Quiver Hecke algebras and 2-Lie algebras

48 Pages·2012·0.596 MB·English

by  Raphael Rouquier

QUIVER HECKE ALGEBRAS AND 2-LIE ALGEBRAS
RAPHAE¨L ROUQUIER
Contents
1. Introduction 1
2. One vertex quiver Hecke algebras 3
2.1. Nil Hecke algebras 3
2.2. Nil affine Hecke algebras 6
2.3. Symmetrizing forms 8
3. Quiver Hecke algebras 11
3.1. Representations of quivers 11
3.2. Quiver Hecke algebras 12
3.3. Half 2-Kac-Moody algebras 18
4. 2-Kac-Moody algebras 20
4.1. Kac-Moody algebras 20
4.2. 2-categories 24
4.3. 2-representation theory 26
4.4. Cyclotomic quiver Hecke algebras 31
5. Geometry 35
5.1. Hall algebras 35
5.2. Functions on moduli stacks of representations of quivers 37
5.3. Flag varieties 38
5.4. Quiver Hecke algebras and geometry 43
5.5. 2-Representations 46
References 47
Abstract. Weprovideanintroductiontothe2-representationtheoryofKac-Moodyalgebras,
starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing
with the resulting monoidal categories, which have a geometric description via quiver vari-
eties, in certain cases. We present basic properties of 2-representations and describe simple
2-representations, via cyclotomic quiver Hecke algebras, and through microlocalized quiver va-
rieties.
1. Introduction
This text provides an introduction and complements to some basic constructions and results
in 2-representation theory of Kac-Moody algebras. We discuss quiver Hecke algebras [Rou2],
which have been introduced independently by Khovanov and Lauda [KhoLau1] and [KhoLau2],
Date: 9 December 2011.
Supported by EPSRC, Project No EP/F065787/1.
1
2 RAPHAE¨L ROUQUIER
and their cyclotomic versions, which have been considered independently in the case of level 2
weights for type A, by Brundan and Stroppel [BrStr]. We discuss the 2-categories associated
with Kac-Moody algebras and their 2-representations: this has been introduced in joint work
with Joe Chuang [ChRou] for sl and implicitly for type A (finite or affine). While the general
2
philosophy of categorifications was older (cf for example [BeFrKho]), the new idea in [ChRou]
was to introduce some structure at the level of natural transformations: an endomorphism of E
and an endomorphism of E2 satisfying Hecke-type relations. The generalization to other types
is based on quiver Hecke algebras, which account for a half Kac-Moody algebra. We discuss
the geometrical construction of the quiver Hecke algebras via quiver varieties, which was our
starting point for the definition of quiver Hecke algebras, and that of cyclotomic quiver Hecke
algebras.
The first chapter gives a gentle introduction to nil (affine) Hecke algebras of type A. We
recall basic properties of Hecke algebras of symmetric groups and provide the construction via
BGG-Demazure operators of the nil Hecke algebras. We also construct symmetrizing forms.
The second chapter is devoted to quiver Hecke algebras. We explain that the more com-
plicated relation involved in the definition is actually a consequence of the other ones, up
to polynomial torsion: this leads to a new, simpler, definition of quiver Hecke algebras. We
construct next the faithful polynomial representation. This generalizes the constructions of
the first chapter, that correspond to a quiver with one vertex. We explain the relation, for
type A quivers, with affine Hecke algebras. Finally, we explain how to put together all quiver
Hecke algebras associated with a quiver to obtain a monoidal category that categorifies a half
Kac-Moody algebra (and its quantum version).
The third chapter introduces 2-categories associated with Kac-Moody algebras and discusses
their integrable representations. We provide various results that reduce the amount of condi-
tions to check that a category is endowed with a structure of an integrable 2-representation,
once the quiver Hecke relations hold: for example, the sl -relations imply all other relations,
2
and it can be enough to check them on K . We explain the universal construction of “simple”
0
2-representations, and give a detailed description for sl . We present a Jordan-Ho¨lder type
2
result. We move next to cyclotomic quiver Hecke algebras, and present Kang-Kashiwara and
Webster’sconstructionof2-representationsoncyclotomicquiverHeckealgebras. Weprovethat
the 2-representation is equivalent to the universal simple 2-representation. Finally, we explain
the construction of Fock spaces from representations of symmetric groups in this framework.
ThelastchapterbringsingeometricalmethodsavailableinthecaseofsymmetricKac-Moody
algebras. We start with a brief recollection of Ringel’s construction of quantum groups via Hall
algebrasandLusztig’s construction ofenveloping algebrasviaconstructiblefunctions. We move
next to the construction of nil affine Hecke algebras in the cohomology of flag varieties. We
introduce Lusztig’s category of perverse sheaves on the moduli space of representations of a
quiver and show that it is equivalent to the monoidal category of quiver Hecke algebras (a result
obtained independently by Varagnolo and Vasserot). As a consequence, the indecomposable
projective modules for quiver Hecke algebras over a field of characteristic 0, and for “geometric”
parameters, correspond to the canonical basis. Finally, we show that Zheng’s microlocalized
categories of sheaves can be endowed with a structure of 2-representation isomorphic to the
universalsimple2-representation. Asaconsequence, theindecomposableprojectivemodulesfor
cyclotomicquiverHeckealgebrasoverafieldofcharacteristic0,andfor“geometric”parameters,
correspond to the canonical basis of simple representations.
QUIVER HECKE ALGEBRAS AND 2-LIE ALGEBRAS 3
This article is based on a series of lectures at the National Taiwan University, Taipei, in
December 2008 and a series of lectures at BICMR, Peking University, in March–April 2010. I
wish to thank Professors Shun-Jen Cheng and Weiqiang Wang, and Professor Jiping Zhang for
their invitations to give these lecture series.
2. One vertex quiver Hecke algebras
The results of this section are classical (cf for example [Rou2, §3]).
2.1. Nil Hecke algebras.
2.1.1. The symmetric group as a Weyl group. Let n ≥ 1. Given i ∈ {1,...,n − 1}, we put
s = (i,i+1) ∈ S .
i n
We define a function r : S → Z . Given w ∈ S , let R = {(i,j)|i < j and w(i) > w(j)}
n ≥0 n w
and let r(w) = |R(w)| be the number of inversions.
The length l(w) of w ∈ S is the minimal integer r such that there exists i ,...,i with
n 1 r
w = s ···s . Such an expression is called a reduced decomposition of w. Proposition 2.1 says
i1 ir
that since s ,...,s generate S , these notions make sense.
1 n−1 n
Note that reduced decompositions are not unique: we have for example (13) = s s s =
1 2 1
s s s . Simpler is s s = s s .
2 1 2 1 3 3 1
Proposition 2.1. The set {s ,...,s } generates S . Given w ∈ S , we have r(w) = l(w).
1 n−1 n n
Proof. Let w ∈ S , w (cid:54)= 1. Note that R (cid:54)=∅. Consider (i,j) ∈ R such that j −i is minimal.
n w w
Assume j (cid:54)= i+1. By the minimality assumption, (i,i+1)(cid:54)∈R and (i+1,j)(cid:54)∈R , so w(j) >
w w
w(i+1) > w(i), a contradiction. So, j = i+1. Let w(cid:48) = ws . We have R = R −{(i,i+1)},
i w(cid:48) w
hence r(w(cid:48)) = r(w)−1. We deduce by induction that there exist i ,...,i ∈ {1,...,n−1}
1 r(w)
such that w = s ···s . In particular, the set {s ,...,s } generates S and l(g) ≤ r(g)
ir(w) i1 1 n−1 n
for all g ∈ S .
n
Let j ∈ {1,...,n} and v = ws . Assume (j,j + 1)(cid:54)∈R . Then, R = R ∪ {(j,j + 1)}. It
j w v w
follows that r(v) = r(w)+1. If (j,j+1) ∈ R , then r(v) = r(w)−1. We deduce by induction
w
that l(g) ≥ r(g) for all g ∈ S . (cid:3)
n
Proposition 2.2. The element w[1,n] = (1,n)(2,n−1)(3,n−2)··· is the unique element of
S with maximal length. We have l(w[1,n]) = n(n−1).
n 2
Proof. Note that R = {(i,j)|i < j} and this contains any set R for w ∈ S , with equality
w[1,n] w n
if and only if w = w[1,n]. The result follows from Proposition 2.1. (cid:3)
The set
C = {1,s = (n−1,n),s s = (n−2,n−1,n),...,s ···s s = (1,2,...,n)}
n n−1 n−2 n−1 1 n−2 n−1
is a complete set of representatives for left cosets S /S . Let w ∈ S and g ∈ C . We
n n−1 n−1 n
(cid:96)
have R(gw) = R(w) R(g), so l(gw) = l(g)+l(w). Consider now w ∈ S . There is a unique
n
decompositionw = c c ···c wherec ∈ C andwehavel(w) = l(c )+···+l(c ). Eachc has
n n−1 2 i i n 2 i
a unique reduced decomposition and that provides us with a canonical reduced decomposition
of w:
w = (s s ···s )(s s ···s )···(s s ···s )
j1 j1+1 i1 j2 j2+1 i2 jr jr+1 ir
where i > i > ··· > i and 1 ≤ j < i .
1 2 r r r
4 RAPHAE¨L ROUQUIER
In the case of the longest element, we obtain
w[1,n] = (s ···s )(s ···s )···s = (s ···s )w[1,n−1].
1 n−1 1 n−2 1 1 n−1
Using canonical reduced decompositions, we can count the number of elements with a given
length and deduce the following result.
Proposition 2.3. We have (cid:80) ql(w) = (1−q)(1−q2)···(1−qn).
w∈Sn (1−q)n
Lemma 2.4. Let w ∈ S . Then, l(w−1) = l(w) and l(w[1,n]w−1) = l(w[1,n])−l(w).
n
Proof. The first statement is clear, since w = s ···s is a reduced expression if and only if
i1 ir
w−1 = s ···s is a reduced expression.
ir i1
We have R = {(i,j)|(i < j) and w(i) < w(j)}. The second statement follows. (cid:3)
w[1,n]w
We recall the following classical result.
Proposition 2.5. The group S has a presentation with generators s ,...,s and relations
n 1 n−1
s s = s s if |i−j| > 1 and s s s = s s s .
i j j i i i+1 i i+1 i i+1
2.1.2. Finite Hecke algebras. Let us recall some classical results about Hecke algebras of sym-
metric groups.
Let R = Z[q ,q ]. Let Hf be the Hecke algebra of S : this is the R-algebra generated by
1 2 n n
T ,...,T , with relations
1 n−1
T T T = T T T , T T = T T if |i−j| > 1 and (T −q )(T −q ) = 0.
i i+1 i i+1 i i+1 i j j i i 1 i 2
There is an isomorphism of algebras
Hf ⊗ R/(q −1,q +1) →∼ Z[S ], T (cid:55)→ s .
n R 1 2 n i i
Let w ∈ S with a reduced decomposition w = s ···s . We put T = T ···T ∈ Hf. One
n i1 ir w i1 ir n
shows that T is independent of the choice of a reduced decomposition of w and that {T }
w w w∈Sn
is an R-basis of the free R-module Hf.
n
Given w,w(cid:48) ∈ S with l(ww(cid:48)) = l(w)+l(w(cid:48)), we have T T = T .
n w w(cid:48) ww(cid:48)
The algebra Hf is a deformation of Z[S ]. At the specialization q = 1, q = −1, the element
n n 1 2
T becomes the group element w.
w
2.1.3. Nil Hecke algebras of type A. Let 0Hf = Hf ⊗ R/(q ,q ). Given w,w(cid:48) ∈ S , we have
n n R 1 2 n
(cid:40)
T if l(ww(cid:48)) = l(w)+l(w(cid:48))
ww(cid:48)
T T =
w w(cid:48)
0 otherwise.
So, the algebra 0Hf is graded with degT = −2l(w). The choice of a negative sign will become
n w
clear soon. The factor 2 comes from the cohomological interpretation.
(cid:76)
Given M = M a graded Z-module and r ∈ Z, we denote by M(cid:104)r(cid:105) the graded module
i∈Z i
given by (M(cid:104)r(cid:105)) = M .
i i+r
We have (0Hf) = (cid:76) ZT . So, (0Hf) = 0 unless i ∈ {0,−2,...,−n(n−1)}.
n i w∈Sn,l(w)=−i/2 w n i
Let k be a field and k0Hf = 0Hf ⊗ k.
n n Z
QUIVER HECKE ALGEBRAS AND 2-LIE ALGEBRAS 5
Proposition 2.6. The Jacobson radical of k0Hf is rad(k0Hf) = (cid:76) kT and k0Hf has
n n w(cid:54)=1 w n
a unique minimal non-zero two-sided ideal soc(k0Hf) = kT . The trivial module k, with
n w[1,n]
0-action of the T ’s, is the unique simple k0Hf-module.
i n
Proof. Let A = k0Hf. Let J = A = (cid:76) kT . We have Jn(n−1)+1 = 0. So, J is a nilpotent
n <0 w(cid:54)=1 w
two-sided ideal of A and A/J (cid:39) k. It follows that J = rad(A): the algebra A is local and k is
the unique simple module.
(cid:80)
Let M be a non-zero left ideal of A. Let m = α T ∈ M be a non-zero element.
w w w
Consider w ∈ S of minimal length such that α (cid:54)= 0. We have T m = α T
n w w[1,n]w−1 w w[1,n]
because T T = 0 if l(w(cid:48)) ≥ l(w) and w(cid:48) (cid:54)= w. It follows that kT ⊂ M. That shows
w[1,n]w−1 w(cid:48) w[1,n]
that kT is the unique minimal non-zero left ideal of A. A similar proof shows it is also the
w[1,n]
unique minimal non-zero right ideal. (cid:3)
Remark 2.7. Let k be a field and A be a finite dimensional graded k-algebra. Assume A = k
0
and A = 0 for i > 0. Then rad(A) = A .
i <0
2.1.4. BGG-Demazure operators. We refer to [Hi, Chapter IV] for a general discussion of the
results below.
Let P = Z[X ,...,X ]. We let S act on P by permutation of the X ’s. We define an
n 1 n n n i
endomorphism of abelian groups ∂ ∈ End (P ) by
i Z n
P −s (P)
i
∂ (P) = .
i
X −X
i+1 i
Note that the operators ∂ are PSn-linear. Note also that im∂ ⊂ Psi = ker∂ .
w n i n i
The following lemma follows from easy calculations.
Lemma 2.8. We have ∂2 = 0, ∂ ∂ = ∂ ∂ for |i−j| > 1 and ∂ ∂ ∂ = ∂ ∂ ∂ .
i i j j i i i+1 i i+1 i i+1
We deduce that we have obtained a representation of the nil Hecke algebra.
Proposition 2.9. The assignment T (cid:55)→ ∂ defines a representation of 0Hf on P .
i i n n
Define a grading of the algebra P by degX = 2. Then, the representation above is com-
n i
patible with the gradings.
Given w ∈ S , we denote by ∂ the image of T .
n w w
The following result is clear.
Lemma 2.10. Let P ∈ P . We have ∂ (P) = 0 for all i if and only if P ∈ PSn.
n i n
If M is a free graded module over a commutative ring k with dim M < ∞ for all i ∈ Z, we
k i
put grdim(M) = (cid:80) qi/2dim(M ).
i∈Z i
Theorem 2.11. The set {∂ (X X2···Xn−1)} is a basis of P over PSn.
w 2 3 n w∈Sn n n
Proof. Let us show by induction on n that
∂ (X X2···Xn−1) = 1.
w[1,n] 2 3 n
We have w[1,n] = s ···s w[2,n] and l(w[1,n]) = l(w[2,n])+n−1. By induction,
n−1 1
∂ (X X2···Xn−1) = X ···X ·∂ (X ···Xn−2) = X ···X .
w[2,n] 2 3 n 2 n w[2,n] 3 n 2 n
6 RAPHAE¨L ROUQUIER
On the other hand, ∂ ···∂ (X ···X ) = 1 and we deduce that ∂ (X X2···Xn−1) = 1.
n−1 1 2 n w[1,n] 2 3 n
Let M be a free PSn-module with basis {b } , with degb = 2l(w[1,n]w−1) = n(n −
n w w∈Sn w
1)−2l(w). Define a morphism of PSn-modules
n
φ : M → P , b (cid:55)→ ∂ (X X2···Xn−1).
n w w 2 3 n
This is a graded morphism.
Let k be a field. Let a = (cid:80) Q b ∈ ker(φ ⊗ k), where Q ∈ k[X ,...,X ]Sn. Assume
w w w w 1 n
a (cid:54)= 0. Consider v ∈ S with Q (cid:54)= 0 and such that l(v) is minimal with this property. We
n v
have ∂ (φ(a)) = Q , hence we have a contradiction. It follows that φ⊗k is injective.
w[1,n]v−1 v
We have grdimP = (1−q)−n. On the other hand, we have PSn = Z[e ,...,e ], where e =
n n 1 n r
e (X ,...,X ) = (cid:80) X ···X . So, grdimPSn = (1−q)−1(1−q2)−1···(1−qn)−1.
r 1 n 1≤i1<···<ir≤n i1 ir n
We deduce that
(cid:88)
grdimM = (1−q)−1···(1−qn)−1 ql(w).
w∈Sn
The formula of Proposition 2.3 shows that grdimM = grdimP . Lemma 2.13 shows that φ ⊗k
n i
is an isomorphism and then Lemma 2.12 shows that φ is an isomorphism for all i. (cid:3)
i
The following two lemmas are clear.
Lemma 2.12. Let f : M → N be a morphism between free finitely generated Z-modules. If
f ⊗ (Z/p) is surjective for all prime p, then f is surjective.
Z
Lemma 2.13. Let k be a field and M,N be two graded k-modules with dimM = dimN finite
i i
for all i. If f : M → N is an injective morphism of graded k-modules, then f is an isomorphism.
Remark 2.14. Note that {Xa2···Xan} is the more classical basis of P over PSn.
2 n 0≤ai≤i−1 n n
2.2. Nil affine Hecke algebras.
2.2.1. Definition. Let 0H be the nil affine Hecke algebra of GL : this is the Z-algebra with
n n
generators X ,...,X ,T ,...,T and relations
1 n 1 n−1
X X = X X , T2 = 0, T T T = T T T , T T = T T if |i−j| > 1,
i j j i i i i+1 i i+1 i i+1 i j j i
T X = X T if j −i (cid:54)= 0,1, T X −X T = 1 and T X −X T = −1.
i j j i i i+1 i i i i i+1 i
It is a graded algebra, with degX = 2 and degT = −2.
i i
The following lemma is easy.
Lemma 2.15. Given P,Q ∈ P , we have ∂ (PQ) = ∂ (P)Q+s (P)∂ (Q).
n i i i i
Lemma 2.15 is the key ingredient to prove the following lemma.
Lemma 2.16. We have a representation ρ of 0H on P given by
n n
ρ(T )(P) = ∂ (P) and ρ(X )(P) = X P.
i i i i
Proposition 2.17. We have a decomposition 0H = P ⊗0Hf as a Z-module and the repre-
n n n
sentation of 0H on P is faithful.
n n
QUIVER HECKE ALGEBRAS AND 2-LIE ALGEBRAS 7
(cid:80)
Proof. Let {P } be a family of elements of P . Let a = P T . If a (cid:54)= 0, there is
w w∈Sn n w w w
w ∈ S of minimal length such that P (cid:54)= 0. We have aT = P T and
n w w−1w[1,n] w w[1,n]
ρ(a)(∂ (X ···Xn−1)) = P ∂ (X ···Xn−1) = P
w−1w[1,n] 2 n w w[1,n] 2 n w
(cf Proof of Proposition 2.11). We deduce that ρ(a) (cid:54)= 0. Consequently, the multiplication
map P ⊗0Hf → 0H is injective and the representation is faithful. On the other hand, the
n n n
multiplication map is easily seen to be surjective. (cid:3)
Note that P and 0Hf are subalgebras of 0H .
n n n
The module P is an induced module: we have an isomorphism of 0H -modules
n n
P →∼ 0H ⊗ Z, P (cid:55)→ P ⊗1.
n n 0Hf
n
Remark 2.18. Given P ∈ P , one shows that T P −s (P)T = PT −T s (P) = ∂ (P).
n i i i i i i i
2.2.2. Description as a matrix ring. Let b = T X X2···Xn−1.
n w[1,n] 2 3 n
Lemma 2.19. We have b2 = b and 0H = 0H b 0H .
n n n n n n
Proof. Note that T is the unique element of End (P ) that sends X X2···Xn−1 to 1
w[1,n] PnSn n 2 3 n
and ∂ (X X2···Xn−1) to 0 for w (cid:54)= 1 (cf Proof of Theorem 2.11 for the first fact). We have
w 2 3 n
ρ(T X X2···Xn−1T )(∂ (X X2···Xn−1)) = 0
w[1,n] 2 3 n w[1,n] w 2 3 n
for w (cid:54)= 1 and
ρ(T X X2···Xn−1T )(X X2···Xn−1) = ∂ (X X2···Xn−1) = 1.
w[1,n] 2 3 n w[1,n] 2 3 n w[1,n] 2 3 n
It follows that b T = T .
n w[1,n] w[1,n]
We show now by induction on n that 1 ∈ 0H T 0H . Given 1 ≤ r ≤ n−1, we have
n w[1,n] n
T ···T T X −X T ···T T = T ···T T ,
r n−1 w[1,n−1] n r r n−1 w[1,n−1] r+1 n−1 w[1,n−1]
(cid:81)
where we use the convention that T ···T = T = 1 if r = n−1. By induction
r+1 n−1 r+1≤j≤n−1 j
on r, we deduce that T ∈ 0H T 0H , since T = T ···T T . By induction
w[1,n−1] n w[1,n] n w[1,n] 1 n−1 w[1,n−1]
on n, it follows that 1 ∈ 0H T 0H = 0H b 0H . (cid:3)
n w[1,n] n n n n
Remark 2.20. Given w ∈ S and P ∈ P , one shows that T PT = ∂ (P)T (a
n n w w[1,n] w w[1,n]
particular case was obtained in the proof of Lemma 2.19).
We have an isomorphism of 0H -modules
n
P →∼ 0H b , P (cid:55)→ Pb
n n n n
This shows that P is a progenerator as a 0H -module: it is a finitely generated projective
n n
0H -module and 0H is a direct summand of a multiple of P , as a 0H -module.
n n n n
Given A a ring, we denote by Aopp the opposite ring: it is A as an abelian group, but the
multiplication of a and b in Aopp is the product ba computed in A.
Proposition 2.21. The action of 0H on P induces an isomorphism of PSn-algebras
n n n
0H →∼ End (P )opp.
n PnSn n
Since P is a free PSn-module of rank n!, the algebra 0H is isomorphic to a (n!×n!)-matrix
n n n
algebra over PSn.
n
8 RAPHAE¨L ROUQUIER
Proof. SinceP isaprogeneratorfor0H ,wededucethatthecanonicalmap0H → End (P )
n n n PnSn n
is a split injection of PSn-modules (Lemma 2.22). The proposition follows from the fact that
n
0H is a free PSn-module of rank (n!)2. (cid:3)
n n
Lemma 2.22. Let R be a commutative ring and A an R-algebra, projective and finitely gen-
erated as an R-module. Let M be a progenerator for A. Then, the canonical map A →
End (M)opp is a split injection of R-modules.
R
Proof. Let f : A → End (M) be the canonical map and L its cokernel. The composition of
R
morphisms of R-modules
m(cid:55)→id⊗m α⊗m(cid:55)→α(m)
M −−−−−→ End (M)⊗ M −−−−−−−→ M
R A
is the identity. So, f ⊗ 1 is a split injection of R-modules, hence L ⊗ M is a projective
A M A
R-module, since End (M) is a projective R-module and M is a projective A-module.
R
By Morita theory, there is N an (End (M),A)-bimodule that is projective as an End (M)-
A A
module and such that M ⊗ N (cid:39) A as (A,A)-bimodules. The R-module L (cid:39) (L ⊗
EndA(M) A
M)⊗ N is projective, since N is a projective End (M)-module. Since L is a projective
EndA(M) A
R-module, we deduce that f is a split injection of R-modules. (cid:3)
We give now a second proof of Proposition 2.21. The proof of the faithfulness of the rep-
resentation P of 0H works also to show that P ⊗ (PSn/m) is a faithful representation
n n n PnSn n
of 0H ⊗ (PSn/m), for any maximal ideal m of PSn. Proposition 2.21 follows now from
n PnSn n n
Lemma 2.23.
Lemma 2.23. Let R be a commutative ring, f : M → N a morphism between free R-modules
of the same finite rank. If f ⊗ 1 is injective for every maximal ideal m of R, then f is an
R R/m
isomorphism.
Proof. Fix bases of M and N and let d be the determinant of f with respect to those bases. Let
I be the ideal of R generated by d. Assume d(cid:54)∈R×. There is a maximal ideal m of R containing
I. Since f ⊗ 1 is an injective map between vector spaces of the same finite rank, it is an
R R/m
isomorphism, so we have d·1 (cid:54)=0, a contradiction. (cid:3)
R/m
2.3. Symmetrizing forms.
2.3.1. Definition and basic properties. LetRbeacommutativeringandAanR-algebra,finitely
generated and projective as an R-module. A symmetrizing form for A is an R-linear map
t ∈ Hom (A,R) such that
R
• t is a trace, i.e., t(ab) = t(ba) for all a,b ∈ A
• the morphism of (A,A)-bimodules
tˆ: A → Hom (A,R), a (cid:55)→ (b (cid:55)→ t(ab))
R
is an isomorphism.
Consider now a commutative ring R(cid:48) such that R is an R(cid:48)-algebra, finitely generated and
projective as an R(cid:48)-module. Consider t ∈ Hom (A,R) a trace and t(cid:48) ∈ Hom (R,R(cid:48)). We have
R R(cid:48)
QUIVER HECKE ALGEBRAS AND 2-LIE ALGEBRAS 9
a commutative diagram
A (cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)tˆt(cid:99)(cid:48)(cid:81)t(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:47)(cid:47) H(cid:81)(cid:81)o(cid:40)(cid:40) mR(A,R) (cid:104)(cid:104)H(cid:104)o(cid:104)m(cid:104)(cid:104)R(cid:104)(a(cid:104)Ad(cid:104)∼j,(cid:104)utˆ(cid:48)(cid:104)n)(cid:104)ct(cid:104)(cid:47)(cid:47)i(cid:104)oH(cid:104)n(cid:104)o(cid:104)m(cid:104)(cid:104)R(cid:51)(cid:51) (A,HomR(cid:48)(R,R(cid:48)))
Hom (A,R(cid:48))
R(cid:48)
We deduce the following result.
Lemma 2.24. If two of the forms t, t(cid:48) and t(cid:48)t are symmetrizing, then so is the third one.
Let now B be another symmetric R-algebra and M an (A,B)-bimodule, finitely generated
and projective as an A-module and as a right B-module. We have isomorphisms of functors
can f(cid:55)→tf
Hom (M,−) ←−− Hom (M,A)⊗ − −−−→ Hom (M,R)⊗ −
A A A R A
∼ ∼
∼
and similarly Hom (Hom (M,R),−) → M ⊗ −. We deduce that M ⊗ − is left and right
B R B B
adjoint to Hom (M,−).
A
2.3.2. Polynomials.
Proposition 2.25. The linear form ∂ is a symmetrizing form for the PSn-algebra P .
w[1,n] n n
We will prove this proposition in §2.3.4: it will be deduced from a corresponding statement
for the nil affine Hecke algebra, that is easier to prove.
Together with Lemma 2.24, Proposition 2.25 provides more general symmetrizing forms.
Corollary 2.26. Given 1 ≤ i ≤ n, then the linear form ∂ ∂ ∂ is a symmetrizing
w[1,n] w[1,i] w[i+1,n]
form for the PnSn-algebra PnS{1,...,i}×S{i+1,...,n}.
2.3.3. Nil Hecke algebras. Define the Z-linear form t : 0Hf → Z by t (T ) = δ .
0 n 0 w w,w[1,n]
Define an algebra automorphism σ (the Nakayama automorphism) of 0Hf by σ(T ) = T .
n i n−i
Note that σ(T ) = T .
w w[1,n]ww[1,n]
The form t is not symmetric, it nevertheless gives rise to a Frobenius algebra structure.
0
Proposition 2.27. Given a,b ∈ 0Hf, we have t (ab) = t (σ(b)a). The form t induces an
n 0 0 0
isomorphism of right 0Hf-modules
n
tˆ : 0Hf →∼ Hom (0Hf,Z),a (cid:55)→ (b (cid:55)→ t (ab)).
0 n Z n 0
Proof. Let w,w(cid:48) ∈ S . We have t (T T ) = 0 unless w(cid:48) = w−1w[1,n], in which case we
n 0 w w(cid:48)
have t (T T ) = 1. We have t (σ(T )T ) = t (T T ). This is 0, unless
0 w w−1w[1,n] 0 w(cid:48) w 0 w[1,n]w(cid:48)w[1,n] w
w = (w[1,n]w(cid:48)w[1,n])−1w[1,n], or equivalently, unless w = w[1,n]w(cid:48)−1. In that case, we get
t (σ(T ,T ) = 1. This shows that t (T T ) = t (σ(T )T ) for all w,w(cid:48) ∈ S .
0 w−1w[1,n] w 0 w w(cid:48) 0 w(cid:48) w n
Let p be a prime number. The kernel I of tˆ ⊗ F is a two-sided ideal of F 0Hf. On the
0 Z p p n
other hand, tˆ(T )(1) = t (T ) = 1, hence T (cid:54)∈I. It follows from Proposition 2.6 that
0 w[1,n] 0 w[1,n] w[1,n]
I = 0. Lemma 2.23 shows now that tˆ is an isomorphism. (cid:3)
0
10 RAPHAE¨L ROUQUIER
2.3.4. Nil affine Hecke algebras. We define a PSn-linear form t on 0H
n n
t : 0H → PSn, t(PT ) = δ ∂ (P) for P ∈ P and w ∈ S .
n n w w,w[1,n] w[1,n] n n
Let γ be the Z-algebra automorphism of 0H defined by
n
γ(X ) = X and γ(T ) = −T .
i n−i+1 i n−i
Lemma 2.28. We have t(ab) = t(γ(b)a) for a,b ∈ 0H .
n
Proof. Let i ∈ {1,...,n}. By induction on l(w), one shows that
(cid:77)
T X −X T ∈ P T .
w i w(i) w n w(cid:48)
w(cid:48)∈Sn,l(w(cid:48))<l(w)
It follows that
(cid:77)
T X −X T ∈ P T .
w[1,n] i n−i+1 w[1,n] n w
w(cid:54)=w[1,n]
We deduce that t(PT X ) = t(PX T ) for w ∈ S and P ∈ P .
w i n−i w n n
Let i ∈ {1,...,n−1} and P ∈ P . Remark 2.18 shows that
n
t(T PT ) = t(s (P)T T )+t(∂ (P)T ).
n−i w n−i n−i w n−i w
We have ∂ (∂ (P)) = 0, so t(∂ (P)T ) = 0. We have ∂ (P + s (P)) = 0, hence
w[1,n] n−i n−i w w[1,n] n−i
∂ (s (P)) = −∂ (P). Since s w = w[1,n] if and only if ws = w[1,n], we deduce
w[1,n] n−i w[1,n] n−i i
that t(s (P)T T ) = −t(PT T ). This shows that t(PT T ) = t(−T PT ) for w ∈ S
n−i n−i w w i w i n−i w n
and P ∈ P . The lemma follows. (cid:3)
n
When viewed as a subalgebra of End (P ), then 0H contains S , since the action of S
Z n n n n
is trivial on PSn (Proposition 2.21). The injection of S in 0H is given explicitly by s (cid:55)→
n n n i
(X −X )T +1.
i i+1 i
The following lemma is an immediate calculation involving endomorphisms of P .
n
Lemma 2.29. We have w[1,n]·a·w[1,n] = γ(a) for all a ∈ 0H .
n
Let t(cid:48) be the linear form on 0H defined by t(cid:48)(a) = t(aw[1,n]).
n
Proposition 2.30. The form t(cid:48) is symmetrizing for the PSn-algebra 0H .
n n
Proof. Lemmas 2.28 and 2.29 show that t(cid:48)(ab) = t(cid:48)(ba) for all a,b ∈ 0H .
n
Let m be a maximal ideal of PSn and k = PSn/m. We have k0H (cid:39) M (k) by Proposition
n n n n!
2.21. We have t(cid:48)(X ···Xn−1T w[1,n]) = 1 (cf the proof of Theorem 2.11), hence the form
2 n w[1,n]
t(cid:48) ⊗ 1 is not zero. As a consequence, it is a symmetrizing form, since k0H (cid:39) M (k) by
PnSn k n n!
Proposition 2.21. We deduce that tˆ(cid:48) ⊗ k is an isomorphism, so tˆ(cid:48) is an isomorphism by
Lemma 2.23. PnSn (cid:3)
Proof of Proposition 2.25. Let m be a maximal ideal of PSn and k = PSn/m. Let P be a non-
n n
zero element of P ⊗ k. By Proposition 2.30, there is a ∈ k0H such that t(cid:48)(PT a) (cid:54)= 0.
n PnSn n w(cid:80)[1,n]
So, t(PT aw[1,n]) (cid:54)= 0. There are elements Q ∈ P ⊗ k such that aw[1,n] = T Q .
w[1,n] w n PnSn w w w
Then
t(PT aw[1,n]) = t(PT Q ) = t(γ(Q )PT ) = ∂ (γ(Q )P) (cid:54)= 0.
w[1,n] w[1,n] 1 1 w[1,n] w[1,n] 1
We deduce that ∂ˆ ⊗ (PSn/m) is injective for any maximal ideal m of PSn. It follows
w[1,n] PnSn n n
from Lemma 2.23 that ∂ˆ is an isomorphism. (cid:3)
w[1,n]