Hall algebras revisited

6 Pages·2005·0.09 MB·English

by  Claus Michael Ringel

Hall algebras revisited
Claus Michael Ringel
Abstract.Foranyfinite-dimensionalalgebraoverafinitefield,thecorresponding
Hallalgebrahasbeenintroducedinordertohandlethepossible filtrationsofmodules
with fixedfactors. For the path algebra of a Dynkindiagram with a fixedorientation,
it has been shown that the Hall algebra satisfies relations which are similar to the
Drinfeld-Jimbo relations defining quantum groups, but they depend on the chosen
orientation. The purpose of this note is to adjust the multiplicationof a Hall algebra
in order to obtain the Drinfeld-Jimbo relations themselves. The additional factor
introduced in our change of multiplication involves the Euler characteristic, in this
way we eliminate the dependence on the orientation.
Givenafinite-dimensionalconnectedhereditaryalgebraAoffiniterepresentation
type, say with Dynkin diagram ∆, the indecomposable A-modules correspond
bijectively to the positive roots of the simple complex Lie-algebra g of type ∆.
Thus, the Grothendieck group G(A,C) of the category of finitely generated A-
modulesrelativetosplitexactsequencesandwithcoefficientsinCmaybeidentified
with n , where g = n ⊕h⊕n is a triangular decomposition of g. Hall algebras
+ − +
have been introduced in order to recover the Lie multiplication on G(A,C) using
the representation theory of A. The Hall algebra H(A,C[q]) is rather similar to the
Drinfeld-Jimbo quantization U (n ) of the universal enveloping algebra U(n ),
q + +
however it depends on the orientation on ∆ given by A. Our aim is to change the
multiplication slightly in order to remove this dependence. We will explain the
change of multiplication dealing with the integral Hall algebra H(A), where A is
any finite-dimensional hereditary k-algebra with center k, with k a finite field.
1. Change of multiplication for graded rings
Let R = R be a graded ring, say with multiplication ·, where G is an
g∈G g
abelian groupL(written additively). Let c be an invertible central element of R of
degree 0, and let α: G×G → Z bea bilinearform. On theunderlying graded group
of R, we define a new multiplication ∗ = ∗ as follows: Given non-zero elements
α
r ∈ R ,s ∈ R , let
g h
r ∗s = cα(g,h)r ·s,
theringobtainedinthiswaywillbedenotedbyR .Itiseasytocheck thatR
[α,c] [α,c]
is again an associative ring, with the same unit element as R. Using induction, one
shows:
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Lemma 1. Let r ∈ R , for 1 ≤ i ≤ n. Then
i g
i
r ∗r ∗···∗r = car ·r ·...·r ,
1 2 m 1 2 m
where a = α(g ,g ).
i<j i j
P
We denote by r∗t = r ∗···∗r the tth power of an element r in R .
[α,c]
2. Hall algebras
Let A be a finite-dimensional hereditary k-algebra, with center k, and let
E ,...,E be the simple A-modules.
1 n
Let K (A) be the Grothendieck group of all finite-dimensional A-modules
0
relative to all exact sequences. For any A-module M, the corresponding element
in K (A) will be denoted by dimM, thus K (A) may be identified with the free
0 0
abelian group with basis dimE ,...,dimE . Let ε be the Euler characteristic on
1 n
K (A), thus given A-modules M ,M ,
0 1 2
ε(dimM ,dimM ) = (−1)tdim Extt(M ,M )
1 2 k 1 2
Xt≥0
= dim Hom(M ,M )−dim Ext1(M ,M ).
k 1 2 k 1 2
Let usassumethatk isafinitefield,letv = |k|.Let H = H(A)⊗Z[v,v−1];it
is a K (A)-graded ring. We consider the ring H∗p= H . We will exhibit a direct
0 [ε,v]
description of H∗ below. Given an A-module M, we denote its isomorphism class
by [M] and the corresponding element in H(A) and in H by u . Let u = u .
[M] i [Ei]
The Hall algebra H∗ may be defined directly as follows: Given A-modules
N ,N ,M, let FM be the number of submodules M′ of M such that M/M′ is
1 2 N N
isomorphic to N ,1w2hereas M′ is isomorphic to N . Let H∗ be the free Z[v,v−1]-
1 2
module with basis (u ) , indexed by the set of isomorphism classes of finite
[M] [M]
A–modules. We define on H∗ a multiplication ∗ by the following rule
u ∗u = vε(dimN1,dimN2) FM u .
[N1] [N2] N1N2 [M]
X
[M]
For any i, let f = dim End(E ). Fix some pair i 6= j, with Ext1(E ,E ) = 0.
i k i i j
Let
a = −dimExt1(E ,E )
ij j i End(Ei)
a = −dim Ext1(E ,E ),
ji End(Ej) j i
thus f a = f a . Let q = v2fi.
i ij j ji i
Recall the Drinfeld-Jimbo relations
n
ρn(q,X,Y) = (−1)t(cid:20)nt (cid:21) q−t(n2−t)XtYXn−t,
Xt=0 q
2
they have been introduced in order to define the quantizations of the universal
enveloping algebras of the semisimple complex Lie algebras, and, more general, of
the Kac-Moody Lie algebras.
Proposition. In H∗, we have
ρ (q ,u ,u ) = 0, and ρ (q ,u ,u ) = 0.
1−aij i i j 1−aji j j i
Proof. We consider a pair i 6= j, with Ext1(E ,E ) = 0. Let
i j
ε = ε(dimE ,dimE ),
ij i j
thus
ε = f , ε = 0, ε = f a = f a .
ii i ij ji i ij j ji
Let m = 1−a . We have to consider products of the form u∗t∗u ∗u∗(m−t), and
ij i j i
Lemma 1 shows that
u∗t ∗u ∗u∗(m−t) = vautu um−t,
i j i i j i
where
m
a = ε +tε +(m−t)ε
ii ij ji
(cid:18)2(cid:19)
m
= f +(m−t)f a
i i ij
(cid:18)2(cid:19)
m(m−1)
= f ( +(m−t)(1−m))
i
2
m
= f (− +tm−t).
i
(cid:18)2(cid:19)
It follows that
m m t
−f t(m−t)+a = f (−tm+t2 − +tm−t) = f (− +2 ).
i i i
(cid:18)2(cid:19) (cid:18)2(cid:19) (cid:18)2(cid:19)
As a consequence
q−t(m2−t)u∗t ∗u ∗u∗(m−t) = q−21(m2)q(2t)utu um−t,
i i j i i i i j i
thus
m
ρ (q ,u ,u ) = (−1)t m q−t(m2−t)u∗t ∗u ∗u∗(m−t)
m i i j (cid:20) t (cid:21) i i j i
Xt=0 qi
m
= (−1)t m q−12(m2)q(2t)utu um−t
(cid:20) t (cid:21) i i i j i
Xt=0 qi
m
= q−21(m2) (−1)t m q(2t)utu um−t,
i (cid:20) t (cid:21) i i j i
Xt=0 qi
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and according to [R3] we know that the latter sum vanishes.
Similarly, let m′ = 1−a . Observe that
ji
u∗(m′−t) ∗u ∗u∗t = vbum′−tu ut,
j i j j i j
with
m′ m′
b = ε +(m′ −t)ε +tε = f (− +tm′ −t),
jj ji ij j
(cid:18) 2 (cid:19) (cid:18) 2 (cid:19)
therefore
ρm′(qj,uj,ui) = (−1)m′ m′ (−1)t(cid:20)mt′(cid:21) qj−t(m2′−t)u∗j(m′−t) ∗ui ∗u∗jt
Xt=0 qj
′
= (−1)m′ m (−1)t m′ q−12(m2′)q(2t)um′−tu ut
(cid:20) t (cid:21) j j j i j
Xt=0 qj
= (−1)m′q−21(m2′) m′ (−1)t m′ q(2t)um′−tu ut,
j (cid:20) t (cid:21) j j i j
Xt=0 qj
and again the latter sum vanishes according to [R3]. This completes the proof.
In a similar way, we may change the multiplication for the generic Hall
algebras, and for the Loewy and composition algebras as defined in [R2,3,4].
3. The Euler characteristic for a quiver
Let Q = (Q ,Q ,s,t) be a quiver, with Q the set of vertices, Q the set of
0 1 0 1
arrows; these arrows are of the form α : s(α) → t(α). If we allow the existence
of cyclic paths, the path algebra kQ will not be finite-dimensional, however the
Hall algebra H(kQ) still is defined provided we assume that there are only finitely
many arrows between any pair of vertices, see [R2]. We consider H(kQ) as a
graded G-ring, where G = ZQ0. Note that dim furnishes a surjective map from
theGrothendieckgroupK (kQ)ofallfinite-dimensionalkQ-modulesmoduloexact
0
sequences onto G, but this map is bijective only in case there are no cyclic paths.
We consider the bilinear form
ε(x,y) = x y − x y ,
i i s(α) t(α)
iX∈Q αX∈Q
0 1
for x,y ∈ G, it satisfies
ε(dimM ,dimM ) = dim Hom(M ,M )−dim Ext1(M ,M ),
1 2 k 1 2 k 1 2
for finite-dimensional kQ-modules M ,M , see [R1]. The quadratic form obtained
1 2
fromtheEulercharacteristicεmaybedescribedalsointermsofalgebraicgeometry:
Givenm ,m ∈ N,letM(m ,m )betheset of(m ×m )-matriceswithentriesink,
1 2 1 2 1 2
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andGl(m )thegroupofinvertible(m ×m )-matrices.Ford ∈ NQ0,theaffinespace
1 1 1
M(d) = M(d ,d ) may be considered as the set of representations M
α∈Q s(α) t(α)
1
withdimLM = dusingfixedvectorspaces;thegroupG(d) = i∈Q Gl(di)operates
0
on M(d) so that the orbits are just the isomorphism classesQ. Then
ε(d,d) = dim G(d)−dim M(d).
k k
One particular quiver should be mentioned explicitly, the quiver with one vertex
and one arrow: its path algebra is the polynomial ring k[T] in one variable, thus
its Hall algebra H(k[T]) is the tensor product of classical Hall algebras (one for
each maximal ideal of k[T]) as studied by Steinitz and Ph. Hall. In this case, the
bilinear form ε is the zero form, thus in forming H∗, the multiplication is not
changed at all.
4. Other bilinear forms
The deviation of the multiplication in the Hall algebra H as compared to
U (n ) was considered by Lusztig in [L1] when he worked with Hall algebras in
q +
ordertoexhibitcanonical bases forU (n ).Theprocess ofchanging multiplication
q +
is implizitly used by Lusztig in [L2], see in particular 10.2. He stresses the cocycle
condition, but does not indicate the nature of the bilinear form. In fact, the
bilinear form he works with differs from the Euler characteristic ε by diagonal
entries. However, bilinear forms α,β which differ only by diagonal entries lead to
isomorphic rings R ,R , as we are going to show.
[α,c] [β,c]
Lemma 2. Let G be a free abelian group with basis e ,...,e . Let α,β be
1 n
bilinear forms on G with values in Z such that α(e ,e ) = β(e ,e ) for all i 6= j.
i j i j
Let R be a G-graded ring, and let c ∈ R be an invertible central element. Then
0
the map η: R → R defined for r ∈ R , g = g e , by
[α,c] [β,c] g i i i
P
g
η(r) = cδ(g)r, with δ(g) = i (β(e ,e )−α(e ,e )),
i i i i
(cid:18)2(cid:19)
Xi
is an isomorphism of rings.
Let us stress that the restriction of η to R is the identity, for all i.
e
i
Proof. Clearly, η is additive, thus, let us consider r ∈ R , s ∈ R where
g h
g = g e , h = h e . We have
i i i i i i
P P
η(r ∗s) = cδ(g+h)r ∗s = cδ(g+h)+α(g,h)r ·s,
α α
η(r)∗η(s) = cδ(g)+δ(h)r ∗s = cδ(g)+δ(h)+β(g,h)r ·s.
β β
The equality
g +h g h
i i i i
− − = g h
i i
(cid:18) 2 (cid:19) (cid:18)2(cid:19) (cid:18)2(cid:19)
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implies that
δ(g +h)−δ(g)−δ(h) = g h (α(e ,e )−β(e ,e ))
i i i i i i
Xi
= α(g,h)−β(g,h)
since we assume that α(e ,e )−β(e ,e ) = 0 for i 6= j.
i j i j
References
[L1] Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Amer. Math.
Soc. 3 (1990), 447-498.
[L2] Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Amer. Math.
Soc. 4 (1991), 365-421.
[R1] Ringel, C.M.: Representations of K-species and bimodules. J. Algebra 41 (1976), 269-302.
[R2] Ringel,C.M.:Hallalgebras.In:TopicsinAlgebra.BanachCenterPubl.26.(1990),433–447.
[R3] Ringel, C.M.: Hall algebras and quantum groups. Inventiones math. 101 (1990), 583–592.
[R4] Ringel, C.M.: From representations of quivers via Hall and Loewy algebras to quantum
groups. In: Proc. Novosibirsk Conf. Algebra 1989 (To appear)
[R5] Ringel, C.M.: Lie algebras arising in representation theory. Proc. Tsukuba Workshop 1990
(to appear).
Fakulta¨t fu¨r Mathematik, Universita¨t, W-4800 Bielefeld 1, Germany
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