Modules over a Hall algebra are unital algebra representations

Let $R$ be a commutative ring with unit (Algebras over a commutative ring, central structure maps, and algebra homomorphisms) and let $H$ be a unital associative $R$-algebra. An $R$-linear representation of $H$ on an $R$-module $M$ is a unital $R$-algebra homomorphism $\rho :H\to {\mathrm{End}}_R(M)$, where ${\mathrm{End}}_R(M)$ is the ring of $R$-linear maps $M\to M$, with composition as multiplication, ${\mathrm{id}}_M$ as unit, and scalar map $r\mapsto r{\mathrm{id}}_M$. It defines the unital left action $h\cdot m:=\rho (h)(m)$; its module axioms are those of Unital left and right modules over a ring; unqualified module means left module. When $H$ is a Hall algebra, this structure is called a Hall-algebra module.

This term does not mean an object of the input category. An input object $A$ contributes a basis symbol $[A]$ to a Hall algebra, but that symbol by itself specifies no representation on an $R$-module. The untwisted Hall algebra instance is supplied by The untwisted Hall algebra of a finitary abelian category, The untwisted Hall product is associative and unital.

The zero $R$-module is allowed; its identity endomorphism is its zero map. This definition gives an algebra action only. An action on a tensor product requires additional structure, such as a specified coproduct.

For a Green-twisted or completed Hall construction, the underlying associative algebra and its coefficient space must first be specified; this definition does not identify different Hall normalizations or completions.