## The Hall algebra

Last update: 16 September 2012

## The Hall algebra

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

$V= ⨁ i∈I Vi and an element x∈ ⨁ (i→j) ∈Ω+ Hom(Vi,Vj),$

which is nilpotent as an element of $\text{End}\phantom{\rule{0.2em}{0ex}}\left(V\right)\text{.}$ The dimension of $V$ is the vector

$dim(V)= (dim(Vi)) i∈I$

in ${\left({Z}_{\ge 0}\right)}^{I}\text{.}$ A morphism $\varphi \in \text{Hom}\phantom{\rule{0.2em}{0ex}}\left(V,W\right)$ is a map

$ϕ∈ ⨁ i∈I Hom(Vi,Wi) such thatϕj xi→j= xi→jϕi,$

for all edges $i\to j\in {\Omega }^{+}.$ An extension $F\in {\text{Ext}}^{1}\phantom{\rule{0.2em}{0ex}}\left(V,W\right)$ is an exact sequence

$0⟶V⟶F⟶W⟶0$

of morphisms of representations.

The following proposition shows that the constants $\text{dimHom}\phantom{\rule{0.2em}{0ex}}\left(M,N\right)$ and ${\text{dimExt}}^{1}\phantom{\rule{0.2em}{0ex}}\left(M,N\right)$ depend only on the dimension vectors of $M$ and $N\text{.}$

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

1. $\text{dimHom}\phantom{\rule{0.2em}{0ex}}\left(M,N\right)=\text{???,}$
2. ${\text{dimExt}}^{1}\phantom{\rule{0.2em}{0ex}}\left(M,N\right)=\text{???}$
3. ${\text{Ext}}^{i}\phantom{\rule{0.2em}{0ex}}\left(M,N\right)=0,\phantom{\rule{0.2em}{0ex}}\text{for all}\phantom{\rule{0.2em}{0ex}}i>1\text{.}$
4. $\chi \left(M,N\right)=\text{dim}\phantom{\rule{0.2em}{0ex}}\left(\text{Hom}\phantom{\rule{0.2em}{0ex}}\left(M,N\right)\right)-\text{dim}\phantom{\rule{0.2em}{0ex}}\left({\text{Ext}}^{1}\phantom{\rule{0.2em}{0ex}}\left(M,N\right)\right)=-\sum _{i\to j\in {\Omega }^{+}}{\mu }_{i}{\nu }_{j}+\sum _{i\in I}{\mu }_{i}{\nu }_{i}\text{.}$

 Proof. $\square$

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

${ Eλ∣Eλ is an isomorphism class of nilpotent representations of (I,Ω+) }$

with multiplication given by

$EμEν=∑λ q -dimHom(μ,μ)- dimHom(ν,ν)+ dimHom(λ,λ)- χ(μ,ν) Fμνλ (q-2)Eλ,$

where

$Fμνλ(q) = Card { γ⊆λ∣γ ≅νand λγ≅μ } = ( # of submodulesγof λof typeν and cotypeν ) .$

### The type A case

The indecomposable representations

$[i,ℓ)k= { 𝔽q , ifi≤k≤i+ℓ-1 , 0 , otherwise ,$

of $\left(I,{\Omega }^{+}\right)$ 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 $\left(I,{\Omega }^{+}\right)$ are identified with multisegments. Let ${E}_{\nu }$ denote the isomorphism class of the representation $\nu$ and let ${E}_{i}={E}_{\left[i,1\right)}\text{.}$ Then

${ Eν∣ν is a multisegment } is a basis of the Hall algebra𝒰-.$

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

1. $\chi \left(M,N\right)=\text{dim}\phantom{\rule{0.2em}{0ex}}\left(\text{Hom}\phantom{\rule{0.2em}{0ex}}\left(M,N\right)\right)-\text{dim}\phantom{\rule{0.2em}{0ex}}\left({\text{Ext}}^{1}\phantom{\rule{0.2em}{0ex}}\left(M,N\right)\right)=-\sum _{i\to j\in {\Omega }^{+}}{\mu }_{i}{\nu }_{j}+\sum _{i\in I}{\mu }_{i}{\nu }_{i}\text{.}$
2. $\text{dimHom}\phantom{\rule{0.2em}{0ex}}\left({E}_{\left[j,\ell \right]},{E}_{\left[i,k\right]}\right)=\left\{\begin{array}{ccc}1,& & \text{if}\phantom{\rule{0.2em}{0ex}}i\le j\le k\le \ell ,\\ 0,& & \text{otherwise},\end{array}$
3. ${\text{dimExt}}^{1}\phantom{\rule{0.2em}{0ex}}\left({E}_{\left[i,k\right]},{E}_{\left[j,\ell \right]}\right)=\left\{\begin{array}{ccc}1,& & \text{if}\phantom{\rule{0.2em}{0ex}}i\le j\le k\le \ell ,\\ 0,& & \text{otherwise},\end{array}$

 Proof. (a) Since $\chi \left(M,N\right)$ 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 $\left\{{m}_{i},{m}_{i+1},\dots ,{m}_{j}\right\}$ be a basis of $M$ such that ${x}_{i\to \left(i+1\right)}\left({n}_{r}\right)={n}_{r+1}\text{.}$ Any homomorphism $\varphi \in \text{Hom}\phantom{\rule{0.2em}{0ex}}\left(M,N\right)$ must have $\text{ker}\phantom{\rule{0.2em}{0ex}}\varphi$ being a submodule of $M$ and $\text{im}\phantom{\rule{0.2em}{0ex}}\varphi$ being a submodule of $N$ and so the only elements of $\text{Hom}\phantom{\rule{0.2em}{0ex}}\left(M,N\right)$ are multiples of the map $\varphi :\phantom{\rule{0.2em}{0ex}}M\to N$ given by $ϕ(mr)= { nr , ifj≤r≤ℓ andi≤r≤k , 0 , otherwise. ,$ $\square$

1. ${F}_{\nu ,i}^{{\nu }^{+}}=\left\{\begin{array}{ccc}{q}^{{\nu }^{+}\left(>\ell ;i\right]}\left(1+q+\dots +{q}^{{\nu }^{+}\left(\ell ;i\right]-1}\right),& & \text{if}\phantom{\rule{0.2em}{0ex}}{\nu }^{+}=\nu -\left(\ell -1;i-1\right]+\left(\ell ;i\right],\\ 0,& & \text{otherwise.}\end{array}$
2. ${F}_{i,\nu }^{{}^{+}\nu }=\left\{\begin{array}{ccc}{q}^{{}^{+}\nu \left[i;<\ell \right)}\left(1+q+\dots +{q}^{{}^{+}\nu \left[i;\ell \right)-1}\right),& & \text{if}\phantom{\rule{0.2em}{0ex}}{}^{+}\nu =\nu -\left[i-1;\ell -1\right)+\left[i;\ell \right),\\ 0,& & \text{otherwise.}\end{array}$

 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 $\nu {\left(\left(\ge \ell ;i\right]\right)}_{i}$ which are not completely contained in $\nu {\left(\left(>\ell ;i\right]\right)}_{i},$ i.e. $P$ $\text{dim}\phantom{\rule{0.2em}{0ex}}\left(P\right)=1,$ $P\subseteq {\nu }_{i}^{+},$ $P\cap {\nu }^{+}\left(\left(<\ell ;i\right]\right)i=0,$ $P⊈{\nu }^{+}\left(\left(>\ell ,i\right]\right)i\text{.}$ The number of such subspaces is $q#(≥ℓ;i]-1 q-1 - q#(>ℓ;i]-1 q-1 = q#(>ℓ;i] ( qν+((ℓ;i])-1 q-1 ) .$ (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 $\left[i;\le \ell \right)$ which is not completely contained in $\left[i;\ell \right)\text{.}$ The number of such subspaces is $(q#[i;≤ℓ)-1) (q#[i;≤ℓ)-q) … ( q#[i;≤ℓ) - q#[i;≤ℓ)-2 ) (q#[i;≤ℓ)-1-1) (q#[i;≤ℓ)-1-q) … ( q#[i;≤ℓ)-1 - q#[i;≤ℓ)-2 ) - (q#[i;<ℓ)-1) (q#[i;<ℓ)-q) … ( q#[i;<ℓ) - q#[i;<ℓ)-2 ) (q#[i;<ℓ)-1-1) (q#[i;<ℓ)-1-q) … ( q#[i;<ℓ)-1 - q#[i;<ℓ)-2 ) = (q#[i;≤ℓ)-1) q-1 - (q#[i;<ℓ)-1) q-1 = q#[i;<ℓ) (q#[i;ℓ)-1) q-1 .$ $\square$

For each multisegment

$ν= ( λ1 λ2 … λr d1 ≥ d2 ≥ … ≥ dr ] place +1 over each λj=i-1, -1 over each λj=i, 0 over each λj≠i,i-1.$

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

$EiEν = ∑c(ν+/ν)=i q ( sum of the±1 beforeν+/ν ) Eν+ and [M][ei] = ∑[M-] q ( sum of the labels after M+/M ) [M+],$

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 ${\nu }^{+}=\nu +\left(\ell ,i\right]-\left(\ell -1;i-1\right]\text{.}$ Then $dimHom(ν+,ν+) -dimHome(ν,ν) = ( dimHom(ν,(ℓ,i]) - dimHom(ν,(ℓ-1,i-1)) ) + ( dimHom((ℓ,i],ν) - dimHom ( (ℓ-1,i-1],ν ) ) + dimHom ( (ℓ,i], (ℓ,i] ) - dimHom ( (ℓ,i], (ℓ-1,i-1] ) + dimHom ( (ℓ-1,i-1], (ℓ,i] ) - dimHom ( (ℓ-1,i-1], (ℓ-1,i-1] ) = ( ν[i,>0)- ν(≤ℓ-1,i-1] ) + (ν(≥ℓ,i]-0) +1-1-0+1 = ν[i,>0)-ν (≤ℓ-1,i-1]+ ν(≥ℓ,i]+1.$ Since $\text{dimHom}\phantom{\rule{0.2em}{0ex}}\left(i,i\right)=1$ and $\chi \left(\nu ,i\right)=-{\nu }_{i-1}+{\nu }_{i}=\nu \left[i,>0\right)-\nu \left(>0,i-1\right],$ $-dimHom(i,i)- dimHom(ν,ν)+ dimHom(ν+,ν+) -χ(ν,i)= ν(≥ℓ,i-1]+ν (≥ℓ,i].$ Thus the coefficient of ${M}_{{\nu }^{+}}$ in ${E}_{\nu }{E}_{i}$ is $q ν(≥ℓ,i-1]+ ν(≥ℓ,i] Fν,iν+ (q-2) = q ν(≥ℓ,i-1]+ ν(≥ℓ,i] q -2ν+ (>ℓ;i] ( 1+q-2+…+ q -2ν+(ℓ,i]+2 ) = q ν(≥ℓ,i-1]+ ν(≥ℓ,i] q -2ν [i,>ℓ) q -2(ν[i,ℓ)+1) q2 ( 1+q2+…+ q2ν[i,ℓ) ) = q ν[i+1,≥ℓ)- ν[i,≥ℓ) ( 1+q2+…+ q2ν[i,ℓ) ) .$ $\square$

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

$Vi { 𝔽q , ifj=i , 0 , ifj≠i ,$

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

$Vk= { 𝔽q , ifk=ior k=j , 0 , otherwise andxij= id𝔽q.$

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

$Ei2Ei+1- (q+q-1) EiEi+1Ei+ Ei+1Ei2=0 and Ei+12Ei- (q+q-1)Ei Ei+1Ei+Ei Ei+12=0,$

in the Hall algebra.

 Proof. Using Proposition ??? we get $Ei2 = q-1 E2(1,i], Ei=12 = q-1 E2(1,i+1], and EiEi+1 = q E (1,i+1] +(1,i] + E(2,i+1], Ei+1Ei = E (1,i] +(1,i+1] ,$ and $Ei2Ei+1 = q E (1,i+1] +2(1,i] + q E (2,i+1] +(1,i] +q-1 E (2,i+1] +(1,i] , EiEi+1Ei = E (1,i+1] +2(1,i] + E (2,i+1] +(1,i] , EiEi+12 = q-1 E (1,i+1] +2(1,i] .$ The result follows. The calculation for the other case is similar. $\square$