The extended Hall algebra and composition subalgebra

Let $\mathcal{A}$ be a finite-field hereditary Hall category and let $K_0=K_0(\mathcal{A})$. Put $([\alpha ],[\beta ])=⟨\alpha ,\beta ⟩+⟨\beta ,\alpha ⟩$. The extended Hall algebra is the vector space $\tilde{H}=\mathbb{Q}(\nu )[K_0]\otimes H_{\mathcal{A},\nu }$ with basis $k_{\alpha }[M]$ and multiplication $(k_{\alpha }[M])(k_{\beta }[N])={\nu }^{-(\beta ,[M])}{k}_{\alpha +\beta }([M]\star [N])\mathrm{.}$ The notation is left-torus normal order: $k_{\alpha }[M]$ is the actual product of the torus and Hall elements. Thus moving $k_{\beta }$ past $[M]$ contributes ${\nu }^{-(\beta ,[M])}$. Equivalently, $k_{\alpha }{k}_{\beta }=k_{\alpha +\beta }$ and $k_{\alpha }[M]k_{\alpha }^{-1}={\nu }^{(\alpha ,[M])}[M]$. Associativity follows from the bilinearity of the symmetric Euler form and the associative Hall product; the toral action is checked in The extended Hall torus action is well-defined ↗.

If $\mathcal{A}$ has a specified set of simple objects $\{S_i\}$, the composition subalgebra is the subalgebra of $H_{\mathcal{A},\nu }$ generated by $[S_i]$. Its extension by all $k_{\alpha }$ is the extended composition subalgebra.