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

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A unital Hall algebra is a left module over itself

Statement

Let $R$ be a commutative ring with unit, and let $H$ be any unital associative $R$ -algebra that is a Hall algebra. The map $ρ : H \to {End}_{R} (H)$ given by $ρ (a) (h) = a h$ is a representation, so the left regular action makes $H$ a Hall-algebra module.

Facts & Assumptions

Given: A unital associative Hall $R$ -algebra $H$ .

[F1]

Modules over a Hall algebra are unital algebra representations: a Hall module is an $R$ -module with an $R$ -bilinear action satisfying the associative and unit laws.

Proof

1.1givenF1construct

Define $ρ (a) (h) = a h$ for $a, h \in H$ . Since multiplication is $R$ -bilinear, for each fixed $a$ the map $h \mapsto a h$ is $R$ -linear and $a \mapsto ρ (a)$ is $R$ -linear, so $ρ : H \to {End}_{R} (H)$ is an $R$ -linear map.

2.1givenF1step 1.1algebra∎

For $a, b, h \in H$ , $ρ (a b) (h) = (a b) h = a (b h) = (ρ (a) \circ ρ (b)) (h)$ by associativity, and $ρ (1_{H}) (h) = 1_{H} h = h$ , so $ρ (1_{H}) = {id}_{H}$ . Thus $ρ$ is a unital $R$ -algebra homomorphism and hence a Hall-algebra representation by [F1].

Depends on

Modules over a Hall algebra are unital algebra representations

Used by

The generic Jordan Hall regular module carries the normalized Heisenberg Fock action Theorem

Dependency tree · two levels

5 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

Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2, §2.3 (standard reference, not scraped)