TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Associative algebra representations are unital modules

Statement

Let $k$ be a field, $A$ a unital associative $k$ -algebra, and $V$ a $k$ -vector space. There is a bijection between unital representations $ρ : A \to {End}_{k} (V)$ of Representations of unital associative algebras and $k$ -bilinear left $A$ -module actions $A \times V \to V$ , $(a, v) \mapsto a \cdot v$ , for which $(a b) \cdot v = a \cdot (b \cdot v)$ and $1_{A} \cdot v = v$ . The maps are $a \cdot v = ρ (a) (v)$ and $ρ (a) (v) = a \cdot v$ . Here $k$ -bilinear means linear in each variable with respect to the given $k$ -vector structures; the module axioms are those of Unital left and right modules over a ring; unqualified module means left module.

Facts & Assumptions

Given: A field $k$ , a unital associative $k$ -algebra $A$ , a $k$ -vector space $V$ , and the representation and action data in the Statement.

[F1]

Representations of unital associative algebras: a representation is a $k$ -linear map $ρ : A \to {End}_{k} (V)$ preserving the unit and multiplication.

[F2]

Unital left and right modules over a ring; unqualified module means left module: a unital left module action is additive in both variables, associative, and satisfies $1_{A} v = v$ .

[F3]

Vector space over a field: scalar multiplication on each $k$ -vector space is compatible with addition and with multiplication in $k$ .

[F4]

The space $L (V, W)$ of linear maps with pointwise addition and scalar multiplication: ${End}_{k} (V)$ has pointwise vector-space operations on its $k$ -linear maps.

[F5]

Algebras over a commutative ring, central structure maps, and algebra homomorphisms: the multiplication and unit of a $k$ -algebra are compatible with its specified $k$ -algebra structure.

Proof

1.1givenF1F3F4F5construct

Given a representation $ρ$ , define $a \cdot v : = ρ (a) (v)$ . For fixed $a$ , this is $k$ -linear in $v$ because $ρ (a) \in {End}_{k} (V)$ ; for fixed $v$ , it is $k$ -linear in $a$ because $ρ$ is $k$ -linear, $A$ has its specified $k$ -vector structure, and evaluation at $v$ respects the pointwise vector-space operations of [F4]. Thus the action is $k$ -bilinear.

1.2givenF2F3F4F5construct

Conversely, given a $k$ -bilinear unital left $A$ -module action, define $ρ (a) (v) : = a \cdot v$ . Bilinearity in $v$ makes each $ρ (a)$ a $k$ -linear endomorphism, and bilinearity in $a$ makes $ρ$ $k$ -linear.

2.1givenF1F2step 1.1step 1.2algebra

For the action constructed in step 1.1, multiplicativity and preservation of the identity in [F1] give $(a b) \cdot v = a \cdot (b \cdot v)$ and $1_{A} \cdot v = v$ . For the map constructed in step 1.2, the unital left-module laws in [F2] give $ρ (a b) = ρ (a) \circ ρ (b)$ and $ρ (1_{A}) = {id}_{V}$ , which are the representation conditions in [F1].

3.1step 1.1step 1.2step 2.1algebra∎

Starting with $ρ$ , the first construction returns $a \cdot v = ρ (a) (v)$ ; starting with an action, the second returns $ρ (a) (v) = a \cdot v$ . Equality for every $a, v$ proves that the constructions are inverse and hence give the stated bijection.

Depends on

Used by

The componentwise formula can fail to define a tensor action Counterexample
The diagonal tensor action of ℂ[C₂] on trivial and sign modules Example

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pavel Etingof et al., Introduction to Representation Theory, §1.1, printed p. 5 (standard reference, not scraped)
Kalle Kytölä, Introduction to Hopf Algebras and Representations, §3.2, printed pp. 29–30 (standard reference, not scraped)