The Hall algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 16 September 2012

The Hall algebra

A quiver $(I,{\Omega }^{+})$ is a directed graph with vertex set $I$ and edge set ${\Omega }^{+}\subseteq I\times I,$ with no loops. A nilpotent representation of quiver $(I,{\Omega }^{+})$ is a pair $(V,x)$ consisting of an $I$–graded vector space over ${𝔽}_q,$

$$V=\underset{i\in I}{⨁}{V}_i\text{and an element}x\in \underset{(i\to j)\in {\Omega }^{+}}{⨁}\text{Hom}(V_i,V_j),$$

which is nilpotent as an element of $\text{End}\left(V\right)\text{.}$ The dimension of $V$ is the vector

$$\text{dim}\left(V\right)={\left(\text{dim}\left(V_i\right)\right)}_{i\in I}$$

in ${\left(Z_{\ge 0}\right)}^I\text{.}$ A morphism $\varphi \in \text{Hom}(V,W)$ is a map

$$\varphi \in \underset{i\in I}{⨁}\text{Hom}(V_i,W_i)\text{such that}{\varphi }_j{x}_{i\to j}=x_{i\to j}{\varphi }_i,$$

for all edges $i\to j\in {\Omega }^{+}.$ An extension $F\in {\text{Ext}}^1(V,W)$ is an exact sequence

$$0⟶V⟶F⟶W⟶0$$

of morphisms of representations.

The following proposition shows that the constants $\text{dimHom}(M,N)$ and ${\text{dimExt}}^1(M,N)$ depend only on the dimension vectors of $M$ and $N\text{.}$

Let $M$ and $N$ be representations of $(I,{\Omega }^{+})$ with dimension vectors $\mu$ and $\nu$ respectively.

1. $\text{dimHom}(M,N)=\text{???,}$
2. ${\text{dimExt}}^1(M,N)=\text{???}$
3. ${\text{Ext}}^i(M,N)=0,\text{for all}i>1\text{.}$
4. ${\displaystyle \chi (M,N)=\text{dim}\left(\text{Hom}(M,N)\right)-\text{dim}\left({\text{Ext}}^1(M,N)\right)=-\sum _{i\to j\in {\Omega }^{+}}{\mu }_i{\nu }_j+\sum _{i\in I}{\mu }_i{\nu }_i\text{.}}$

		Proof.
	$\square$

The Ringel–Hall algebra is the vector space ${𝒰}^{-}$ with basis

$$\{E_{\lambda }\mid E_{\lambda }\text{is an isomorphism class of nilpotent representations of}(I,{\Omega }^{+})\}$$

with multiplication given by

$$E_{\mu }{E}_{\nu }=\sum _{\lambda }{q}^{-\text{dimHom}(\mu ,\mu )-\text{dimHom}(\nu ,\nu )+\text{dimHom}(\lambda ,\lambda )-\chi (\mu ,\nu )}{F}_{\mu \nu }^{\lambda }\left(q^{-2}\right)E_{\lambda },$$

where

$$\begin{array}{ccc}{F}_{\mu \nu }^{\lambda }\left(q\right)& =& \text{Card}\{\gamma \subseteq \lambda \mid \gamma \cong \nu \text{and}\raisebox{1ex}{$\lambda $}\!\left/ \!\raisebox{-1ex}{$\gamma $}\right.\cong \mu \}\\ & =& \left(\text{\# of submodules}\gamma \text{of}\lambda \text{of type}\nu \text{and cotype}\nu \right)\text{.}\end{array}$$

The type A case

The indecomposable representations

$${[i,\ell )}_k=\{\begin{array}{ccc}{𝔽}_q,& & \text{if}i\le k\le i+\ell -1,\\ 0,& & \text{otherwise},\end{array}$$

of $(I,{\Omega }^{+})$ are identified with segments. By the analogue of the Krull–Schmidt theorem for representations of quivers every representation is isomorphic to a direct sum of indecomposable representations and so the isomorphism classes of representations of $(I,{\Omega }^{+})$ are identified with multisegments. Let $E_{\nu }$ denote the isomorphism class of the representation $\nu$ and let $E_i=E_{[i,1)}\text{.}$ Then

$$\{E_{\nu }\mid \nu \text{is a multisegment}\}\text{is a basis of the Hall algebra}{𝒰}^{-}\text{.}$$

Let $M$ and $N$ be representations of $(I,{\Omega }^{+})$ with dimension vectors $\mu$ and $\nu$ respectively.

1. $\chi (M,N)=\text{dim}\left(\text{Hom}(M,N)\right)-\text{dim}\left({\text{Ext}}^1(M,N)\right)=-\sum _{i\to j\in {\Omega }^{+}}{\mu }_i{\nu }_j+\sum _{i\in I}{\mu }_i{\nu }_i\text{.}$
2. $\text{dimHom}(E_{[j,\ell ]},E_{[i,k]})=\{\begin{array}{ccc}1,& & \text{if}i\le j\le k\le \ell ,\\ 0,& & \text{otherwise},\end{array}$
3. ${\text{dimExt}}^1(E_{[i,k]},E_{[j,\ell ]})=\{\begin{array}{ccc}1,& & \text{if}i\le j\le k\le \ell ,\\ 0,& & \text{otherwise},\end{array}$

		Proof.
	(a) Since $\chi (M,N)$ is linear in $M$ and linear in $N$ it is sufficient to check that the formula is correct on indecomposable modules. This follows directly from parts (b) and (c). Let $\{m_i,m_{i+1},\dots ,m_j\}$ be a basis of $M$ such that $x_{i\to (i+1)}\left(n_r\right)=n_{r+1}\text{.}$ Any homomorphism $\varphi \in \text{Hom}(M,N)$ must have $\text{ker}\varphi$ being a submodule of $M$ and $\text{im}\varphi$ being a submodule of $N$ and so the only elements of $\text{Hom}(M,N)$ are multiples of the map $\varphi :M\to N$ given by $$\varphi \left(m_r\right)=\{\begin{array}{ccc}{n}_r,& & \text{if}j\le r\le \ell \text{and}i\le r\le k,\\ 0,& & \text{otherwise.},\end{array}$$ $\square$

1. $F_{\nu ,i}^{{\nu }^{+}}=\{\begin{array}{ccc}{q}^{{\nu }^{+}(>\ell ;i]}(1+q+\dots +q^{{\nu }^{+}(\ell ;i]-1}),& & \text{if}{\nu }^{+}=\nu -(\ell -1;i-1]+(\ell ;i],\\ 0,& & \text{otherwise.}\end{array}$
2. $F_{i,\nu }^{{}^{+}\nu }=\{\begin{array}{ccc}{q}^{{}^{+}\nu [i;<\ell )}(1+q+\dots +q^{{}^{+}\nu [i;\ell )-1}),& & \text{if}{}^{+}\nu =\nu -[i-1;\ell -1)+[i;\ell ),\\ 0,& & \text{otherwise.}\end{array}$

		Proof.
	(a) Count submodules of type $i$ in ${\nu }^{+}$ such that the quotient is of type $\nu \text{.}$ These are 1–dimensional spaces $P$ in $\nu {\left((\ge \ell ;i]\right)}_i$ which are not completely contained in $\nu {\left((>\ell ;i]\right)}_i,$ i.e. $P$ $\text{dim}\left(P\right)=1,$ $P\subseteq {\nu }_i^{+},$ $P\cap {\nu }^{+}\left((<\ell ;i]\right)i=0,$ $P⊈{\nu }^{+}\left((>\ell ,i]\right)i\text{.}$ The number of such subspaces is $$\frac{q^{\#(\ge \ell ;i]}-1}{q-1}-\frac{q^{\#(>\ell ;i]}-1}{q-1}=q^{\#(>\ell ;i]}\left(\frac{q^{{\nu }^{+}\left((\ell ;i]\right)}-1}{q-1}\right)\text{.}$$ (b) Count submodules of ${\nu }^{}$ of type $\nu$ such that the quotient is of type $i\text{.}$ Choosing such a submodule amounts to choosing a codimension 1 space in the $i$th graded part of $[i;\le \ell )$ which is not completely contained in $[i;\ell )\text{.}$ The number of such subspaces is $$\begin{array}{cc}\multicolumn{2}{c}{\frac{(q^{\#[i;\le \ell )}-1)(q^{\#[i;\le \ell )}-q)\dots (q^{\#[i;\le \ell )}-q^{\#[i;\le \ell )-2})}{(q^{\#[i;\le \ell )-1}-1)(q^{\#[i;\le \ell )-1}-q)\dots (q^{\#[i;\le \ell )-1}-q^{\#[i;\le \ell )-2})}}\\ & -\frac{(q^{\#[i;<\ell )}-1)(q^{\#[i;<\ell )}-q)\dots (q^{\#[i;<\ell )}-q^{\#[i;<\ell )-2})}{(q^{\#[i;<\ell )-1}-1)(q^{\#[i;<\ell )-1}-q)\dots (q^{\#[i;<\ell )-1}-q^{\#[i;<\ell )-2})}\\ =& \frac{(q^{\#[i;\le \ell )}-1)}{q-1}-\frac{(q^{\#[i;<\ell )}-1)}{q-1}=q^{\#[i;<\ell )}\frac{(q^{\#[i;\ell )}-1)}{q-1}\text{.}\end{array}$$ $\square$

For each multisegment

$$\nu =\left(\begin{array}{ccccccc}{\lambda }_1& & {\lambda }_2& & \dots & & {\lambda }_r\\ d_1& \ge & d_2& \ge & \dots & \ge & d_r\end{array}\right]\text{place}\begin{array}{cc}+1& \text{over each}{\lambda }_j=i-1,\\ -1& \text{over each}{\lambda }_j=i,\\ 0& \text{over each}{\lambda }_j\ne i,i-1\text{.}\end{array}$$

Let $\left[M\right]$ be the class of the representation indexed by the multisegment $M\text{.}$ Then

$$\begin{array}{ccc}{E}_i{E}_{\nu }& =& \sum _{c({\nu }^{+}/\nu )=i}{q}^{(\text{sum of the}\pm 1\text{before}{\nu }^{+}/\nu )}{E}_{{\nu }^{+}}\text{and}\\ \left[M\right]\left[e_i\right]& =& \sum _{[M-]}{q}^{(\text{sum of the labels after}{}^{+}M/M)}\left[{}^{+}M\right],\end{array}$$

where the first sum is over all multisegments ${\nu }^{+}$ which are obtained from $\nu$ by adding a box of content $i$ to the end of a row of $\nu ,$ and the second sum is over all multisegments ${}^{+}M$ obtained from $M$ by adding a box of content $i$ to the beginning of a row of $M\text{.}$

		Proof.
	Let ${\nu }^{+}=\nu +(\ell ,i]-(\ell -1;i-1]\text{.}$ Then $$\begin{array}{ccc}\multicolumn{3}{c}{\text{dimHom}({\nu }^{+},{\nu }^{+})-\text{dimHome}(\nu ,\nu )}\\ & \multicolumn{2}{c}{=(\text{dimHom}(\nu ,(\ell ,i])-\text{dimHom}(\nu ,(\ell -1,i-1)))}\\ & & +(\text{dimHom}((\ell ,i],\nu )-\text{dimHom}((\ell -1,i-1],\nu ))\\ & & +\text{dimHom}((\ell ,i],(\ell ,i])-\text{dimHom}((\ell ,i],(\ell -1,i-1])\\ & & +\text{dimHom}((\ell -1,i-1],(\ell ,i])-\text{dimHom}((\ell -1,i-1],(\ell -1,i-1])\\ & \multicolumn{2}{c}{=(\nu [i,>0)-\nu (\le \ell -1,i-1])+(\nu (\ge \ell ,i]-0)+1-1-0+1}\\ & \multicolumn{2}{c}{=\nu [i,>0)-\nu (\le \ell -1,i-1]+\nu (\ge \ell ,i]+1\text{.}}\end{array}$$ Since $\text{dimHom}(i,i)=1$ and $\chi (\nu ,i)=-{\nu }_{i-1}+{\nu }_i=\nu [i,>0)-\nu (>0,i-1],$ $$-\text{dimHom}(i,i)-\text{dimHom}(\nu ,\nu )+\text{dimHom}({\nu }^{+},{\nu }^{+})-\chi (\nu ,i)=\nu (\ge \ell ,i-1]+\nu (\ge \ell ,i]\text{.}$$ Thus the coefficient of $M_{{\nu }^{+}}$ in $E_{\nu }{E}_i$ is $$\begin{array}{ccc}{q}^{\nu (\ge \ell ,i-1]+\nu (\ge \ell ,i]}{F}_{\nu ,i}^{{\nu }^{+}}\left(q^{-2}\right)& =& q^{\nu (\ge \ell ,i-1]+\nu (\ge \ell ,i]}{q}^{-2{\nu }^{+}(>\ell ;i]}(1+q^{-2}+\dots +q^{-2{\nu }^{+}(\ell ,i]+2})\\ & =& q^{\nu (\ge \ell ,i-1]+\nu (\ge \ell ,i]}{q}^{-2\nu [i,>\ell )}{q}^{-2(\nu [i,\ell )+1)}{q}^2(1+q^2+\dots +q^{2\nu [i,\ell )})\\ & =& q^{\nu [i+1,\ge \ell )-\nu [i,\ge \ell )}(1+q^2+\dots +q^{2\nu [i,\ell )})\text{.}\end{array}$$ $\square$

For each vertex $i\in I$ let $\left[e_i\right]$ be the class of the representation given by

$$V_i\{\begin{array}{ccc}{𝔽}_q,& & \text{if}j=i,\\ 0,& & \text{if}j\ne i,\end{array}$$

and, for each edge $i\to j$ in ${\Omega }^{+}$ let $e_{ij}$ be the representation given by

$$V_k=\{\begin{array}{ccc}{𝔽}_q,& & \text{if}k=i\text{or}k=j,\\ 0,& & \text{otherwise}\end{array}\text{and}{x}_{ij}={\text{id}}_{{𝔽}_q}\text{.}$$

Let $(I,{\Omega }^{+})$ be a type $A$ quiver with the canonical orientation. Then

$$\begin{array}{cc}{E}_i^2{E}_{i+1}-(q+q^{-1})E_i{E}_{i+1}{E}_i+E_{i+1}{E}_i^2=0& \text{and}\\ E_{i+1}^2{E}_i-(q+q^{-1})E_i{E}_{i+1}{E}_i+E_i{E}_{i+1}^2=0,\end{array}$$

in the Hall algebra.

		Proof.
	Using Proposition ??? we get $$\begin{array}{ccc}{E}_i^2& =& q^{-1}{E}_{2(1,i]},\\ E_{i=1}^2& =& q^{-1}{E}_{2(1,i+1]},\end{array}\text{and}\begin{array}{ccc}{E}_i{E}_{i+1}& =& q{E}_{(1,i+1]+(1,i]}+E_{(2,i+1]},\\ E_{i+1}{E}_i& =& E_{(1,i]+(1,i+1]},\end{array}$$ and $$\begin{array}{ccc}{E}_i^2{E}_{i+1}& =& q{E}_{(1,i+1]+2(1,i]}+q{E}_{(2,i+1]+(1,i]}+q^{-1}{E}_{(2,i+1]+(1,i]},\\ E_i{E}_{i+1}{E}_i& =& E_{(1,i+1]+2(1,i]}+E_{(2,i+1]+(1,i]},\\ E_i{E}_{i+1}^2& =& q^{-1}{E}_{(1,i+1]+2(1,i]}\text{.}\end{array}$$ The result follows. The calculation for the other case is similar. $\square$

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