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 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.


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 quantum 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 A V , the isomorphism to the group algebra of the symmetric group in d letters kS d gives the latter a graded presentation.The grading on kS d 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.
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.
paq Are there DG-enhanced versions of affine Hecke algebras that are free resolutions of cyclotomic 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 qaffine 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 N (where 0 N) and a field k that for simplicity we consider to be algebraically closed.We consider the Z-graded algebra s H d generated by T 1 , . . ., T d¡1 and X 1 , . . ., X d in degree zero and θ in degree 1.The generators T 1 , . . ., T d¡1 and X 1 , . . ., X d satisfy the relations of the degenerate affine Hecke algebra s H d .The generator θ commutes with the X r 's and with T 2 , . . ., T d¡1 and satisfies θ 2 0 and T 1 θT 1 θ θT 1 θT 1 0. This implies that the subalgebra of s H d concentrated in degree zero is isomorphic to s H d .For Q pQ 1 , . . ., Q ℓ q k ℓ , we introduce a differential f Q by declaring that it acts as zero on s We denote by x Ha the completion of the algebra s H d at a sequence of ideals depending on a k d .
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 Q r I for each r.We fix a I d and we set ν and Λ such that ν i and Λ i are the multiplicities of i in respectively a and Q.We have There is a similar construction for the affine q-Hecke algebra, which we do in Section 2.3 and Section 4.3.Fix q k, q $ 0, 1 and denote by pH d , f Q q and by p H a , f Q ¨the DG-enhanced version of the affine q-Hecke and its completion.The construction of H d also adds a variable θ in degree 1 that also satisfies θ 2 0 and commutes with all generators but T 1 the relation being In a nutshell, fix Q pQ 1 , . . ., Q ℓ q pk ¢ q ℓ .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 Q 1 , . . ., Q ℓ and fix a I d .We define ν and Λ in the same way as above.Let pRpνq, d Λ q be the DG-enhanced version of the KLR algebra of type Γ with ν and Λ as above and let p Rpνq, d Λ ¨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 p Rpνq, The two main results above imply that we have a family of isomorphisms p Rpνq p H a 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 Hd and H d in the following way.It is already proved in [16,Proposition 4.14] that the homology of the DG-algebra pRpνq, d Λ q 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 s 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 s Proposition 4.18.The homology of the DG-algebra pH d , f Q q is concentrated in degree 0 and is isomorphic to 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 Hd and H d 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.
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 representation 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 Hd and H d 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 completion).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 Hd and H d are free resolutions of the corresponding cyclotomic Hecke algebras.

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.Fix an algebraically closed field k, q k, q $ 0, 1 and d N once and for all.

The polynomial rings Pol d and Poll d
Set Pol d krX 1 , . . ., X d s.Let S d be the symmetric group on d letters, which we view as a Coxeter group with generators s 1 , . . ., s d¡1 .These correspond to the simple transpositions pi i 1q, and we use these two descriptions interchangeably throughout.As usual, we let S d act from the left on Pol d by permuting the variables: for w S d we have wpX i q X wpiq , and wpf gq wpf qwpgq for f, g Pol d .
Using the S d -action above, one defines the Demazure operators f i on P d for all 1 ¤ i ¤ d ¡ 1 in the usual way, as We have s i f i pfq f i pfq and f i ps i f q ¡f i pfq for all i, so f i is in fact an operator from Pol d to the subring Pol for all f, g Pol d and for 1 ¤ i ¤ n ¡ 1, and the relations where pθq is the exterior k-algebra in the variables θ.Here P d is a subring concentrated in parity zero.Introduce an additional Z-grading on P d denoted λpq and defined as λpX i q 0 and λpθ i q 1.This grading is half the grading deg λ introduced in [14, Section 3.1].If we forget the grading, the algebra P d is the symmetric algebra corresponding to a superspace of dimension pd|dq.
As explained in [14,Section 8.3], the action of S d on Pol d extends to an action on P d by setting This action respects the grading, as one easily checks, and allows extending the action of the Demazure operators in (2.1) to P d .We denote the extensions of the Demazure operators to P d by the same symbols.Similarly to the operators above, f i is an operator from P d to the subring As in the case of P d above, we form the supercommutative ring This ring is also endowed with the grading λpq, which is defined in the same way as in P d .Moreover, the S d -action on Poll d can be obviously extended to a S d -action on P l d .This means that the action of the Demazure operators on Poll d also extends to operators on P l d that satisfy the relations in (2.2) (for f and g in P l d ) and (2.3)-(2.5).

Degenerate affine Hecke algebra
The degenerate affine Hecke algebra s H d is the k-algebra generated by T 1 , . . ., T d¡1 and X 1 , . . ., X d , with relations (2.9)For w s i 1 . . .s i k S d a reduced expression, we put T w T i 1 . . .T i k .Then T w is independent of the choice of the reduced expression of w and the set There is a faithful representation of s Let ℓ be a positive integer and Q pQ 1 , . . ., Q ℓ q be an ℓ-tuple of elements of the field k.
Definition 2.1.The degenerate cyclotomic Hecke algebra is the quotient

The algebra Hd
Definition 2.2.Define the algebra Hd as the k-algebra generated by T 1 , . . ., T d¡1 and X 1 , . . ., X d in λ-degree zero, and an extra generator θ in λ-degree 1, with relations (2.7) to (2.9) and The algebra Hd contains the degenerate affine Hecke algebra Hd as a subalgebra concentrated in λ-degree zero.
Lemma 2.3.The algebra Hd acts on P d by T r pfq s r pfq ¡ f r pfq, X r pfq X r f, θpf q θ 1 f, for all f P d and where s r pfq and f r pfq are as in (2.6) and (2.1).
Proof .The defining relations of Hd can be checked by a straightforward computation.■ Define ξ 1 , . . ., ξ d Hd by the rules ξ 1 θ, ξ i 1 T i ξ i T i .The following is straightforward.Lemma 2.4.The elements ξ r satisfy for all r t1, . . ., d ¡ 1u and all ℓ t1, . . ., du, 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: Abusing the notation, we will write θ r for the operator on P d that multiplies each element of P d by θ r .Set M t0, Proof .Let H °bM h b θ b be an operator that acts by zero.Assume that H has a nonzero coefficient.Let b 0 be such that h b 0 $ 0 and such that |b 0 | is minimal with this property.Then for each element P P d , we have H θ b 0 P ¨ ¨hb 0 θ 1 P .This shows that h b 0 acts by zero on θ , is a basis of the k-vector space H¤k d .Proof .It is clear that the given set spans.Linear independence follows from Lemma 2. Proof .We prove by induction on k.The case k 1 is trivial.Now, assume that d k is not a right zero divisor and let us show that d k 1 is not a right zero divisor.Since we have we get It is enough to check that the element ppX k 1 ¡ X k qT k ¡ 1q is not a right zero divisor.This follows from the fact that it acts on P d by the operator ■ It is not hard to write a basis of Hd in terms of the ξ r 's.
Proposition 2.10.The set , is a basis of the k-vector space Hd .
Proof .We start by showing that this set spans Hd .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 ξ 1 and we move all ξ's to the right by using Lemma 2.4.This shows that the set above spans Hd .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 Hd has a triangular decomposition (only as a vector space) Hd !krX 1 , . . ., X d s kS d pξ 1 , . . ., ξ d q.

DG-enhancement of Hd
Let ℓ and Q be as in Section 2.2.1.
Definition 2.13.Define an operator f Q on s H d by declaring that f Q acts as zero on s and it respects the graded Leibniz rule: for a, b s Proof .We prove something slightly more general.Let P krX 1 s be a polynomial.Define H d by declaring that d P acts as zero on s H d , while d P pθq P , together with the graded Leibniz rule.Then d P is a differential on s H d .To prove the claim is suffices to check that d P pT 1 θT 1 θ θT 1 θT 1 q 0. We have T 1 P s 1 pPqT 1 ¡ f 1 pPq and P T 1 T 1 s 1 pPq ¡ f 1 pPq, where f 1 is the Demazure operator.This also implies T 1 P T 1 s 1 pPq ¡ f 1 pPqT 1 .Note also that f 1 pPq is a symmetric polynomial with respect to X 1 , X 2 , so it commutes with T 1 .So, we have which proves the claim.■ We will prove in Proposition 4.17 that the homology of the DG-algebra s

Completions of Hd
Consider the algebra of symmetric polynomials Sym d Pol S d d .We consider it as a (central) subalgebra of Hd .
For each d-tuple a pa 1 , . . ., a d q k d we have a character χ a : Sym d Ñ k given by the evaluation X r Þ Ñ a r .It is obvious from the definition that if the d-tuple a I is a permutation of the d-tuple a then the characters χ a and χ a I are the same.Denote by m a the kernel of χ a .

Definition 2.15. Denote by x
Ha the completion of the algebra Hd with respect to m a .
Since m a is in the kernel of f Q , we can extend f Q to x Ha .Set also where 1 b is just a formal idempotent projecting on the corresponding direct factor.Here S d a is the S d -orbit of a with respect to the obvious S d -action on k d .We can obviously extend the action of Hd on P d to an action of x Ha on p P a .Each finite dimensional x Ha -module M decomposes into its generalized eigenspaces M À bS d a M b , where For each b S d a the algebra x Ha contains an idempotent 1 b that projects onto M b when applied to M .pbq The representation p P a of x Ha is faithful.
Proof .It is clear that the elements from the statement generate the x Pol a -module x Ha .To see that they form a basis, it is enough to remark that they act by linear independent over x Pol a öperators on the representation p P a .This proves paq.Then pbq also holds because a basis acts on p P a by linearly independent operators.■ The algebra HQ d has a decomposition HQ a (with a finite number of nonzero terms) such that Sym d acts on each finite dimensional HQ a -module with a generalized character χ a .

Affine q-Hecke algebra
The affine q-Hecke algebra H d is the k-algebra generated by T 1 , . . ., T d¡1 and X ¨1 1 , . . ., X ¨1 d , with relations (2.12) Note that relation (2.11) implies that the element T i is invertible.For w s i 1 . . .s i k S d a reduced decomposition, we put T w T i 1 . . .T i k .Then T w is independent of the choice of the reduced decomposition of w and the set is a basis of the k-vector space H d .There is a faithful representation of H d on Poll d , where Let ℓ be a positive integer.Let Q pQ 1 , . . ., Q ℓ q be an ℓ-tuple of nonzero elements of the field k.
Definition 2.17.The cyclotomic q-Hecke algebra is the quotient The algebra H d contains the affine q-Hecke algebra H d as a subalgebra concentrated in λdegree zero.
Lemma 2.19.The algebra H d acts on P l d by T r pfq qs r pfq ¡ pq ¡ 1qX r 1 f r pfq, X ¨1 r pfq X ¨1 r f, θpf q θ 1 f, for all f P d and where s r pfq and f r pfq are as in (2.6) and (2.1).
Proof .The defining relations of H d can be checked by a straightforward computation.■ Define ξ 1 , . . ., ξ d H d by the rules ξ 1 θ, ξ i 1 T i ξ i T ¡1 i .The following is straightforward.
Proposition 2.21.The set , is a basis of the k-vector space H d . Proof Proof .Similarly to the proof of Lemma 2.14, we consider a more general differential d P .We have to check d P pT 1 θT 1 θ θT 1 θT 1 q d P ppq ¡ 1qθT 1 θq.
We have T 1 P s 1 pPqT 1 ¡ pq ¡ 1qX 2 f 1 pPq and P T 1 T 1 s 1 pPq ¡ pq ¡ 1qX 2 f 1 pPq, where f 1 is the Demazure operator.Note also that f 1 pPq is a symmetric polynomial with respect to X 1 , X 2 , so it commutes with T 1 .So, we have which proves the claim.■ We will prove in Proposition 4.18 that the homology of the DG-algebra pH d , f Q q is concentrated in degree 0 and is isomorphic to pbq The representation p P a of p H a is faithful.
The algebra with a finite number of nonzero terms) such that Syml d acts on each finite dimensional H Q a -module with a generalized character χ a .

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.
and set |ν| °i ν i .We allow the quiver to have infinite number of vertices.In this case, only a finite number of ν i is nonzero.
For each i, j I, we denote by h i,j the number of arrows in the quiver Γ going from i to j, and define for i $ j the polynomials Q i,j pu, vq pu ¡ vq h i,j pv ¡ uq h j,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 algebra Rpνq
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 I.
The KLR relations, for all i, j, k I: And the additional relations, for all i, j I: We now define a Z ¢ Z-grading in Rpνq.Contrary to [16], we work with a single homological degree λ.The homological nature of this degree justified by the DG-structure defined in Section 3.5.We declare p¡2, 0q if i j, p¡1, 0q if h i,j 1, p0, 0q otherwise, and deg where the second grading is called λ-grading, which we write λpq.The defining relations of Rpνq are homogeneous with respect to this bigrading.
Remark 3.3.The subalgebra of Rpνq in λ-degree zero coincides with the usual KLR algebra 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 i 1 . . .i d , define the idempotent and let Seqpνq be the set of all ordered sequences i i 1 i 2 . . .i d with each i k I and i appearing ν i times in the sequence.For i, j Seqpνq the idempotents 1 i and 1 j are orthogonal iff i $ j, we have 1 Rpνq °iSeqpνq 1 i , where 1 Rpνq denotes the identity element in Rpνq, and Finally, the algebra R is defined as R à νNrIs Rpνq.

Polynomial action of Rpνq
We now describe a faithful action of Rpνq on a supercommutative ring, which was defined in [ Here we mean that the algebra P R ν is a direct sum of copies of the algebra P R d , labelled by Seqpνq.We denote by 1 i the idempotent projecting to the ith copy.
For each i I, 1 ¤ r ¤ ν i and i pi 1 , i 2 , . . ., i d q Seqpνq, we denote by r I r I pr, i, iq the rth index r I t1, 2, . . ., du (counting from the left) among the indices such that i r I i.
The algebra P R ν is bigraded supercommutative with gradings degpY t q p2, 0q, degpω r,i q p¡2r, 1q and degp1 i q p0, 0q, where the variables ω r,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 P R ν given by For each i, j I, i $ j, we consider the polynomial P ij pu, vq pu ¡ vq h i,j , where h i,j denotes as above the number of arrows from i to j.Note that we have Q i,j pu, vq P i,j pu, vqP j,i pv, uq.
In the sequel, it is useful to have an algebraic presentation of Rpνq as in [2, equations (1.7)- (1.15)].We set We declare that a e k Rpνqe j acts as zero on P R I 1 i whenever j $ i. Otherwise 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 P R ν .We would like to construct a representation structure of p Rpνq in the vector space y P R ν .The S |ν| -action on P R ν extends obviously to an S |ν| -action on y P R ν .Moreover, the action of Rpνq on P R ν yields an action of p Rpνq on y P R ν .

Completion of Rpνq
Lemma 3.6.The representation y P R ν of p Rpνq is faithful.
Proof .An explicit PolR d -basis of R ν is constructed in [16,Section 3.2].We would like to check that the same set forms a z PolR d -basis of p R ν .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 on y P R ν by linearly independent operators.Then, this proves automatically the faithfulness of the representation.■

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 Λ i ).Let I Λ be the 2-sided ideal of Rpνq generated by Y 1 1 i with i Seqpνq.In terms of diagrams, this is the 2-sided ideal generated by all diagrams of the form , with i Seqpνq.Definition 3.7.The cyclotomic KLR algebra is the quotient R Λ pνq Rpνq{I Λ .

DG-enhancements of Rpνq
We turn Rpνq into a DG-algebra by introducing a differential d Λ given by together with the Leibniz rule 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 d Λ , we can extend d Λ to p Rpνq.The following is [16,Proposition 4.14].
Proposition 3.8.The homology of the DG-algebra pRpνq, d Λ q is concentrated in degree 0 and is isomorphic to the cyclotomic KLR algebra R Λ pνq.

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 I, i $ j, there is at most one arrow from i to j.
Let PolR d be as in Section 3.3.Set PolR ν À iSeqpνq PolR d 1 i .Here, similarly to (3.8), the element 1 i is the idempotent projecting to the ith component of the direct sum.Let P A ν be a PolR ν -algebra free over PolR ν (the most interesting examples for us are P A ν P R ν and P A ν PolR ν ).Set also y P A ν z PolR ν PolRν P A ν .
Fix an action of S |ν| on y P A ν (by ring automorphisms) that extends the obvious S |ν| -action on z PolR ν .We assume that such an extension exists.We make additionally the following assumption.
Assumption 4.1.For each simple generator s r of S |ν| , each i Seqpνq such that i r i r 1 and each f y P A ν , we have pf ¡ s r pfqq1 i pY r ¡ Y r 1 q y P A ν .
This assumption implies that the Demazure operator 1¡sr Yr¡Y r 1 is well defined on y P A ν 1 i .Fix a subalgebra y P A I ν of y P A ν .Assume now that we have an algebra p Apνq that has a faithful representation on y P A ν .We make the following assumption.
Assumption 4.2.The action of p Apνq on y P A ν is generated by multiplication by elements of y P A I ν and by the operators τ r , r t1, 2, . . ., |ν| ¡ 1u given by if i r i r 1 , then τ r acts on f 1 i by a (nonzero scalar) multiple of the Demazure operator, i.e., τ r sends f 1 i to a multiple of pf¡srpfqq1 i Yr¡Y r 1 , if i r $ i r 1 , then τ r sends f 1 i to P ir,i r 1 pY r , Y r 1 qs r pf1 i q.
The goal for this section is to give non-trivial sufficient conditions for an algebra to be isomorphic to p Apνq, 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) We have only included the degenerate version of the Hecke algebra in the column on the right, the q-version being very similar.
Let I be a subset of k that contains Q 1 , . . ., Q ℓ .We construct the quiver Γ with the vertex set I using the following rule: for i, j I we have an edge i Ñ j if and only if we have j 1 i.
Note that this convention for Γ is opposite to [20].Let d be a positive integer.Fix a I d (see Section 2.2.4).Finally, we consider ν such that ν i is the multiplicity of i in a.In particular, we see that |ν| d is the length of a.Note that we have Seqpνq S d a.
For each i I, denote by Λ i the multiplicity of i in pQ 1 , . . ., Q ℓ q.In particular, this implies By construction, we have the isomorphism Moreover, this isomorphism commutes with the action of S d .We assume the following.We get the following proposition (if the Assumptions 4.1-4.5 are satisfied).
Proposition 4.6.There is an algebra isomorphism p Apνq p Ba that intertwines the representation in y P A ν y P B a .
Proof .We only have to show that we can write the operator τ r in terms of T r (and multiplication by elements of y P A First of all, note that the element pY r ¡ Y r 1 cq krrY 1 , . . ., Y d ss is invertible for each nonzero c k and that its inverse is c ¡1 °n¥0 c ¡n pY r 1 ¡ Y r q ¨.Now, since we have pX r ¡ X r 1 q1 i pY r ¡ Y r 1 i r ¡ i r 1 q under the isomorphism z PolR ν x Pol a , we see that the element pX r ¡ X r 1 q ¡1 1 i x Pol a is well First, we express τ r in terms of T r .We can rewrite the operator T r in the following way: commute with the image of x Pol a (i.e., with z PolR ν ).Moreover, we want to make this choice in such a way that α is bijective and S d -invariant.
First, we set 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 θ r in the following way αpθ r q p¡1q r¡1 f r¡1 ¤ ¤ ¤ f 2 f 1 pαpθ 1 qq.This choice is motivated by the fact that we want α to be S d -invariant and we have that θ r ¡f r¡1 pθ r¡1 q.Since we have s r 1 ¡ pX r ¡ X r 1 qf r , equation (4.4) implies immediately αps r pθ r qq s r pαpθ r qq.
for each i Seqpνq, each r r1; ds and each k 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 θ 1 and αpθ 1 q are s k -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 θ r is s k -invariant.So (4.6) is equivalent to the s k -invariance of αpθ r q.
Assume that k ¡ r or k r ¡ 2. This assumption implies that s k commutes with s r¡1 .Moreover, we already know by induction hypothesis that αpθ r¡1 q is s k -invariant.So, the s kinvariance of αpθ r¡1 q together with (4.4) implies the s k -invariance of αpθ r q.Now, assume k r ¡ 1.In this case the s r¡1 -invariance of αpθ r q is obvious from (4.4).Finally, assume k r ¡ 2. To prove the s r¡2 -invariance of αpθ r q, we have to show that f r¡2 pαpθ r qq 0. We have f r¡2 pαpθ r qq f r¡2 f r¡1 f r¡2 pαpθ r¡2 qq f r¡1 f r¡2 f r¡1 pαpθ r¡2 qq.
This is equal to zero because f r¡1 pαpθ r¡2 qq 0 by the s r¡1 -invariance of αpθ r¡2 q.
This completes the proof of the S d -invariance of α.
Now, let us prove that α is an isomorphism.It is easy to see from (4.3) and (4.4) that αpθ r 1 i q is of the form where P t y P R ν 1 i for r t1, 2, . . ., ru and P r is invertible in y P R ν 1 i .Then the bijectivity is clear from (4.7) and from the fact that α restricts to a bijection x Pol a z PolR ν .■ We get the following theorem.
Theorem 4.13.There is an isomorphism of DG-algebras p p Rpνq, d Λ q p x Ha , f Q q.
Proof .Note that (4.3) implies that the isomorphism α (see Lemma 4.12) identifies the subalgebra y P A I ν of y P A ν with the subalgebra y P B I a of y P B a .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 : x Ha Ñ p Rpνq.It is obvious that γ preserves the λ-grading.We claim that for each h x Ha , we have γpf Q phqq d Λ pγphqq.P B a in the same way as in Section 4.2).This can be done in the same way as in the degenerate case.However, some formulas 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 S d -invariant homomorphism α I : x Pol a Ñ y P R ν Now, we extend α I to a homomorphism α : p P a Ñ y P R ν in the following way: As in the previous section, we can show that α is a S d -invariant isomorphism.
We get the following theorem.Pol a .Then we get (the completion version of) the usual Brundan-Kleshchev-Rouquier isomorphism.

The homology of Hd and H d
We now have the tools to prove the following two propositions.Proposition 4.17.The homology of the DG-algebra s H d , f Q ¨is concentrated in degree 0 and is isomorphic to s H Q d .
Proposition 4.18.The homology of the DG-algebra pH d , f Q q is concentrated in degree 0 and is isomorphic to First, we start from a similar statement for the KLR algebra.
Proposition 4.19.The homology of the DG-algebra p Rpνq, d Λ ¨is concentrated in degree 0 and is isomorphic to R Λ pνq.
Proof .It is proved in [16,Proposition 4.14] that the homology of the DG-algebra pRpνq, d Λ q is concentrated in degree 0 and is isomorphic to R Λ pνq.Assume, that for some i ¡ 0, we have H i Hd , f Q ¨$ 0 and consider it as a Pol d -module.The annihilator of this Pol d -module is contained in some maximal ideal M Pol d .The ideal M is of the form M pX 1 ¡ a 1 , . . ., X d ¡ a d q for some a pa 1 , . . ., a d q k d .

Then the completion of H i
Hd , f Q ¨$ 0 with respect to the ideal M is nonzero.This leads to a contradiction because H i x Ha , f Q ¨ 0 together with Künneth formula implies Proposition 4.18 is proved in the same way.■

2. 1
The polynomial rings Pol d and Poll d and the rings P d and P l d 1u d .Denote by 1 the sequence 1 p1, 1, . . ., 1q M .For each sequence b pb 1 , . . ., b d q M , we set θ b θ b 1 Lemma 2.6.The operators 2 θ b | b M @ acting on P d are linearly independent over Hd .More precisely, if we have °bM h b θ b 0 with h b Hd , then we have h b 0 for each b M .
6. ■ Similarly to the notation θ b above, we set ξ b ξ b 1 1 . . .ξ b d d .For two elements b, b I M , we write b I b if there is an index r r1; ds such that b I r b r and b I t b t for t ¡ r.For b M , write maxpbq for the maximal index r r1; ds such that b r 1. Lemma 2.8.The element ξ k acts on P d by an operator of the form c k d k θ k , where c k H¤k¡1 d , d k H¤k¡1 d , λpc k q 1 and d k is not a right zero divisor in Hd .

9 .
The element ξ b Hd acts on P d by an operator of the form c b d b θ b , where d b H¤maxpbq¡1 d and d b is not a right zero divisor in Hd and c b is of the form °bI b h b Iθ b I with h b I H¤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 b 1 M be such that θ b θ b 1 θ p .By the induction assumption, the element ξ b ξ b 1 ξ p acts on P d by an operator of the form (up to sign) pc p d p θ p q c b 1 d b 1 θ b 1 ¨.This operator can be written as c b d b θ b for d b d p d b 1 and c b c p pc b 1 d b 1 θ b 1 q d p θ p c b 1 .Now, we obviously get d b H¤p¡1 d because it is a product of two elements of H¤p¡1 d and it is not a right zero divisor as a product of two right non-zero divisors.Moreover, the element c b is of the form °bI b h b Iθ b I because d p θ p c b 1 d p c b 1 θ p is of the required form and because c p pc b 1 d b 1 θ b 1 q H¤p¡1 d (and then it is also of the required form).

Definition 3 . 1 .
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].For each ν NrIs, we define the k-algebra Rpνq by the data:It is generated by the KLR generators for i, j I, where each diagram contains ν i strands labeled i, together with floating dots that are confined to a region immediately to the right of the left-most strand, i ¤ ¤ ¤ .

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 ppY 1 , Y 2 q °r,s c r,s Y r 1 Y s 2 with c r,s k, we have ppY 1 , Y 2 q

Definition 3 . 5 .
We will consider PolR d krY 1 , Y 2 , . . ., Y d s as a subalgebra of Rpνq.Let m be the ideal of PolR d generated by all Y p , 1 ¤ p ¤ d.Denote by p Rpνq the completion of the algebra Rpνq at the sequence of ideals Rpνqm j Rpνq.Let y P R d krrY 1 , . ., Y d ss xΩ 1 , . . ., Ω d y be the similar completion of P R d and let y P R ν À iSeqpνq y P R d 1 i be the similar completion of P R ν .

PolR ν À iSeqpνq krY 1 P B a À bS d a krrX 1 ¡
, . . . ,Y d s1 i Pol d krX 1 , . . ., X d s P A ν : a PolR ν -algebra P B d : a Pol d -algebra z PolR ν À iSeqpνq krrY 1 , . . . ,Y d ss1 i x Pol a À bS d a krrX 1 ¡ b 1 , . . ., X d ¡ b d ss1 b y P A ν z PolR ν PolRν P A ν y b 1 , . . .,X d ¡ b d ss Pol d P B d ¨1b y Endp y P A ν q p Ba xT r , y P B I a y Endp y P B a q

Assumption 4 . 4 .
As above, we set Pol d krX 1 , ¤ ¤ ¤ , X d s.Let P B d be a Pol d -algebra free over Pol d .The most interesting examples are P B d P d and P B d Pol d .Set x Pol a à bS d a krrX 1 ¡ b 1 , . . ., X d ¡ b d ss1 b , y P B a à bS d a pkrrX 1 ¡ b 1 , . . ., X d ¡ b d ss Pol d P B d q1 b .Then y P B a is a x Pol a -algebra.Fix an action of S d on P B d (by ring automorphisms) that extends the obvious S d -action on Pol d .We assume that such an extension exists.We assume additionally the following.Assumption 4.3.For each simple generator s r of S d and each f P B d , we have f ¡ s r pfq pX r ¡ X r 1 qPB d .In particular, this assumption implies that the Demazure operator f r 1¡sr Xr¡X r 1 is well defined on P B d .The action of S d on Pol d and P B d can be obviously extended to an action on x Pol a and y P B a .Fix a subalgebra y P BI a of y P B a .We make the following assumption.There is an algebra p Ba that has a faithful representation in y P B a that is generated by multiplication by elements of y P B I a and by the operators T r s r ¡ f r .

Assumption 4 . 5 .x
We can extend the isomorphism z PolR ν Pol a in (4.1) to an S d -invariant isomorphism y P A ν y P B a .This extension restricts to an isomorphism y RA

(4. 5 )
Lemma 4.12.The homomorphism α : p P a Ñ y P R ν given by (4.3) and (4.4) is an isomorphism and it is S d -invariant.Proof .Since the homomorphism α I : x Pol a Ñ y P R ν is obviously S d -invariant, to show the S dinvariance of α, we have to show

(4. 8 )Remark 4 . 14 .
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.■We could also take yPol a .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 p Apνq p B a for some algebras p Apνq and p B a that satisfy some list of properties.Let us show that we can apply Proposition 4.10 to the special situation p Apνq p Rpνq and p B a p H a .We assume that ν and a are related as in Section 4.1.2.In this case, we can take y P A ν y P R ν and y P B a p P a .To be able to apply Proposition 4.10, we only have to construct a S d -invariant isomorphism α : y P R ν p P a extending the isomorphism (4.2) such that α restricts to an isomorphism y

Theorem 4 . 15 .Remark 4 . 16 .
There is an isomorphism of DG-algebras p Rpνq,d Λ ¨ p H a , f Q ¨.We could also take y 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 p Rpνq, d Λ DG-enhanced version of the KLR algebra of type Γ with parameters ν and Λ as above and p Rpνq, d Λ ¨its completion.
s i d Pol d of invariants under the transposition pi i 1q.It is well known that the action of the Demazure operators on Pol d satisfy the Leibniz rule which is the localization of Pol d obtained by adding the inverses of X 1 , . .., X d .Moreover, the S d -action on Pol d can be obviously extended to a S d -action on Poll d .This means that the action of the Demazure operators on Pol d also extends to operators on Poll d that satisfy the relations in (2.2) (for f and g in Poll d ) and (2.3)-(2.5).2.1.2The rings P d and P l d Let θ tθ 1 , . . ., θ d u be odd variables and form the supercommutative ring P d Pol d pθq, invariants under the transposition pi i 1q.It was proved in [15, Lemma 2.2] that the Demazure operators on P d satisfy the Leibniz rule (2.2) (for f, g P d ), the relations (2.3)-(2.5)and the following relations: H d on Pol d , where T i pfq s i pfq ¡ f i pfq and X i s H d acts as multiplication by X i .It is immediate that s H d contains kS d and Pol d as subalgebras and that for p Pol d , T i p ¡ s i ppqT i ¡f i ppq.
1 P d θ 1 Pol d .But this implies h b 0 0 because the polynomial representation of Hd on Pol d is faithful, see [20, Section 3.1.2].■ For each, k t0, 1, . . ., du we denote by H¤k d the subalgebra of the algebra of operators on P d generated by X i , θ i for i ¤ k and T r for r k.Denote also by H¤k Since Hd acts faithfully on P d , we can see H¤k d the subalgebra of Hd generated by X i for i ¤ k and T r for r k.
. 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 algebra H d has a triangular decomposition (only as a vector space) H fin d is the (finite dimensional) Hecke algebra of the group S d .Explicitly, the algebra H fin d is defined by generators T 1 , . . ., T d¡1 and the relations in (2.11).
d $ H fin d pξ 1 , . . ., ξ d q, where d Let ℓ and Q be as in Section 2.3.1.Definition 2.24.Define an operator f Q on H d by declaring that f Q acts as zero on H d , while .3.4 Completions of H d Similarly to Section 2.2.4,we want to define a completion of the algebra H d .Consider the algebra of symmetric Laurent polynomials Syml d kFor each d-tuple a pa 1 , . . ., a d q pk ¢ q n , we have a character χ a : Syml d Ñ k given by the evaluation X r Þ Ñ a r .Denote by m a the kernel of χ a .Definition 2.26.Denote by p H a the completion of the algebra H d at the sequence of ideals H d m j a H d .Since m a is in the kernel of f Q , we can extend f Q to p H a .Set also p P a krrX 1 ¡ a 1 , . . . ,X d ¡ a d ss pθq.can obviously extend the action of H d on P d to an action of p H a on p P a .Similarly to x Ha , the algebra p H a has idempotents 1 b , b S d a that are defined in the same way as in Section 2.2.4.Similar to Proposition 2.16, we have the following.Pol a -module p H a is free with basis d $ S d .We consider it as a (central) subalgebra of H d .
16, Section 3.2] and extends the polynomial action of KLR algebras from [6, Section 2.3].We fix ν NrIs with |ν| d.Set P R d krY 1 , . . ., Y d s xΩ 1 , . . ., Ω d y.Now consider 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 x Ha , f Q ¨and p H a , f Q ¨are concentrated in degree 0 and are isomorphic to HQ a and H Q 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 Hd , f Q ¨in degree zero is HQ d .We only have to check that the homology groups in other degrees are zero.