𝐴$ is a module over itself.

Explanation

Since a $𝕂$ -monoid $𝐴$ over a field $𝕂$ is itself a ring, it is possible to form a module $𝑉$ over $𝐴$ . The action of $𝕂$ on $𝐴$ and $𝐴$ on $𝑉$ induces an action of $𝕂$ on $𝑉$ , thus the module $𝑉$ inherits the $𝕂$ -linear structure of the underlying ring $𝐴$ . Therefore $𝑉$ is a vector space over $𝕂$ .

Proof

Let $𝟙 \in 𝐴$ be the identity element of the associative algebra $𝐴$ . Then a distributive and linear field action is given by

(\cdot) : 𝕂 \times 𝑉 \to 𝑉 (𝜆, 𝑣) \mapsto 𝜆 𝟙 𝑣

since for any $𝑢, 𝑣 \in 𝑉$ and $𝜇, 𝜆 \in 𝕂$ :

1 𝟙 𝑣 = 𝑣

satisfying ^V4;

(𝜇 𝜆) 𝟙 𝑣 = 𝜇 𝟙 (𝜆 𝟙 𝑣)

satisfying ^V5;

𝜆 𝟙 (𝑢 + 𝑣) = 𝜆 𝟙 𝑢 + 𝜆 𝟙 𝑣

satisfying ^V6; and

(𝜇 + 𝜆) 𝟙 𝑣 = 𝜇 𝟙 𝑣 + 𝜆 𝟙 𝑣

satisfying ^V7.

Such a module coïncides exactly with the notion of a Group representation of the algebra $𝐴$ over $𝕂$ .

Module over a 𝕂-monoid

Properties and further terminology

Explanation

See also

Module over a $𝕂$ -monoid