The Hall algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 16 September 2012

The Hall algebra

A quiver (I,Ω+) is a directed graph with vertex set I and edge set Ω+I×I, with no loops. A nilpotent representation of quiver (I,Ω+) is a pair (V,x) consisting of an I–graded vector space over 𝔽q,

V= iI Vi and an element x (ij) Ω+ Hom(Vi,Vj),

which is nilpotent as an element of End(V). The dimension of V is the vector

dim(V)= (dim(Vi)) iI

in (Z0)I. A morphism ϕHom(V,W) is a map

ϕ iI Hom(Vi,Wi) such thatϕj xij= xijϕi,

for all edges ijΩ+. An extension FExt1(V,W) is an exact sequence


of morphisms of representations.

The following proposition shows that the constants dimHom(M,N) and dimExt1(M,N) depend only on the dimension vectors of M and N.

Let M and N be representations of (I,Ω+) with dimension vectors μ and ν respectively.

  1. dimHom(M,N)= ???,
  2. dimExt1(M,N) =???
  3. Exti(M,N)=0, for alli>1.
  4. χ(M,N)= dim (Hom(M,N))- dim (Ext1(M,N)) =-ijΩ+ μiνj+ iIμiνi.


The Ringel–Hall algebra is the vector space 𝒰- with basis

{ EλEλ 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,


Fμνλ(q) = Card { γλγ νand λγμ } = ( # of submodulesγof λof typeν and cotypeν ) .

The type A case

The indecomposable representations

[i,)k= { 𝔽q , ifiki+-1 , 0 , otherwise ,

of (I,Ω+) 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,Ω+) are identified with multisegments. Let Eν denote the isomorphism class of the representation ν and let Ei=E[i,1). Then

{ Eνν is a multisegment } is a basis of the Hall algebra𝒰-.

Let M and N be representations of (I,Ω+) with dimension vectors μ and ν respectively.

  1. χ(M,N)= dim (Hom(M,N)) -dim (Ext1(M,N)) =-ijΩ+ μiνj+ iIμiνi.
  2. dimHom ( E[j,], E[i,k] ) = { 1 , ifij k , 0 , otherwise ,
  3. dimExt1 ( E[i,k], E[j,] ) = { 1 , ifij k , 0 , otherwise ,


(a) Since χ(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 { mi,mi+1 ,,mj } be a basis of M such that xi(i+1) (nr)=nr+1. Any homomorphism ϕHom(M,N) must have kerϕ being a submodule of M and imϕ being a submodule of N and so the only elements of Hom(M,N) are multiples of the map ϕ:MN given by

ϕ(mr)= { nr , ifjr andirk , 0 , otherwise. ,

  1. Fν,iν+= { q ν+ (>;i] ( 1+q++ qν+(;i]-1 ) , ifν+=ν- (-1;i-1]+ (;i] , 0 , otherwise.
  2. Fi,νν+= { q ν+ [i;<) ( 1+q++ qν+ [i;)-1 ) , ifν+ =ν- [i-1;-1) +[i;) , 0 , otherwise.


(a) Count submodules of type i in ν+ such that the quotient is of type ν. These are 1–dimensional spaces P in ν((;i])i which are not completely contained in ν((>;i])i, i.e. P

  1. dim(P)=1,
  2. Pνi+,
  3. Pν+ ((<;i])i =0,
  4. Pν+ ((>,i])i.

The number of such subspaces is

q#(;i]-1 q-1 - q#(>;i]-1 q-1 = q#(>;i] ( qν+((;i])-1 q-1 ) .

(b) Count submodules of ν+ of type ν such that the quotient is of type i. Choosing such a submodule amounts to choosing a codimension 1 space in the ith graded part of [i;) which is not completely contained in [i;). The number of such subspaces is

(q#[i;)-1) (q#[i;)-q) ( q#[i;) - q#[i;)-2 ) (q#[i;)-1-1) (q#[i;)-1-q) ( q#[i;)-1 - q#[i;)-2 ) - (q#[i;<)-1) (q#[i;<)-q) ( q#[i;<) - q#[i;<)-2 ) (q#[i;<)-1-1) (q#[i;<)-1-q) ( q#[i;<)-1 - q#[i;<)-2 ) = (q#[i;)-1) q-1 - (q#[i;<)-1) q-1 = q#[i;<) (q#[i;)-1) q-1 .

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 λji,i-1.

Let [M] be the class of the representation indexed by the multisegment M. 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 ν+ which are obtained from ν by adding a box of content i to the end of a row of ν, 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.

  1. Let ν+=ν+(,i]- (-1;i-1]. Then

    dimHom(ν+,ν+) -dimHome(ν,ν) = ( dimHom(ν,(,i]) - dimHom(ν,(-1,i-1)) ) + ( dimHom((,i],ν) - dimHom ( (-1,i-1],ν ) ) + dimHom ( (,i], (,i] ) - dimHom ( (,i], (-1,i-1] ) + dimHom ( (-1,i-1], (,i] ) - dimHom ( (-1,i-1], (-1,i-1] ) = ( ν[i,>0)- ν(-1,i-1] ) + (ν(,i]-0) +1-1-0+1 = ν[i,>0)-ν (-1,i-1]+ ν(,i]+1.

    Since dimHom(i,i)=1 and χ(ν,i)=- νi-1+νi= ν[i,>0)-ν (>0,i-1],

    -dimHom(i,i)- dimHom(ν,ν)+ dimHom(ν+,ν+) -χ(ν,i)= ν(,i-1]+ν (,i].

    Thus the coefficient of Mν+ in EνEi is

    q ν(,i-1]+ ν(,i] Fν,iν+ (q-2) = q ν(,i-1]+ ν(,i] q -2ν+ (>;i] ( 1+q-2++ q -2ν+(,i]+2 ) = q ν(,i-1]+ ν(,i] q -2ν [i,>) q -2(ν[i,)+1) q2 ( 1+q2++ q2ν[i,) ) = q ν[i+1,)- ν[i,) ( 1+q2++ q2ν[i,) ) .

For each vertex iI let [ei] be the class of the representation given by

Vi { 𝔽q , ifj=i , 0 , ifji ,

and, for each edge ij in Ω+ let eij be the representation given by

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

Let (I,Ω+) 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.


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] ,


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.

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