DG-Enhanced Hecke and KLR Algebras
We construct DG-enhanced versions of the degenerate affine Hecke algebra and of the affine Hecke algebra. We extend Brundan-Kleshchev and Rouquier's isomorphism and prove that after completion, DG-enhanced versions of affine Hecke algebras (degenerate or nondegenerate) are isomorphic to complet...
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| description | We construct DG-enhanced versions of the degenerate affine Hecke algebra and of the affine Hecke algebra. We extend Brundan-Kleshchev and Rouquier's isomorphism and prove that after completion, DG-enhanced versions of affine Hecke algebras (degenerate or nondegenerate) are isomorphic to completed DG-enhanced versions of KLR algebras for suitably defined quivers. As a byproduct, we deduce that these DG-algebras have homologies concentrated in degree zero. These homologies are isomorphic respectively to the degenerate cyclotomic Hecke algebra and the cyclotomic Hecke algebra.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 095, 24 pages
DG-Enhanced Hecke and KLR Algebras
Ruslan MAKSIMAU a and Pedro VAZ b
aq Laboratoire Analyse Géométrie Modélisation, CY Cergy Paris Université,
2 av. Adolphe Chauvin (Bat. E, 5ème étage), 95302 Cergy-Pontoise, France
E-mail: ruslmax@gmail.com, ruslan.maksimau@cyu.fr
bq Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain,
Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
E-mail: pedro.vaz@uclouvain.be
URL: https://perso.uclouvain.be/pedro.vaz
Received March 30, 2023, in final form November 15, 2023; Published online November 22, 2023
https://doi.org/10.3842/SIGMA.2023.095
Abstract. We construct DG-enhanced versions of the degenerate affine Hecke algebra and
of the affine Hecke algebra. We extend Brundan–Kleshchev and Rouquier’s isomorphism
and prove that after completion DG-enhanced versions of affine Hecke algebras (degenerate
or nondegenerate) are isomorphic to completed DG-enhanced versions of KLR algebras for
suitably defined quivers. As a byproduct, we deduce that these DG-algebras have homologies
concentrated in degree zero. These homologies are isomorphic respectively to the degenerate
cyclotomic Hecke algebra and the cyclotomic Hecke algebra.
Key words: Hecke algebra; KLR algebra; DG-algebra
2020 Mathematics Subject Classification: 20C08; 16E45
1 Introduction
Hecke algebras and their affine versions are fundamental objects in mathematics and have a rich
representation theory (see, for example, the review [9]). The representation theory of finite
dimensional Hecke algebras also carries interesting symmetries which occur in categorification
of Fock spaces and Heisenberg algebras [5, 11].
In a series of outstanding papers, Lauda [10], Khovanov–Lauda [6, 7, 8] and independently
Rouquier [20], have constructed categorifications of quantum groups. They take the form of
2-categories whose Grothendieck groups are isomorphic to the idempotent version of the quan-
tum enveloping algebra of a Kac–Moody algebra. Both constructions were later proved to be
equivalent by Brundan [1]. As a main ingredient of the constructions of Khovanov–Lauda and
Rouquier there is a certain family of algebras, nowadays known as KLR algebras, that are
constructed using actions of symmetric groups on polynomial spaces.
It turns out that in type A the KLR algebras are closely related to affine Hecke algebras.
It was proved by Rouquier [20, Section 3.2] that KLR algebras of type A become isomorphic
to affine Hecke algebras after a suitable localization of both algebras. Independently, Brundan
and Kleshchev [2] have proved a similar result for cyclotomic quotient algebras. This endows
cyclotomic Hecke algebras with a presentation as graded idempotented algebras. In particular,
in the case of KLR for the quiver of type A8, the isomorphism to the group algebra of the
symmetric group in d letters kSd gives the latter a graded presentation. The grading on kSd
was already known to exist (see [19]) but transporting the grading from the KLR algebras
allowed to construct it explicitly. This gave rise to a new approach to the representation theory
of symmetric groups and Hecke algebras [3]. These results are valid over an arbitrary field k.
mailto:ruslmax@gmail.com
mailto:ruslan.maksimau@cyu.fr
mailto:pedro.vaz@uclouvain.be
https://perso.uclouvain.be/pedro.vaz
https://doi.org/10.3842/SIGMA.2023.095
2 R. Maksimau and P. Vaz
The BKR (Brundan–Kleshchev–Rouquier) isomorphism was later extended to isomorphisms
between families of other KLR-like algebras and Hecke-like algebras. A similar isomorphism
between the Dipper–James–Mathas cyclotomic q-Schur algebra and the cyclotomic quiver Schur
algebra is given in [21]. The authors of [12] and [23] have constructed a higher level version of the
affine Hecke algebra and have proved that after completion they are isomorphic to a completion
of Webster’s tensor product algebras [22]. A weighted version of this isomorphism is also given
in [23]. A similar relation between quiver Schur algebras and affine Schur algebras is given in [13].
Also in [12] the authors have constructed a higher level version of the affine Schur algebra and
have proved that after completion it is isomorphic to a completion of the higher level quiver
Schur algebras.
The BKR isomorphism was also generalized to other algebras. For example, in [18] it is
used to show that cyclotomic Yokonuma–Hecke algebras are particular cases of cyclotomic KLR
algebras for certain cyclic quivers, and in [17] the BKR isomorphism is extended to connect affine
Hecke algebras of type B and a generalization of KLR algebras for a Weyl group of type B.
Motivated by the work of Khovanov–Lauda [6, 8], Rouquier [20], and Kang–Kashiwara [4],
the second author and Naisse introduced in [16] a family of KLR-like DG-algebras. These are
referred to as DG-enhanced KLR algebras” because they are obtained from free resolutions of
cyclotomic KLR algebras over (non-cyclotomic) KLR algebras, where the cyclotomic condition
is in some sense replaced by a differential. The algebras underlying these DG-algebras also
provide categorification of universal Verma modules.
It seems natural to ask the following questions.
Questions 1.1.
paq Are there DG-enhanced versions of affine Hecke algebras that are free resolutions of cyclo-
tomic Hecke algebras over affine Hecke algebras?
pbq In this case, does the BKR isomorphism extend to an isomorphism between (completions
of) DG-enhanced versions of KLR algebras and DG-enhanced versions of Hecke algebras?
In this article, we answer these questions affirmatively.
Remark 1.2. In this paper, we work with two versions of affine Hecke algebras, usual affine
Hecke algebra, which is an affinization of the Hecke algebra for the symmetric group, and its
degenerate version. We slightly simplify the terminology and refer to these algebras as the q-
affine Hecke algebra, and the degenerate affine Hecke algebra. In fact, our “affine” always means
“extended affine”.
Let us give an overview of our Hecke algebras and the main results in this article. Fix d P N
(where 0 P N) and a field k that for simplicity we consider to be algebraically closed. We
consider the Z-graded algebra sHd generated by T1, . . . , Td�1 and X1, . . . , Xd in degree zero and θ
in degree 1. The generators T1, . . . , Td�1 and X1, . . . , Xd satisfy the relations of the degenerate
affine Hecke algebra sHd. The generator θ commutes with the Xr’s and with T2, . . . , Td�1 and
satisfies θ2 � 0 and T1θT1θ � θT1θT1 � 0. This implies that the subalgebra of sHd concentrated
in degree zero is isomorphic to sHd. For Q � pQ1, . . . , Qℓq P kℓ, we introduce a differential BQ
by declaring that it acts as zero on sHd while BQpθq �
±ℓ
r�1pX1 �Qrq. We denote by x̄Ha the
completion of the algebra sHd at a sequence of ideals depending on a P kd.
In order to make the connection to DG-enhanced versions of KLR algebras we consider
a quiver Γ with a vertex set I � k and with an edge i Ñ j iff j � 1 � i. We assume that Qr P I
for each r. We fix a P Id and we set ν and Λ such that νi and Λi are the multiplicities of i
in respectively a and Q. We have
±ℓ
r�1pX1 � Qrq �
±
iPIpX1 � iqΛi . Let pRpνq, dΛq be the
DG-enhanced version of the KLR algebra of type Γ with parameters ν and Λ as above and� pRpνq, dΛ� its completion.
DG-Enhanced Hecke and KLR Algebras 3
The first main result in this article is a DG-enhanced version of the BKR isomorphism for
the degenerate affine Hecke algebra:
Theorem 4.13. There is an isomorphism of DG-algebras
� pRpνq, dΛ� � �x̄Ha, BQ
�
.
There is a similar construction for the affine q-Hecke algebra, which we do in Section 2.3
and Section 4.3. Fix q P k, q � 0, 1 and denote by pHd, BQq and by
� pHa, BQ
�
the DG-enhanced
version of the affine q-Hecke and its completion. The construction of Hd also adds a variable θ
in degree 1 that also satisfies θ2 � 0 and commutes with all generators but T1 the relation being
T1θT1θ � θT1θT1 � pq � 1qθT1θ.
In a nutshell, fix Q � pQ1, . . . , Qℓq P pk�qℓ. We consider a quiver Γ with a vertex set I � k�
and with an edge i Ñ j iff qj � i. We assume that I contains Q1, . . . , Qℓ and fix a P Id. We
define ν and Λ in the same way as above. Let pRpνq, dΛq be the DG-enhanced version of the
KLR algebra of type Γ with ν and Λ as above and let
� pRpνq, dΛ� be its completion. The second
main result in this article is the DG-enhanced version of the BKR isomorphism for the affine
q-Hecke algebra:
Theorem 4.15. There is an isomorphism of DG-algebras
� pRpνq, dΛ� � � pHa, BQ
�
.
The two main results above imply that we have a family of isomorphisms pRpνq � pHa between
the underlying algebras parameterized by integral dominant weights.
The DG-enhanced versions of BKR isomorphisms above allow us to compute the homology of
the DG-algebras H̄d and Hd in the following way. It is already proved in [16, Proposition 4.14]
that the homology of the DG-algebra pRpνq, dΛq is concentrated in degree 0 and is isomorphic to
the cyclotomic KLR algebra. The most difficult part of this proof is to show that the homology
is concentrated in degree zero. The proof of this fact is quite technical and there is no obvious
way to rewrite it for Hecke algebras. So we use the following strategy: we deduce the statement
for Hecke algebras from the statement for KLR algebras using the DG-enhanced version of the
BKR isomorphism.
As a corollary of Theorems 4.13 and 4.15 and [16, Proposition 4.14], the DG-algebras
� sHd, BQ
�
and pHd, BQq are resolutions of the cyclotomic Hecke algebras sHQ
d andHQ
d . These are cyclotomic
quotients of the degenerate affine Hecke algebras and of the affine q-Hecke algebras, respectively.
Proposition 4.17. The homology of the DG-algebra
� sHd, BQ
�
is concentrated in degree 0 and
is isomorphic to sHQ
d .
Proposition 4.18. The homology of the DG-algebra pHd, BQq is concentrated in degree 0 and
is isomorphic to HQ
d .
To our knowledge, the DG-enhanced versions of Hecke algebras we introduce are new. We
would also like to emphasize the fact that the algebras H̄d andHd have triangular decompositions
(see Remarks 2.12 and 2.23). This looks like an analogy with the triangular decomposition in
the Cherednik algebras, see also Remark 2.5.
Plan of the paper
In Section 2, we introduce DG-enhanced versions of the degenerate affine Hecke algebra and of
the affine q-Hecke algebra and their completions, that will be used in the BKR isomorphism.
The material in this section is new.
In Section 3, we review the DG-enhanced version of the KLR algebra introduced in [16]. We
give the presentation of this algebra as in [16, Corollary 3.16] which is more convenient to us,
and present its completion, which is involved in the BKR isomorphism.
4 R. Maksimau and P. Vaz
Section 4 contains the main results. We first generalize the BKR isomorphism to a class of
algebras satisfying some properties. The most important point is that to have a generalization
of the BKR isomorphism we need an isomorphism between the completed polynomial represen-
tation of the Hecke-like algebra and the completed polynomial representation of the KLR-like
algebra, and this isomorphism must intertwine the action of the symmetric group. Our main
results, Theorems 4.13 and 4.15, are then proved by showing that our DG-enhanced versions of
Hecke algebras H̄d and Hd on one side, and the DG-enhanced versions of KLR algebras Rpνq
on the other side satisfy the properties that are required for them to be isomorphic (after com-
pletion). We then use the DG-enhanced version of the BKR isomorphism and the fact that the
DG-algebra Rpνq is a free resolution of the cyclotomic KLR algebra to show in Corollary 4.20
that the algebras H̄d and Hd are free resolutions of the corresponding cyclotomic Hecke algebras.
2 DG-enhanced versions of Hecke algebras
For integers a and b such that a ¤ b we write ra; bs � ta, a� 1, . . . , b� 1, bu.
2.1 The polynomial rings Pold and Polld and the rings Pd and Pld
Fix an algebraically closed field k, q P k, q � 0, 1 and d P N once and for all.
2.1.1 The polynomial rings Pold and Polld
Set Pold � krX1, . . . , Xds. Let Sd be the symmetric group on d letters, which we view
as a Coxeter group with generators s1, . . . , sd�1. These correspond to the simple transposi-
tions pi i�1q, and we use these two descriptions interchangeably throughout. As usual, we
let Sd act from the left on Pold by permuting the variables: for w P Sd we have wpXiq � Xwpiq,
and wpfgq � wpfqwpgq for f, g P Pold.
Using the Sd-action above, one defines the Demazure operators Bi on Pd for all 1 ¤ i ¤ d� 1
in the usual way, as
Bipfq �
f � sipfq
Xi �Xi�1
. (2.1)
We have siBipfq � Bipfq and Bipsifq � �Bipfq for all i, so Bi is in fact an operator from Pold
to the subring Polsid � Pold of invariants under the transposition pi i�1q. It is well known that
the action of the Demazure operators on Pold satisfy the Leibniz rule
Bipfgq � Bipfqg � sipfqBipgq, (2.2)
for all f, g P Pold and for 1 ¤ i ¤ n� 1, and the relations
B2i � 0, BiBi�1Bi � Bi�1BiBi�1, (2.3)
BiBj � BjBi for |i� j| ¡ 1, (2.4)
XiBi � BiXi�1 � 1, BiXi �Xi�1Bi � 1. (2.5)
Set Polld � k
�
X�1
1 , . . . , X�1
d
�
, which is the localization of Pold obtained by adding the inverses
of X1, . . . , Xd. Moreover, the Sd-action on Pold can be obviously extended to a Sd-action
on Polld. This means that the action of the Demazure operators on Pold also extends to operators
on Polld that satisfy the relations in (2.2) (for f and g in Polld) and (2.3)–(2.5).
DG-Enhanced Hecke and KLR Algebras 5
2.1.2 The rings Pd and Pld
Let θ � tθ1, . . . , θdu be odd variables and form the supercommutative ring
Pd � Poldb
�
pθq,
where
�
pθq is the exterior k-algebra in the variables θ. Here Pd is a subring concentrated in
parity zero. Introduce an additional Z-grading on Pd denoted λp
q and defined as λpXiq � 0
and λpθiq � 1. This grading is half the grading degλ introduced in [14, Section 3.1]. If we
forget the grading, the algebra Pd is the symmetric algebra corresponding to a superspace of
dimension pd|dq.
As explained in [14, Section 8.3], the action of Sd on Pold extends to an action on Pd by
setting
sipθjq � θj � δi,jpXi �Xi�1qθi�1. (2.6)
This action respects the grading, as one easily checks, and allows extending the action of the
Demazure operators in (2.1) to Pd. We denote the extensions of the Demazure operators to Pd
by the same symbols. Similarly to the operators above, Bi is an operator from Pd to the subring
P si
d � Pd of invariants under the transposition pi i�1q. It was proved in [15, Lemma 2.2] that the
Demazure operators on Pd satisfy the Leibniz rule (2.2) (for f, g P Pd), the relations (2.3)–(2.5)
and the following relations:
Biθk � θkBi for k � i,
Bipθi �Xi�1θi�1q � pθi �Xi�1θi�1qBi,
for all i � 1, . . . , d� 1.
As in the case of Pd above, we form the supercommutative ring
Pld � Polldb
�
pθq.
This ring is also endowed with the grading λp
q, which is defined in the same way as in Pd.
Moreover, the Sd-action on Polld can be obviously extended to a Sd-action on Pld. This means
that the action of the Demazure operators on Polld also extends to operators on Pld that satisfy
the relations in (2.2) (for f and g in Pld) and (2.3)–(2.5).
2.2 Degenerate version
2.2.1 Degenerate affine Hecke algebra
The degenerate affine Hecke algebra sHd is the k-algebra generated by T1, . . . , Td�1 and X1, . . . ,
Xd, with relations
T 2
i � 1, TiTj � TjTi if |i� j| ¡ 1, TiTi�1Ti � Ti�1TiTi�1, (2.7)
XiXj � XjXi, (2.8)
TiXi �Xi�1Ti � �1, TiXj � XjTi if j � i � 0, 1. (2.9)
For w � si1 . . . sik P Sd a reduced expression, we put Tw � Ti1 . . . Tik . Then Tw is independent
of the choice of the reduced expression of w and the set
Xm1
1 . . . Xmd
d Tw
(
wPSd,miPZ¥0
is a basis of the k-vector space sHd.
6 R. Maksimau and P. Vaz
There is a faithful representation of sHd on Pold, where Tipfq � sipfq � Bipfq and Xi P sHd
acts as multiplication by Xi. It is immediate that sHd contains kSd and Pold as subalgebras and
that for p P Pold,
Tip� sippqTi � �Bippq.
Let ℓ be a positive integer and Q � pQ1, . . . , Qℓq be an ℓ-tuple of elements of the field k.
Definition 2.1. The degenerate cyclotomic Hecke algebra is the quotient
H̄Q
d � H̄d{
ℓ¹
r�1
pX1 �Qrq.
2.2.2 The algebra H̄d
Definition 2.2. Define the algebra H̄d as the k-algebra generated by T1, . . . , Td�1 and X1, . . . ,
Xd in λ-degree zero, and an extra generator θ in λ-degree 1, with relations (2.7) to (2.9) and
θ2 � 0,
Xrθ � θXr for r � 1, . . . , d,
Trθ � θTr for r ¡ 1,
T1θT1θ � θT1θT1 � 0.
The algebra H̄d contains the degenerate affine Hecke algebra H̄d as a subalgebra concentrated
in λ-degree zero.
Lemma 2.3. The algebra H̄d acts on Pd by
Trpfq � srpfq � Brpfq, Xrpfq � Xrf, θpfq � θ1f,
for all f P Pd and where srpfq and Brpfq are as in (2.6) and (2.1).
Proof. The defining relations of H̄d can be checked by a straightforward computation. ■
Define ξ1, . . . , ξd P H̄d by the rules ξ1 � θ, ξi�1 � TiξiTi. The following is straightforward.
Lemma 2.4. The elements ξr satisfy for all r P t1, . . . , d� 1u and all ℓ P t1, . . . , du,
ξ2ℓ � 0, ξrξℓ � ξℓξr � 0, Trξℓ � ξsrpℓqTr.
Remark 2.5. It is easy to give the relations between T ’s and X’s and between T ’s and ξ’s.
However, X’s and ξ’s satisfy more elaborate relations, which is similar to what happens with
two polynomial rings in Cherednik (double affine Hecke) algebras. For example, the following
commutation relations can be checked easily:
rXr, ξ1s � 0, rX1, ξ2s � �rX2, ξ2s � rξ2, T1s � rT1, ξ1s, rX1, ξ3s � T2rT1, ξ1sT2.
Abusing the notation, we will write θr for the operator on Pd that multiplies each element
of Pd by θr. Set M � t0, 1ud. Denote by 1 the sequence 1 � p1, 1, . . . , 1q P M . For each
sequence b � pb1, . . . , bdq P M , we set θb � θb11 . . . θbdd . For each b P M , we set b � 1 � b. In
particular, we have θb � θb � �θ1θ2 . . . θd � �θ1. Set also |b| � b1 � b2 � � � � � bd.
Lemma 2.6. The operators
θb | b P M
(
acting on Pd are linearly independent over H̄d. More
precisely, if we have
°
bPM hbθ
b � 0 with hb P H̄d, then we have hb � 0 for each b P M .
DG-Enhanced Hecke and KLR Algebras 7
Proof. Let H �
°
bPM hbθ
b be an operator that acts by zero. Assume that H has a nonzero
coefficient. Let b0 be such that hb0 � 0 and such that |b0| is minimal with this property.
Then for each element P P Pd, we have H
�
θb0P
�
� �hb0θ
1P . This shows that hb0 acts by
zero on θ1Pd � θ1 Pold. But this implies hb0 � 0 because the polynomial representation of H̄d
on Pold is faithful, see [20, Section 3.1.2]. ■
For each, k P t0, 1, . . . , du we denote by H̄¤k
d the subalgebra of the algebra of operators on Pd
generated by Xi, θi for i ¤ k and Tr for r k. Denote also by H̄¤k
d the subalgebra of H̄d
generated by Xi for i ¤ k and Tr for r k. Since H̄d acts faithfully on Pd, we can see H̄¤k
d as
a subalgebra of H̄¤k
d . We mean that for k � 0 we have H̄¤0
d � H̄¤0
d � k. The λ-grading on Pd
induces a grading on H̄¤k
d that we also call λ-grading.
Lemma 2.7. The set
Xa1
1 . . . Xak
k Twθ
b1
1 . . . θbkk | w P Sk, pa1, . . . , akq P Nk, pb1, . . . , bkq P t0, 1u
k
(
,
is a basis of the k-vector space H̄¤k
d .
Proof. It is clear that the given set spans. Linear independence follows from Lemma 2.6. ■
Similarly to the notation θb above, we set ξb � ξb11 . . . ξbdd . For two elements b,b1 P M , we
write b1 b if there is an index r P r1; ds such that b1r br and b1t � bt for t ¡ r. For b P M ,
write maxpbq for the maximal index r P r1; ds such that br � 1.
Lemma 2.8. The element ξk acts on Pd by an operator of the form ck�dkθk, where ck P H̄¤k�1
d ,
dk P H̄¤k�1
d , λpckq � 1 and dk is not a right zero divisor in H̄d.
Proof. We prove by induction on k. The case k � 1 is trivial. Now, assume that dk is not
a right zero divisor and let us show that dk�1 is not a right zero divisor. Since we have
TkdkθkTk � TkdkTkpθk � pXk �Xk�1qθk�1q � Tkdkθk�1
� TkdkTkθk � pTkdkTkpXk �Xk�1q � Tkdkqθk�1,
we get
dk�1 � TkdkTkpXk �Xk�1q � Tkdk � TkdkpTkpXk �Xk�1q � 1q
� TkdkppXk�1 �XkqTk � 1q.
It is enough to check that the element ppXk�1 � XkqTk � 1q is not a right zero divisor. This
follows from the fact that it acts on Pd by the operator pXk�1 �Xk � 1qsk. ■
Lemma 2.9. The element ξb P H̄d acts on Pd by an operator of the form cb � dbθ
b, where
db P H̄
¤maxpbq�1
d and db is not a right zero divisor in H̄d and cb is of the form
°
b1 b hb1θ
b1
with hb1 P H̄
¤maxpbq�1
d .
Proof. We prove the statement by induction on |b| � r. The case r � 1 follows immediately
from the lemma above. Now, for r ¡ 1, assume that the statement is true for r� 1, let us prove
it for r.
Set p � maxpbq. Let b1 P M be such that θb � θb1θp. By the induction assumption,
the element ξb � ξb1ξp acts on Pd by an operator of the form (up to sign) pcp � dpθpq
�
cb1 �
db1θ
b1
�
. This operator can be written as cb � dbθ
b for db � dpdb1 and cb � cppcb1 � db1θ
b1q �
dpθpcb1 . Now, we obviously get db P H̄¤p�1
d because it is a product of two elements of H̄¤p�1
d
and it is not a right zero divisor as a product of two right non-zero divisors. Moreover, the
element cb is of the form
°
b1 b hb1θ
b1 because dpθpcb1 � dpcb1θp is of the required form and
because cppcb1 � db1θ
b1q P H̄¤p�1
d (and then it is also of the required form). ■
8 R. Maksimau and P. Vaz
It is not hard to write a basis of H̄d in terms of the ξr’s.
Proposition 2.10. The set
Xa1
1 . . . Xad
d Twξ
b1
1 . . . ξbdd | w P Sd, pa1, . . . , adq P Nd, pb1, . . . , bdq P t0, 1u
d
(
,
is a basis of the k-vector space H̄d.
Proof. We start by showing that this set spans H̄d. First, each monomial on θ, X’s and T ’s
can be rewritten as a linear combination of similar monomials with all X’s on the left. After
that, we replace θ by ξ1 and we move all ξ’s to the right by using Lemma 2.4. This shows that
the set above spans H̄d. Linear independence follows from Lemmas 2.6 and 2.9. ■
Corollary 2.11. The representation defined in Lemma 2.3 is faithful.
Proof. We see from the proof of the proposition above that the elements of the basis act by
linearly independent operators. ■
Remark 2.12. We see from Proposition 2.10 that the algebra H̄d has a triangular decomposition
(only as a vector space)
H̄d � krX1, . . . , Xds b kSd b
�
pξ1, . . . , ξdq.
2.2.3 DG-enhancement of H̄d
Let ℓ and Q be as in Section 2.2.1.
Definition 2.13. Define an operator BQ on sHd by declaring that BQ acts as zero on sHd � sHd,
while
BQpθq �
ℓ¹
r�1
pX1 �Qrq,
and it respects the graded Leibniz rule: for a, b P sHd, BQpabq � BQpaqb� p�1qλpaqaBQpbq.
Lemma 2.14. The operator BQ is a differential on sHd.
Proof. We prove something slightly more general. Let P P krX1s be a polynomial. Define
dP : sHd Ñ sHd by declaring that dP acts as zero on sHd, while dP pθq � P , together with the
graded Leibniz rule. Then dP is a differential on sHd. To prove the claim is suffices to check
that dP pT1θT1θ � θT1θT1q � 0.
We have T1P � s1pP qT1 � B1pP q and PT1 � T1s1pP q � B1pP q, where B1 is the Demazure
operator. This also implies T1PT1 � s1pP q � B1pP qT1. Note also that B1pP q is a symmetric
polynomial with respect to X1, X2, so it commutes with T1. So, we have
dP pT1θT1θ � θT1θT1q � T1PT1θ � T1θT1P � PT1θT1 � θT1PT1
� ps1pP qθ � B1pP qT1θq � pT1θs1pP qT1 � T1θB1pP qq
� pT1s1pP qθT1 � B1pP qθT1q � pθs1pP q � θB1pP qT1q � 0,
which proves the claim. ■
We will prove in Proposition 4.17 that the homology of the DG-algebra
� sHd, BQ
�
is concen-
trated in degree 0 and is isomorphic to sHQ
d .
DG-Enhanced Hecke and KLR Algebras 9
2.2.4 Completions of H̄d
Consider the algebra of symmetric polynomials Symd � PolSd
d . We consider it as a (central)
subalgebra of H̄d.
For each d-tuple a � pa1, . . . , adq P kd we have a character χa : Symd Ñ k given by the
evaluation Xr ÞÑ ar. It is obvious from the definition that if the d-tuple a1 is a permutation of
the d-tuple a then the characters χa and χa1 are the same. Denote by ma the kernel of χa.
Definition 2.15. Denote by x̄Ha the completion of the algebra H̄d with respect to ma.
Since ma is in the kernel of BQ, we can extend BQ to x̄Ha. Set also
xPola � à
bPSda
krrX1 � b1, . . . , Xd � bdss1b,
pPa �
à
bPSda
pkrrX1 � b1, . . . , Xd � bdss b
�
pθqq1b,
where 1b is just a formal idempotent projecting on the corresponding direct factor. Here Sda
is the Sd-orbit of a with respect to the obvious Sd-action on kd. We can obviously extend the
action of H̄d on Pd to an action of x̄Ha on pPa. Each finite dimensional x̄Ha-module M decomposes
into its generalized eigenspaces M �
À
bPSda
Mb, where
Mb �
m P M | DN P N such that pXr � brq
Nm � 0 @r
(
.
For each b P Sda the algebra x̄Ha contains an idempotent 1b that projects onto Mb when applied
to M .
Proposition 2.16.
paq The xPola-module x̄Ha is free with basis
Twξ
b1
1 . . . ξbdd | w P Sd, pb1, . . . , bdq P t0, 1u
d
(
.
pbq The representation pPa of x̄Ha is faithful.
Proof. It is clear that the elements from the statement generate the xPola-module x̄Ha. To see
that they form a basis, it is enough to remark that they act by linear independent
�
over xPola�
operators on the representation pPa. This proves paq. Then pbq also holds because a basis acts
on pPa by linearly independent operators. ■
The algebra H̄Q
d has a decomposition H̄Q
d � `aH̄
Q
a (with a finite number of nonzero terms)
such that Symd acts on each finite dimensional H̄Q
a -module with a generalized character χa.
2.3 q-version
2.3.1 Affine q-Hecke algebra
The affine q-Hecke algebra Hd is the k-algebra generated by T1, . . . , Td�1 and X�1
1 , . . . , X�1
d ,
with relations
XrX
�1
r � X�1
r Xr � 1, X�1
i X�1
j � X�1
j X�1
i , (2.10)
pTi � qqpTi � 1q � 0, TiTj � TjTi if |i� j| ¡ 1, TiTi�1Ti � Ti�1TiTi�1, (2.11)
TiXj � XjTi for j � i � 0, 1, TiXiTi � qXi�1. (2.12)
10 R. Maksimau and P. Vaz
Note that relation (2.11) implies that the element Ti is invertible. For w � si1 . . . sik P Sd
a reduced decomposition, we put Tw � Ti1 . . . Tik . Then Tw is independent of the choice of the
reduced decomposition of w and the set
Xm1
1 . . . Xmd
d Tw
(
wPSd,miPZ
is a basis of the k-vector space Hd. There is a faithful representation of Hd on Polld, where
Tipfq � qsipfq � pq � 1qXi�1Bipfq.
Let ℓ be a positive integer. Let Q � pQ1, . . . , Qℓq be an ℓ-tuple of nonzero elements of the
field k.
Definition 2.17. The cyclotomic q-Hecke algebra is the quotient
HQ
d � Hd{
ℓ¹
r�1
pX1 �Qrq.
2.3.2 The algebra Hd
Definition 2.18. The algebra Hd is the k-algebra generated by T1, . . . , Td�1 and X�1
1 , . . . , X�1
d
in λ-degree zero, and an extra generator θ in λ-degree 1, with relations (2.10) to (2.12) and
θ2 � 0, X�1
r θ � θX�1
r for r � 1, . . . , d,
Trθ � θTr for r ¡ 1,
T1θT1θ � θT1θT1 � pq � 1qθT1θ.
The algebra Hd contains the affine q-Hecke algebra Hd as a subalgebra concentrated in λ-
degree zero.
Lemma 2.19. The algebra Hd acts on Pld by
Trpfq � qsrpfq � pq � 1qXr�1Brpfq, X�1
r pfq � X�1
r f, θpfq � θ1f,
for all f P Pd and where srpfq and Brpfq are as in (2.6) and (2.1).
Proof. The defining relations of Hd can be checked by a straightforward computation. ■
Define ξ1, . . . , ξd P Hd by the rules ξ1 � θ, ξi�1 � TiξiT
�1
i . The following is straightforward.
Lemma 2.20. The elements ξr satisfy for all r � 1, . . . , d� 1 and all ℓ � 1, . . . , d,
ξ2ℓ � 0, ξrξℓ � ξℓξr � 0
and
Tℓξr �
$'&'%
ξrTℓ if r � ℓ, ℓ� 1,
ξℓTℓ � pq � 1qpξℓ�1 � ξℓq if r � ℓ� 1,
ξℓ�1Tℓ if r � ℓ.
It is not hard to write a basis of Hd in terms of the ξr’s.
Proposition 2.21. The set
Xa1
1 . . . Xad
d Twξ
b1
1 . . . ξbdd | w P Sd, pa1, . . . , adq P Zd, pb1, . . . , bdq P t0, 1u
d
(
,
is a basis of the k-vector space Hd.
DG-Enhanced Hecke and KLR Algebras 11
Proof. Imitate the proof of Proposition 2.10. ■
Corollary 2.22. The representation defined in Lemma 2.19 is faithful.
Remark 2.23. We see from Proposition 2.21 that the algebraHd has a triangular decomposition
(only as a vector space)
Hd � k
�
X�1
1 , . . . , X�1
d
�
bHfin
d b
�
pξ1, . . . , ξdq,
where Hfin
d is the (finite dimensional) Hecke algebra of the group Sd. Explicitly, the algebra Hfin
d
is defined by generators T1, . . . , Td�1 and the relations in (2.11).
2.3.3 DG-enhancement of Hd
Let ℓ and Q be as in Section 2.3.1.
Definition 2.24. Define an operator BQ on Hd by declaring that BQ acts as zero on Hd, while
BQpθq �
ℓ¹
r�1
pX1 �Qrq,
and for a, b P sHd, BQpabq � BQpaqb� p�1qλpaqaBQpbq.
Lemma 2.25. The operator BQ is a differential on Hd.
Proof. Similarly to the proof of Lemma 2.14, we consider a more general differential dP . We
have to check
dP pT1θT1θ � θT1θT1q � dP ppq � 1qθT1θq.
We have T1P � s1pP qT1�pq� 1qX2B1pP q and PT1 � T1s1pP q� pq� 1qX2B1pP q, where B1 is
the Demazure operator. Note also that B1pP q is a symmetric polynomial with respect to X1, X2,
so it commutes with T1. So, we have
dP pT1θT1θ � θT1θT1q � T1PT1θ � T1θT1P � PT1θT1 � θT1PT1
�
�
T 2
1 s1pP qθ � pq � 1qB1pP qT1X2θ
�
� pT1θs1pP qT1
� pq � 1qT1θX2B1pP qq � pT1s1pP qθT1 � pq � 1qX2B1pP qθT1q
� pθs1pP qT
2
1 � pq � 1qθB1pP qX2T1q
� T 2
1 s1pP qθ � θs1pP qT
2
1
� pq � 1qPT1θ � pq � 1qθT1P
� dP ppq � 1qθT1θq,
which proves the claim. ■
We will prove in Proposition 4.18 that the homology of the DG-algebra pHd, BQq is concen-
trated in degree 0 and is isomorphic to HQ
d .
12 R. Maksimau and P. Vaz
2.3.4 Completions of Hd
Similarly to Section 2.2.4, we want to define a completion of the algebraHd. Consider the algebra
of symmetric Laurent polynomials Symld � k
�
X�1
1 , . . . , X�1
d
�Sd . We consider it as a (central)
subalgebra of Hd.
For each d-tuple a � pa1, . . . , adq P pk�qn, we have a character χa : Symld Ñ k given by the
evaluation Xr ÞÑ ar. Denote by ma the kernel of χa.
Definition 2.26. Denote by pHa the completion of the algebra Hd at the sequence of ideals
Hdm
j
aHd.
Since ma is in the kernel of BQ, we can extend BQ to pHa. Set also
pPa � krrX1 � a1, . . . , Xd � adss b
�
pθq.
We can obviously extend the action of Hd on Pd to an action of pHa on pPa. Similarly to x̄Ha, the
algebra pHa has idempotents 1b, b P Sda that are defined in the same way as in Section 2.2.4.
Similar to Proposition 2.16, we have the following.
Proposition 2.27.
paq The xPola-module pHa is free with basis
Twξ
b1
1 . . . ξbdd | w P Sd, pb1, . . . , bdq P t0, 1u
d
(
.
pbq The representation pPa of pHa is faithful.
The algebra HQ
d has a decomposition HQ
d � `aH
Q
a (with a finite number of nonzero terms)
such that Symld acts on each finite dimensional HQ
a -module with a generalized character χa.
3 DG-enhanced versions of KLR algebras
DG-enhanced versions of KLR algebras were introduced in [16] as one of the main ingredients
in the categorification of Verma modules for symmetrizable quantum Kac–Moody algebras.
Let Γ � pI, Aq be a quiver without loops with set of vertices I and set of arrows A. We call
elements in I labels. Let also NrIs be the set of formal N-linear combinations of elements of I.
Fix ν P NrIs,
ν �
¸
iPI
νi � i, νi P N, i P I,
and set |ν| �
°
i νi. We allow the quiver to have infinite number of vertices. In this case, only
a finite number of νi is nonzero.
For each i, j P I, we denote by hi,j the number of arrows in the quiver Γ going from i to j,
and define for i � j the polynomials
Qi,jpu, vq � pu� vqhi,j pv � uqhj,i .
3.1 The algebra Rpνq
We give a diagrammatic definition of the algebras R � RpΓq from [16, Section 3]. The definition
we give corresponds to the presentation in [16, Corollary 3.16].
Definition 3.1. For each ν P NrIs, we define the k-algebra Rpνq by the data:
DG-Enhanced Hecke and KLR Algebras 13
� It is generated by the KLR generators
i
. . . . . . and
i j
� � � � � �
for i, j P I, where each diagram contains νi strands labeled i, together with floating dots
that are confined to a region immediately to the right of the left-most strand,
i
� � �
.
Diagrams are taken modulo isotopies that do not allow triple crossings of strands, do not
allow a dot going through a crossing, and do not allow two floating dots at the same level.
� The multiplication is given by gluing diagrams on top of each other1 whenever the labels
of the strands agree, and zero otherwise, subject to the local relations (3.1) to (3.7) below,
for all i, j, k P I.
� The KLR relations, for all i, j, k P I:
i i
� 0 and
i j
� Qi,jpY1, Y2q
i j
if i � j, (3.1)
i j
�
i j
,
i j
�
i j
if i � j, (3.2)
i i
�
i i
�
i i
�
i i
�
i i
, (3.3)
ji k
�
ji k
unless i � k � j, (3.4)
ji i
�
ji i
�
Qi,jpY3, Y2q �Qi,jpY1, Y2q
Y3 � Y1
i j i
if i � j. (3.5)
� And the additional relations, for all i, j P I:
i
. . . � 0, (3.6)
1We follow the usual (and useful) convention that m dots on the same strand are depicted as a single dot with
an exponent m.
14 R. Maksimau and P. Vaz
i j
� �
i j
. (3.7)
Remark 3.2. A diagram with a box containing a polynomial means a polynomial in dots.
The indices in the variables indicate the strands carrying the corresponding dots. For example,
for ppY1, Y2q �
°
r,s cr,sY
r
1 Y
s
2 with cr,s P k, we have
ppY1, Y2q
i j
�
¸
r,s
cr,s
i
r
j
s .
We now define a Z � Z-grading in Rpνq. Contrary to [16], we work with a single homolog-
ical degree λ. The homological nature of this degree is justified by the DG-structure defined
in Section 3.5. We declare
deg
�
i
�
� p2, 0q, deg
�
i j
�
�
$'&'%
p�2, 0q if i � j,
p�1, 0q if hi,j � 1,
p0, 0q otherwise,
and
deg
�
i
� � �
�
�
�
�2, 1
�
,
where the second grading is called λ-grading, which we write λp
q. The defining relations ofRpνq
are homogeneous with respect to this bigrading.
Remark 3.3. The subalgebra of Rpνq in λ-degree zero coincides with the usual KLR alge-
bra Rpνq defined in [6] and [20]. More precisely, the algebra Rpνq is defined by the first two
types of generators in Definition 3.1 and relations (3.1)–(3.5).
For i � i1 . . . id, define the idempotent
1i �
i1 i2
. . .
id
and let Seqpνq be the set of all ordered sequences i � i1i2 . . . id with each ik P I and i appearing νi
times in the sequence. For i, j P Seqpνq the idempotents 1i and 1j are orthogonal iff i � j, we
have 1Rpνq �
°
iPSeqpνq 1i, where 1Rpνq denotes the identity element in Rpνq, and
Rpνq �
à
j,iPSeqpνq
1jRpνq1i.
Finally, the algebra R is defined as
R �
à
νPNrIs
Rpνq.
DG-Enhanced Hecke and KLR Algebras 15
3.2 Polynomial action of Rpνq
We now describe a faithful action of Rpνq on a supercommutative ring, which was defined in [16,
Section 3.2] and extends the polynomial action of KLR algebras from [6, Section 2.3].
We fix ν P NrIs with |ν| � d. Set PRd � krY1, . . . , Yds b
�
xΩ1, . . . ,Ωdy. Now consider
PRν �
à
iPSeqpνq
PRd1i. (3.8)
Here we mean that the algebra PRν is a direct sum of copies of the algebra PRd, labelled
by Seqpνq. We denote by 1i the idempotent projecting to the ith copy.
For each i P I, 1 ¤ r ¤ νi and i � pi1, i2, . . . , idq P Seqpνq, we denote by r1 � r1pr, i, iq
the rth index r1 P t1, 2, . . . , du (counting from the left) among the indices such that ir1 � i.
Set ωr,i1i � Ωr11i.
The algebra PRν is bigraded supercommutative with gradings degpYtq � p2, 0q, degpωr,iq �
p�2r, 1q and degp1iq � p0, 0q, where the variables ωr,i are odd while the polynomial variables and
the idempotents are even. Note that we consider a λ-grading that is one half the one considered
in [16]. This is to agree with the analogous degrees on Hecke algebras in Section 2.1.
Now, similarly to [16, Section 3.2.1], we consider the action of S|ν| on PRν given by
sk : PRd1i Ñ PRd1ski,
sends Yp1i ÞÑ Yskppq1ski and
Ωp1i ÞÑ
$'&'%
pΩk � pYk � Yk�1qΩk�1q 1i if p � k and ik � ik�1,
Ωp1i if p � k � 1 and ik � ik�1,
Ωskppq1ski otherwise.
For each i, j P I, i � j, we consider the polynomial Pijpu, vq � pu� vqhi,j , where hi,j denotes
as above the number of arrows from i to j. Note that we have Qi,jpu, vq � Pi,jpu, vqPj,ipv, uq.
In the sequel, it is useful to have an algebraic presentation of Rpνq as in [2, equations (1.7)–
(1.15)]. We set
i1
. . .
ir
. . .
id
� Yr1i,
i1 ir ir�1 id
. . . . . . � τr1i,
i1 i2
. . .
id
� Ω1i.
We declare that a P ekRpνqej acts as zero on PRI1i whenever j � i. Otherwise
Yr1i ÞÝÑ f1i ÞÑ Yrf1i, Ω1i ÞÝÑ f1i ÞÑ Ω1f1i,
and
τr1i ÞÝÑ f1i ÞÑ
$&%
f1i � srpf1iq
Yr � Yr�1
if ir � ir�1,
Pir,ir�1pYr, Yr�1qsrpf1iq if ir � ir�1.
The following is Proposition 3.8 and Theorem 3.15 in [16].
Proposition 3.4. The rules above define a faithful action of Rpνq on PRν .
16 R. Maksimau and P. Vaz
3.3 Completion of Rpνq
We will consider PolRd � krY1, Y2, . . . , Yds as a subalgebra of Rpνq. Let m be the ideal of PolRd
generated by all Yp, 1 ¤ p ¤ d.
Definition 3.5. Denote by pRpνq the completion of the algebra Rpνq at the sequence of ide-
als RpνqmjRpνq. Let yPRd � krrY1, . . . , Ydss b
�
xΩ1, . . . ,Ωdy be the similar completion of PRd
and let yPRν �
À
iPSeqpνq
yPRd1i be the similar completion of PRν .
We would like to construct a representation structure of pRpνq in the vector space yPRν . The
S|ν|-action on PRν extends obviously to an S|ν|-action on yPRν . Moreover, the action of Rpνq
on PRν yields an action of pRpνq on yPRν .
Lemma 3.6. The representation yPRν of pRpνq is faithful.
Proof. An explicit PolRd-basis of Rν is constructed in [16, Section 3.2]. We would like to check
that the same set forms a zPolRd-basis of pRν . The fact that this is a spanning set can be proved
by the same argument. The linear independence follows from the fact that the elements act onyPRν by linearly independent operators. Then, this proves automatically the faithfulness of the
representation. ■
3.4 Cyclotomic KLR algebras
Let Λ be a dominant integral weight of type Γ (i.e., for each vertex i of Γ we fix a nonnegative
integer Λi). Let IΛ be the 2-sided ideal of Rpνq generated by Y
Λi1
1 1i with i P Seqpνq. In terms
of diagrams, this is the 2-sided ideal generated by all diagrams of the form
i1
Λi1
i2
� � �
i|ν|
,
with i P Seqpνq.
Definition 3.7. The cyclotomic KLR algebra is the quotient RΛpνq � Rpνq{IΛ.
3.5 DG-enhancements of Rpνq
We turn Rpνq into a DG-algebra by introducing a differential dΛ given by
dΛp1iq � dΛpYrq � dΛpτkq � 0, dΛpΩ1iq � p�Y1q
Λi11i,
together with the Leibniz rule
dΛpabq � dΛpaqb� p�1qλpaqdΛpbq.
This algebra is differential graded with respect to the homological degree given by counting the
number of floating dots. Since m is in the kernel of dΛ, we can extend dΛ to pRpνq.
The following is [16, Proposition 4.14].
Proposition 3.8. The homology of the DG-algebra pRpνq, dΛq is concentrated in degree 0 and
is isomorphic to the cyclotomic KLR algebra RΛpνq.
DG-Enhanced Hecke and KLR Algebras 17
4 The isomorphism theorems
4.1 A generalization of the Brundan–Kleshchev–Rouquier isomorphisms
Choose I, Γ and ν as in Section 3. Assume additionally that for i, j P I, i � j, there is at most
one arrow from i to j.
Let PolRd be as in Section 3.3. Set PolRν �
À
iPSeqpνq PolRd 1i. Here, similarly to (3.8),
the element 1i is the idempotent projecting to the ith component of the direct sum. Let PAν
be a PolRν-algebra free over PolRν (the most interesting examples for us are PAν � PRν
and PAν � PolRν). Set also yPAν � zPolRν bPolRν PAν .
Fix an action of S|ν| on yPAν (by ring automorphisms) that extends the obvious S|ν|-action
on zPolRν . We assume that such an extension exists. We make additionally the following
assumption.
Assumption 4.1. For each simple generator sr of S|ν|, each i P Seqpνq such that ir � ir�1 and
each f P yPAν , we have pf � srpfqq1i P pYr � Yr�1qyPAν .
This assumption implies that the Demazure operator 1�sr
Yr�Yr�1
is well defined on yPAν1i. Fix
a subalgebra yPA
1
ν of yPAν . Assume now that we have an algebra pApνq that has a faithful
representation on yPAν . We make the following assumption.
Assumption 4.2. The action of pApνq on yPAν is generated by multiplication by elements of yPA
1
ν
and by the operators τr, r P t1, 2, . . . , |ν| � 1u given by
� if ir � ir�1, then τr acts on f1i by a (nonzero scalar) multiple of the Demazure operator,
i.e., τr sends f1i to a multiple of pf�srpfqq1i
Yr�Yr�1
,
� if ir � ir�1, then τr sends f1i to Pir,ir�1pYr, Yr�1qsrpf1iq.
The goal for this section is to give non-trivial sufficient conditions for an algebra to be
isomorphic to pApνq, generalizing the BKR isomorphism.
The table below summarizes the various rings appearing on the KLR side and on the Hecke
side of the picture.
The KLR side The Hecke side (degenerate version)
PolRν �
À
iPSeqpνq
krY1, . . . , Yds1i Pold � krX1, . . . , Xds
PAν : a PolRν-algebra PBd: a Pold-algebrazPolRν �
À
iPSeqpνq
krrY1, . . . , Ydss1i xPola � À
bPSda
krrX1 � b1, . . . , Xd � bdss1b
yPAν � zPolRν bPolRν PAν
yPBa �
À
bPSda
�
krrX1 � b1, . . . , Xd � bdss bPold PBd
�
1b
yPA
1
ν � yPAν
yPB
1
a � yPBapApνq � xτr,yPA
1
νy � EndpyPAνq
p̄Ba � xTr,yPB
1
ay � EndpyPBaq
We have only included the degenerate version of the Hecke algebra in the column on the right,
the q-version being very similar.
4.1.1 Degenerate version
Fix Q � pQ1, . . . , Qℓq P kℓ, as in Section 2.2.1. Now we fix some special choice of Γ and ν.
Let I be a subset of k that contains Q1, . . . , Qℓ. We construct the quiver Γ with the vertex set I
using the following rule: for i, j P I we have an edge i Ñ j if and only if we have j � 1 � i.
18 R. Maksimau and P. Vaz
Note that this convention for Γ is opposite to [20]. Let d be a positive integer. Fix a P Id
(see Section 2.2.4). Finally, we consider ν such that νi is the multiplicity of i in a. In particular,
we see that |ν| � d is the length of a. Note that we have Seqpνq � Sda.
For each i P I, denote by Λi the multiplicity of i in pQ1, . . . , Qℓq. In particular, this im-
plies
±ℓ
r�1pX1 �Qrq �
±
iPIpX1 � iqΛi .
As above, we set Pold � krX1, � � � , Xds. Let PBd be a Pold-algebra free over Pold. The most
interesting examples are PBd � Pd and PBd � Pold. SetxPola � à
bPSda
krrX1 � b1, . . . , Xd � bdss1b,
yPBa �
à
bPSda
pkrrX1 � b1, . . . , Xd � bdss bPold PBdq1b.
Then yPBa is a xPola-algebra.
Fix an action of Sd on PBd (by ring automorphisms) that extends the obvious Sd-action
on Pold. We assume that such an extension exists. We assume additionally the following.
Assumption 4.3. For each simple generator sr of Sd and each f P PBd, we have
f � srpfq � pXr �Xr�1qPBd.
In particular, this assumption implies that the Demazure operator Br � 1�sr
Xr�Xr�1
is well
defined on PBd. The action of Sd on Pold and PBd can be obviously extended to an action onxPola and yPBa. Fix a subalgebra yPB
1
a of yPBa. We make the following assumption.
Assumption 4.4. There is an algebra p̄Ba that has a faithful representation in yPBa that is
generated by multiplication by elements of yPB
1
a and by the operators Tr � sr � Br.
By construction, we have the isomorphism
zPolRν � xPola, Yr1i ÞÑ pXr � irq1i. (4.1)
Moreover, this isomorphism commutes with the action of Sd. We assume the following.
Assumption 4.5. We can extend the isomorphism zPolRν � xPola in (4.1) to an Sd-invariant
isomorphism yPAν � yPBa. This extension restricts to an isomorphism yRA
1
ν � yPB
1
a.
We get the following proposition (if the Assumptions 4.1–4.5 are satisfied).
Proposition 4.6. There is an algebra isomorphism pApνq � p̄Ba that intertwines the representa-
tion in yPAν � yPBa.
Proof. We only have to show that we can write the operator τr in terms of Tr (and multiplication
by elements of yPA
1
ν � yPB
1
a) and vice versa.
First of all, note that the element pYr � Yr�1 � cq P krrY1, . . . , Ydss is invertible for each
nonzero c P k and that its inverse is c�1
�°
n¥0 c
�npYr�1 � Yrq
�
. Now, since we have
pXr �Xr�1q1i � pYr � Yr�1 � ir � ir�1q
under the isomorphism zPolRν � xPola, we see that the element pXr �Xr�1q
�11i P xPola is well
defined if ir � ir�1 and the element pXr �Xr�1 � 1q�11i P Pola is well defined if ir � 1 � ir�1.
First, we express τr in terms of Tr. We can rewrite the operator Tr in the following way:
Tr � 1�
Xr �Xr�1 � 1
Xr �Xr�1
psr � 1q.
DG-Enhanced Hecke and KLR Algebras 19
Fix i P Seqpνq � Sda. Assume ir � ir�1. Then the action of the operator pXr �Xr�1 � 1q�11i
on yPAν � yPBa is well defined. The element �pXr �Xr�1� 1q�1pTr � 1q1i acts on yPBa by the
same operator as τr1i.
Now, assume that we have ir � ir�1. If additionally we have no arrow ir Ñ ir�1, we can
write sr1i �
� Xr�Xr�1
Xr�Xr�1�1pTr � 1q � 1
�
1i. We need the condition ir�1 � 1 � ir to be able to
divide by pXr�Xr�1�1q here. The operator sr1i acts on yPAν � yPBa in the same way as τr1i.
Finally, if we have ir Ñ ir�1, then the operator
pXr �Xr�1 � 1qsr1i � rpXr �Xr�1qpTr � 1q � pXr �Xr�1 � 1qs1i
acts on yPBa in the same way as τr1i.
Now, we express Tr in terms of τr. The operator Tr1i acts by
�
1� pXr�Xr�1�1q
Xr�Xr�1
psr� 1q
�
1i. In
the case ir � ir�1, we are allowed to divide by Xr�Xr�1 here. If we additionally have no arrow
ir Ñ ir�1, then the element sr1i acts in the same way as τr1i. If we have an arrow ir Ñ ir�1,
then pXr �Xr�1 � 1qsr1i acts in the same way as τr1i. It remains to treat the case ir � ir�1.
In this case, the element sr�1
Xr�Xr�1
acts in the same way as �τr1i. ■
4.1.2 q-version
Fix q P k, q � 0, 1. Fix also Q � pQ1, . . . , Qℓq P pk�qℓ, as in Section 2.3.1. Now we fix some
special choice of Γ and ν. Let I be a subset of k� that contains Q1, . . . , Qℓ. We construct the
quiver Γ with the vertex set I using the following rule: for i, j P I we have an edge i Ñ j if and
only if we have qj � i. Note that this convention for Γ is opposite to [12] and [20]. Fix a P Id
(see Section 2.3.4). Finally, we consider ν such that νi is the multiplicity of i in a. In particular,
we see that |ν| � d is the length of a. Note that we have Seqpνq � Sda. As in the degenerate
case, for each i P I we denote by Λi the multiplicity of i in pQ1, . . . , Qℓq.
Set Polld � k
�
X�1
1 , � � � , X�1
d
�
. Let PBd be a Polld-algebra, free over Polld. The most inter-
esting examples are PBd � Pd and PBd � Polld. Set xPola �bPSda
krrX1�b1, . . . , Xd�bdss1b
and yPBa �
À
bPSda
pkrrX1 � b1, . . . , Xd � bdss bPolld PBdq1b. Then yPBa is a xPola-algebra.
Fix an action of Sd on PBd (by ring automorphisms) that extends the obvious Sd-action on
Poll. We assume additionally the following.
Assumption 4.7. For each simple generator sr of Sd and each f P PBd, we have
f � srpfq � pXr �Xr�1qPld.
In particular, this assumption implies that the Demazure operator 1�sr
Xr�Xr�1
is well defined
on Pld. The action of Sd on Polld and Pld can be obviously extended to an action on yPolla
and yPBa.
Fix a subalgebra yPB
1
a of yPBa. We make the following assumption.
Assumption 4.8. There is an algebra pBa that has a faithful representation in yPBa that is
generated by multiplication by elements of yPB
1
a and by the operators
Tr � q �
pqXr �Xr�1q
Xr �Xr�1
psr � 1q.
By construction, we have the isomorphism
zPolRν � xPola, Yr1i ÞÑ i�1
r pXr � irq1i. (4.2)
Moreover, this isomorphism commutes with the action of Sd. We assume the following.
20 R. Maksimau and P. Vaz
Assumption 4.9. We can extend the isomorphism zPolRν � xPola in (4.2) to an Sd-invariant
isomorphism yPAν � yPBa. This extension restricts to an isomorphism yPA
1
ν � yPB
1
a.
Then we have the following (if Assumptions 4.1, 4.2, 4.7, 4.8, 4.9 are satisfied).
Proposition 4.10. There is an algebra isomorphism pApνq � pBa that intertwines the represen-
tation in yPAν � yPBa.
Proof. We only have to show that we can write the operator τr in terms of Tr (and multiplication
by elements of yPA
1
ν � yPB
1
a) and vice versa. First, we express τr in terms of Tr. Fix i P Seqpνq �
Sda.
Assume ir � ir�1. Then the action of the operator pqXr �Xr�1q
�11i on yPAν � yPBa is well
defined. The element �pqXr �Xr�1q
�1pTr � qq1i acts on yPBa by the same operator as τr1i.
Now, assume that we have ir � ir�1. If moreover we have no arrow ir Ñ ir�1, we can write
sr1i �
� Xr�Xr�1
qXr�Xr�1
pTr � qq � 1
�
1i (we need the condition qir�1 � ir to be able to divide by
pqXr �Xr�1q here). The operator sr1i acts on yPAν � yPBa in the same way as τr1i. Finally, if
we have ir Ñ ir�1, then the operator pqXr�Xr�1qsr1i � rpXr�Xr�1qpTr�qq�pqXr�Xr�1qs1i
acts on yPBa in the same way as τr1i up to scalar.
Now, we express Tr in terms of τr. The operator Tr1i acts by
�
q � pqXr�Xr�1q
Xr�Xr�1
psr � 1q
�
1i.
In the case ir � ir�1, we are allowed to divide by Xr � Xr�1 here. If we additionally have
no arrow ir Ñ ir�1, then the element sr1i acts in the same way as τr1i. If we have an ar-
row ir Ñ ir�1, then pqXr �Xr�1qsr1i acts up to scalar in the same way as τr1i. It remains to
treat the case ir � ir�1. In this case, the element sr�1
Xr�Xr�1
acts in the same way as �τr1i. ■
4.2 The DG-enhanced isomorphism theorem: the degenerate version
In Proposition 4.6, we proved that we have an isomorphism of algebras pApνq � p̄Ba for some
algebras pApνq and p̄Ba that satisfy some list of properties. Let us show that we can apply Propo-
sition 4.10 to the special situation pApνq � pRpνq and p̄Ba �
x̄Ha. We assume that ν and a are
related as in Section 4.1.1. In this case we can take yPAν � yPRν and yPBa � pPa. We consider
the subalgebra yPA
1
ν of yPAν generated by zPolRν and Ω1, and the subalgebra yPB
1
a of yPBa
generated by xPola and θ1.
To be able to apply Proposition 4.6, we only have to construct a Sd-invariant isomor-
phism α : pPa � yPRν extending the isomorphism (4.1) such that α restricts to an isomorphismyPB
1
a � yPA
1
ν . First, we consider the following homomorphism α1 : xPola Ñ yPRν .
1i ÞÑ 1i, Xr1i ÞÑ pYr � irq1i.
This homomorphism is obviously Sd-invariant.
Remark 4.11. For each 1 ¤ r d, the Demazure operator Br �
1�sr
Xr�Xr�1
is well defined on pPa.
Now, using the isomorphism xPola � zPolRν , we can consider it as an operator on yPRν . The
action of Br on yPRν can be given explicitly by
Brpf1iq �
f1i � srpfq1srpiq
Yr � Yr�1 � ir � ir�1
, f P krrY1, . . . , Ydss.
Attention, the operator Br on yPRν should not be confused with 1�sr
Yr�Yr�1
, which is not well
defined. The Demazure operators Br on yPRν satisfy relations (2.3), (2.4), (2.5).
Now, we want to extend α1 to a homomorphism α : pPa Ñ yPRν . To do this, we have to choose
the images of θ1, θ2, . . . , θd in yPRν such that these images anticommute with each other and
DG-Enhanced Hecke and KLR Algebras 21
commute with the image of xPola (i.e., with zPolRν). Moreover, we want to make this choice in
such a way that α is bijective and Sd-invariant.
First, we set
αpθ11iq �
� ¹
iPI,i�i1
pY1 � i1 � iqΛi
p�1qΛi1Ω11i. (4.3)
This choice is motivated by the fact that we will want α to be compatible with the DG-structure.
For r ¡ 1, we construct the images of other θr in the following way
αpθrq � p�1qr�1Br�1 � � � B2B1pαpθ1qq. (4.4)
This choice is motivated by the fact that we want α to be Sd-invariant and we have that
θr � �Br�1pθr�1q. Since we have sr � 1� pXr �Xr�1qBr, equation (4.4) implies immediately
αpsrpθrqq � srpαpθrqq. (4.5)
Lemma 4.12. The homomorphism α : pPa Ñ yPRν given by (4.3) and (4.4) is an isomorphism
and it is Sd-invariant.
Proof. Since the homomorphism α1 : xPola Ñ yPRν is obviously Sd-invariant, to show the Sd-
invariance of α, we have to show
skpαpθr1iqq � αpskpθr1iqq (4.6)
for each i P Seqpνq, each r P r1; ds and each k P r1; d � 1s. We give a proof by induction on r.
First, we prove (4.6) for r � 1. If k ¡ 1 and r � 1, then (4.6) is obvious because θ1 and αpθ1q
are sk-invariant. The case k � r � 1 follows from (4.5).
Now, assume that r ¡ 1 and that (4.6) is already proved for smaller values of r. The
case k � r follows from (4.5).
For k � r, the element θr is sk-invariant. So (4.6) is equivalent to the sk-invariance of αpθrq.
Assume that k ¡ r or k r � 2. This assumption implies that sk commutes with sr�1.
Moreover, we already know by induction hypothesis that αpθr�1q is sk-invariant. So, the sk-
invariance of αpθr�1q together with (4.4) implies the sk-invariance of αpθrq.
Now, assume k � r � 1. In this case the sr�1-invariance of αpθrq is obvious from (4.4).
Finally, assume k � r � 2. To prove the sr�2-invariance of αpθrq, we have to show that
Br�2pαpθrqq � 0. We have
Br�2pαpθrqq � Br�2Br�1Br�2pαpθr�2qq � Br�1Br�2Br�1pαpθr�2qq.
This is equal to zero because Br�1pαpθr�2qq � 0 by the sr�1-invariance of αpθr�2q.
This completes the proof of the Sd-invariance of α.
Now, let us prove that α is an isomorphism. It is easy to see from (4.3) and (4.4) that αpθr1iq
is of the form
αpθr1iq �
ŗ
t�1
PtΩt1i, (4.7)
where Pt P yPRν1i for r P t1, 2, . . . , ru and Pr is invertible in yPRν1i. Then the bijectivity is clear
from (4.7) and from the fact that α restricts to a bijection xPola � zPolRν . ■
We get the following theorem.
Theorem 4.13. There is an isomorphism of DG-algebras p pRpνq, dΛq � px̄Ha, BQq.
22 R. Maksimau and P. Vaz
Proof. Note that (4.3) implies that the isomorphism α (see Lemma 4.12) identifies the subal-
gebra yPA
1
ν of yPAν with the subalgebra yPB
1
a of yPBa. Then the isomorphism of algebras follows
immediately from Proposition 4.6. We only have to check the DG-invariance.
Denote by γ the isomorphism of algebras γ : x̄Ha Ñ pRpνq. It is obvious that γ preserves the
λ-grading. We claim that for each h P x̄Ha, we have
γpBQphqq � dΛpγphqq. (4.8)
Indeed, it is enough to check (4.8) for h � θ. This follows directly from (4.3). In fact, this is
exactly the reason why we define (4.3) in such a way. ■
Remark 4.14. We could also take yPAν � yPA
1
ν �
zPolRν and yPBa � yPB
1
a �
xPola. Then we
get (the completion version of) the usual Brundan–Kleshchev–Rouquier isomorphism.
4.3 The DG-enhanced isomorphism theorem: the q-version
In Proposition 4.6, we proved that we have an isomorphism of algebras pApνq � pBa for some
algebras pApνq and pBa that satisfy some list of properties. Let us show that we can apply Propo-
sition 4.10 to the special situation pApνq � pRpνq and pBa � pHa. We assume that ν and a are
related as in Section 4.1.2. In this case, we can take yPAν � yPRν and yPBa � pPa.
To be able to apply Proposition 4.10, we only have to construct a Sd-invariant isomor-
phism α : yPRν � pPa extending the isomorphism (4.2) such that α restricts to an isomorphismyPA
1
ν � yPB
1
a (we choose the subalgebras yPA
1
ν � yPAν and yPB
1
a � yPBa in the same way as
in Section 4.2). This can be done in the same way as in the degenerate case. However, some for-
mulas in this case are different from the previous section because of the difference between (4.1)
and (4.2). Here, we only give the modified formulas. The proofs are the same as in the previous
section.
We consider the Sd-invariant homomorphism α1 : xPola Ñ yPRν
1i ÞÑ 1i, Xr1i ÞÑ irpYr � 1q1i.
Now, we extend α1 to a homomorphism α : pPa Ñ yPRν in the following way:
αpθ11iq �
� ¹
iPI, i�i1
pi1pY1 � 1q � iqΛi
p�i1q
Λi1Ω11i,
αpθrq � p�1qr�1Br�1 � � � B2B1pαpθ1qq.
As in the previous section, we can show that α is a Sd-invariant isomorphism.
We get the following theorem.
Theorem 4.15. There is an isomorphism of DG-algebras
� pRpνq, dΛ� � � pHa, BQ
�
.
Remark 4.16. We could also take yPAν � yPA
1
ν �
zPolRν and yPBa � yPB
1
a �
xPola. Then we
get (the completion version of) the usual Brundan–Kleshchev–Rouquier isomorphism.
4.4 The homology of H̄d and Hd
We now have the tools to prove the following two propositions.
Proposition 4.17. The homology of the DG-algebra
� sHd, BQ
�
is concentrated in degree 0 and
is isomorphic to sHQ
d .
Proposition 4.18. The homology of the DG-algebra pHd, BQq is concentrated in degree 0 and
is isomorphic to HQ
d .
DG-Enhanced Hecke and KLR Algebras 23
First, we start from a similar statement for the KLR algebra.
Proposition 4.19. The homology of the DG-algebra
� pRpνq, dΛ� is concentrated in degree 0 and
is isomorphic to RΛpνq.
Proof. It is proved in [16, Proposition 4.14] that the homology of the DG-algebra pRpνq, dΛq is
concentrated in degree 0 and is isomorphic to RΛpνq. The same proof with minor modifications
applies to our case. We just have to replace polynomials by power series. ■
Corollary 4.20. The homologies of the DG-algebras
�x̄Ha, BQ
�
and
� pHa, BQ
�
are concentrated
in degree 0 and are isomorphic to H̄Q
a and HQ
a , respectively.
Proof. The statement follows from Theorems 4.13 and 4.15, Proposition 4.19 and from the
usual Brundan–Kleshchev–Rouquier isomorphism. ■
Proof of Propositions 4.17 and 4.18. It is obvious that the homology group of
�
H̄d, BQ
�
in
degree zero is H̄Q
d . We only have to check that the homology groups in other degrees are zero.
Assume, that for some i ¡ 0, we have H i
�
H̄d, BQ
�
� 0 and consider it as a Pold-module. The
annihilator of this Pold-module is contained in some maximal ideal M � Pold. The ideal M is
of the form M � pX1 � a1, . . . , Xd � adq for some a � pa1, . . . , adq P kd.
Then the completion of H i
�
H̄d, BQ
�
� 0 with respect to the ideal M is nonzero. This leads
to a contradiction because H i
�x̄Ha, BQ
�
� 0 together with Künneth formula implies
krrX1 � a1, . . . , Xd � adss bPold H
i
�
H̄d, BQ
�
� 0.
Proposition 4.18 is proved in the same way. ■
Acknowledgements
We thank Jonathan Grant for useful discussions and the anonymous referees for the careful
reading of our document. PV was supported by the Fonds de la Recherche Scientifique – FNRS
under Grant no. MIS-F.4536.19.
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1 Introduction
2 DG-enhanced versions of Hecke algebras
2.1 The polynomial rings Pol_d and Poll_d and the rings P_d and Pl_d
2.1.1 The polynomial rings Pol_d and Poll_d
2.1.2 The rings P_d and Pl_d
2.2 Degenerate version
2.2.1 Degenerate affine Hecke algebra
2.2.2 The algebra bar H_d
2.2.3 DG-enhancement of bar H_d
2.2.4 Completions of bar H_d
2.3 q-version
2.3.1 Affine q-Hecke algebra
2.3.2 The algebra H_d
2.3.3 DG-enhancement of H_d
2.3.4 Completions of H_d
3 DG-enhanced versions of KLR algebras
3.1 The algebra R(nu)
3.2 Polynomial action of R(nu)
3.3 Completion of R(nu)
3.4 Cyclotomic KLR algebras
3.5 DG-enhancements of R(nu)
4 The isomorphism theorems
4.1 A generalization of the Brundan–Kleshchev–Rouquier isomorphisms
4.1.1 Degenerate version
4.1.2 q-version
4.2 The DG-enhanced isomorphism theorem: the degenerate version
4.3 The DG-enhanced isomorphism theorem: the q-version
4.4 The homology of bar H_d and H_d
References
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| id | nasplib_isofts_kiev_ua-123456789-212036 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T21:24:54Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Maksimau, Ruslan Vaz, Pedro 2026-01-23T10:09:52Z 2023 DG-Enhanced Hecke and KLR Algebras. Ruslan Maksimau and Pedro Vaz. SIGMA 19 (2023), 095, 24 pages 1815-0659 2020 Mathematics Subject Classification: 20C08; 16E45 arXiv:1906.03055 https://nasplib.isofts.kiev.ua/handle/123456789/212036 https://doi.org/10.3842/SIGMA.2023.095 We construct DG-enhanced versions of the degenerate affine Hecke algebra and of the affine Hecke algebra. We extend Brundan-Kleshchev and Rouquier's isomorphism and prove that after completion, DG-enhanced versions of affine Hecke algebras (degenerate or nondegenerate) are isomorphic to completed DG-enhanced versions of KLR algebras for suitably defined quivers. As a byproduct, we deduce that these DG-algebras have homologies concentrated in degree zero. These homologies are isomorphic respectively to the degenerate cyclotomic Hecke algebra and the cyclotomic Hecke algebra. We thank Jonathan Grant for useful discussions and the anonymous referees for the careful reading of our document. The Fonds de la Recherche Scientifique supported PV– FNRS under Grant No. MIS-F.4536.19. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications DG-Enhanced Hecke and KLR Algebras Article published earlier |
| spellingShingle | DG-Enhanced Hecke and KLR Algebras Maksimau, Ruslan Vaz, Pedro |
| title | DG-Enhanced Hecke and KLR Algebras |
| title_full | DG-Enhanced Hecke and KLR Algebras |
| title_fullStr | DG-Enhanced Hecke and KLR Algebras |
| title_full_unstemmed | DG-Enhanced Hecke and KLR Algebras |
| title_short | DG-Enhanced Hecke and KLR Algebras |
| title_sort | dg-enhanced hecke and klr algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212036 |
| work_keys_str_mv | AT maksimauruslan dgenhancedheckeandklralgebras AT vazpedro dgenhancedheckeandklralgebras |