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Categorification of quantum Kac-Moody superalgebras

We introduce a non-degenerate bilinear form and use it to provide a new characterization of quantum Kac-Moody superalgebras with no isotropic odd simple roots. We show that the spin quiver Hecke algebras introduced by Kang-Kashiwara-Tsuchioka provide a categorification of half the quantum Kac-Moody superalgebras, using the recent work of Ellis-Khovanov-Lauda. A new idea here is that a super sign is categorified as spin (i.e., the parity-shift functor).

Several years ago, partly motivated by Nazarov's construction of affine Hecke-Clifford algebras [Naz], the second author [Wa] introduced spin Hecke algebras and studied them in a series of papers with Khongsap starting in [KW]. The spin Hecke algebras are associated to spin Weyl groups, and they afford many variations (e.g., affine, double affine, degenerate, nil, etc.). A distinct new feature in [Wa, KW] is the appearance of the skew-polynomial algebra as a subalgebra of the spin Hecke algebra, in contrast to the polynomial algebra for the stardard affine/double Hecke algebras. The spin Hecke algebras are naturally superalgebras, and are Morita super-equivalent to Hecke-Clifford algebras (though, we prefer to suppress the prefix "super" for such associative algebras in contrast to Lie superalgebras). A straightforward modification of the spin Hecke algebra is the spin nilHecke algebra, which has recently been rediscovered and studied in depth in [EKL] for the affine type A case.
Almost as recently, Kang-Kashiwara-Tsuchioka [KKT] utilized the spin nilHecke algebra to generalize the KLR construction to several new families of algebras, including the spin quiver Hecke algebras (called "quiver Hecke superalgebras" in loc. cit.) and quiver Hecke-Clifford algebras, starting from a generalized Cartan matrix (GCM) A parametrized by an index set I = I0 ∪ I1 subject to some natural conditions (see §2.1). Roughly speaking, to each i ∈ I0 is attached the usual nilHecke algebra and to each i ∈ I1 is attached the spin nilHecke algebra; when I1 = ∅, the KKT construction reduces to the original KLR construction. It is suggested in [KKT] that these new algebras can be used to categorify the quantum Kac-Moody algebra associated to the GCM A with the Z 2 -parity forgotten (denoted by A + in this paper). Their expectation was partly motivated by [BK1] where affine Kac-Moody algebra of type A (2) 2ℓ arises (also cf. [Ts]).
1.2. What to categorify? It has been known much earlier [Kac] that a Kac-Moody superalgebra can be associated to a GCM A exactly as specified in §2.1. This class of Kac-Moody superalgebras is distinguished among the Lie superalgebras in the following sense: the odd simple roots are all non-isotropic, the notion of integrable modules is defined as usual, and the super Weyl-Kac character formula for integrable modules holds. Note however that the only finite-dimensional simple Lie superalgebras in this class, which are not Lie algebras, are osp(1|2n). This class of Lie superalgebras and beyond have been quantized in [Ya, BKM], and a generalization of Lusztig's theorem [Lu1] on deforming integrable modules was obtained in [BKM].
We propose a connection between the spin quiver Hecke algebras and quantum Kac-Moody superalgebras, despite the major difficulty caused by additional signs appearing in the superalgebras. These signs show up in the super quantum integers (3.1), which may specialize to zero if one naively takes q → 1, and also appear in the super quantum Serre relations (3.2). Another major conceptual obstacle is that no canonical basisà la Lusztig-Kashiwara has ever been constructed (even conjecturally) for superalgebras, in spite of various works generalizing the corresponding crystal basis theory.
1.3. The main results. In this paper, we introduce a twisted bialgebra A f π , called the (quantum) covering Kac-Moody algebra, with an additional parameter π satisfiying π 2 = 1. This algebra is associated to the GCM A with an extra natural condition (C6) in §2.2 which is assumed throughout this paper unless otherwise specified.
The assumption (C6) plays an essential role in the paper. First, we introduce an apparently novel bar involution on A f π , which is the identity on the Chevalley generators:θ i = θ i (i ∈ I), as usual, butq = πq −1 . Assumption (C6) guarantees that the quantum integers in A f π , and therefore the divided powers, θ (a) i (i ∈ I), are bar invariant. Additionally, following Rouquier [Ro1], we define a family of (skew-) polynomials Q = (Q ij (u, v)) i,j∈I from a quiver with compatible automorphism and show in Lemma 2.1 that assumption (C6) implies these polynomials satisfy the necessary conditions to construct an associated spin quiver Hecke algebra, [KKT,(3.1)].
The two specializations, π → 1 and π → −1, of A f π become half of the Kac-Moody algebra associated to A + and half of the Kac-Moody superalgebra associated to A, respectively. The main result of this paper is that the spin quiver Hecke algebras defined by the family of polynomials, Q, naturally categorify the algebra A f π and, consequently, we obtain a categorification of halves of the corresponding Kac-Moody bialgebra and bisuperalgebra simultaneously.
Our key new idea in this paper is to distinguish two types of signs occuring in (quantum) Kac-Moody superalgebras. Signs common to Lie algebras and superalgebras, or ordinary signs, are denoted by (−1) as usual, while the signs arising from exchanges of odd elements are replaced by the parameter π. The ordinary signs are categorified using complexes as usual following [KL1,KL2,Ro1], while π is the shadow of a parity-shift functor (called spin). Just as the parameter q is categorified by an integer grading shift, π is categorified by a spin. Forgetting the spin corresponds to the specialization π → 1, and in this way one ends up with the usual quantum Kac-Moody algebras (as suggested in [KKT]).
We introduce a bilinear form on the algebra A f π (which specializes to a form on the superalgebra A f − ) and establish its non-degeneracy, following [Lu2]. With this in place, the necessary categorical constructions can be obtained within the framework of [KL1,KL2] using the spin quiver Hecke algebras. In the process we find the detailed structures of spin nilHecke algebras worked out in [EKL] handy to use. As a consequence of our categorification, we prove a conjecture of [KKT] that all simple modules of spin quiver Hecke algebras are of type M (that is, they remain simple with the Z 2 -grading forgotten).
In the simplest case when I = I1 consists of an (odd) singleton, the spin quiver Hecke algebra reduces to the spin nilHecke algebra of the second author. In this case, our assertion is that the spin nilHecke algebras categorify half of the quantum osp(1|2) (which is new) as well as half of the quantum sl(2) (which was already proved in [EKL]; see also [KKT]).
1.4. Future work. The ideas of this paper are expected to have several ramifications. The results here can be rephrased in terms of 2-Kac-Moody superalgebras in the sense of [Ro1,Ro2]. One can also imitate [Web] to formulate a (conjectural) categorification of tensor products of integrable modules of quantum Kac-Moody superalgebras. Following the algebraic construction of [KK] it should allow one to show that the cyclotomic spin quiver Hecke algebras categorify the integrable modules of the quantum Kac-Moody (super)algebras. The idea here can also be combined with [KOP] to categorify the more general quantum Borcherds superalgebras and their integrable modules studied in [BKM].
Another main point of this paper is the introduction for the first time of a bar-involution on quantum superalgebras such thatq = −q −1 , and the assumption (C6) is again perfect for this purpose. A remarkable property of this bar-involution is its compatibility with the categorification. The canonical basis for the modified or idempotented quantum osp(1|2) is constructed in [CW]. In a work in preparation joint with Sean Clark, we are undertaking a construction of the canonical bases for quantum Kac-Moody superalgebras. It will be interesting to compare these canonical bases with those coming from categorification.
We hope that our work helps to clarify the right framework for categorifying the odd Khovanov homology (cf. [EKL] and references therein). A new idea was suggested in [Kh2] on how to categorify a superalgebra with an isotropic odd simple root. It is natural to expect, though remains highly non-trivial, that the categorification of the more general Kac-Moody superalgebras will have to combine all these ideas of categorifying the even simple roots, the isotropic odd simple roots, and the non-isotropic odd simple roots.
1.5. Organization. The layout of this paper is as follows. In Section 2 we collect the necessary Lie theoretic data. In Section 3 we define the covering Kac-Moody algebra, realize it as the quotient of a free algebra by the radical of a bilinear form, and define a new bar involution. In Section 4 we recall the definition of the spin quiver Hecke algeba and describe some of its basic properties. In Section 5 we introduce the category of finitely generated graded projective modules over the spin quiver Hecke algebra that will categorify the covering Kac-Moody algebra, and then establish the categorical Serre relations. From these results, we deduce in Section 6 the categorification of the covering Kac-Moody algebra.

Conventions.
A module over a superalgebra R in this paper is understood as a left module with Z 2 -grading compatible with that of R.

Acknowledgement.
We are indebted to the authors of [KKT, EKL], as this paper grew out of our readings of their papers and relies heavily on their results. We thank Sean Clark for stimulating discussions and collaborations. The second author is grateful to Jinkui Wan for her collaboration in 2009 when we were motivated by [Naz,BK1,Wa,KW] to make an unsuccessful attempt to construct the spin/Clifford generalization of the KLR algebras. The research of the second author is partially supported by NSF grant DMS-1101268.
After the completion of this paper, Kang-Kashiwara-Oh [KKO] used [KKT] to categorify the Kac-Moody algebra g = g(A + ) and its integrable representations following [KK], but the connection with Kac-Moody superalgebras was not pursued.
Condition (C4) was imposed first in [Kac] so that the corresponding Kac-Moody superalgebras possess similar favorable properties as osp(1|2n), i.e., the odd simple roots are all non-isotropic and the Weyl-Kac character formula for integrable modules expressed in terms of the Weyl group holds.

2.2.
The assumption (C6). We will impose an additional condition on a GCM A introduced in §2.1 for the Kac-Moody superalgebras considered in this paper: (C6) I1 = ∅, and the integer s i is odd if and only if i is odd (i.e., i ∈ I1).
The case I1 = ∅ has been well studied, and we have nothing new to add. There is precisely one Kac-Moody superalgebra of finite type satisfying (C1)-(C6). Namely, osp(1|2n) (or, B(0, n)): In Table 1 below, we list the affine Dynkin diagrams satisfying the parity assumption (C6). The nodes labeled by I0 are drawn as hollow circles •; the nodes labeled by I1 are drawn as solid dots •. A complete list of affine Lie superalgebras and Dynkin diagrams can be found in [vdL] and we observe that there is exactly one family of affine superalgebras excluded by (C6).
3. Quivers with compatible automorphism. Let K be a field, charK = 2. We continue to work under the assumptions of §2.1 and §2.2 throughout the paper.
Let Γ be a graph without loops. We construct a Dynkin diagram Γ by giving Γ the structure of a graph with compatible automorphism in the sense of [Lu2,§12,14]. To define the quiver Hecke algebra, we will use the notion of a quiver with compatible automorphism as described in [Ro1,§3.2.4].
Let I be the labelling set for Γ, and H be the (multi)set of edges. An automorphism a : Γ → Γ is said to be compatible with Γ if, whenever (i, j) ∈ H is an edge, i is not in the orbit of j under a (so the quotient graph has no loops).
Fix a compatible automorphism a : Γ → Γ, and set I to be a set of representatives of the obits of I under a and let Γ = Γ/a be the corresponding diagram with nodes labeled by I. For each i ∈ I, let α i ∈ I/a be the corresponding orbit. For i, j ∈ I with i = j, let For all i, j ∈ I, let a ij = (α i , α j )/s i . Then, by [Lu2, Proposition 14.1.2] A = (a ij ) i,j∈I is a GCM and every GCM arises in this way. Moreover, the symmetrizing constants s i are nonnegative integers by definition, and we may assume that gcd{s i |i ∈ I} = 1 (otherwise, let ℓ = lcm{s i | i ∈ I}/ gcd{s i | i ∈ I}, and repeat the construction above with Γ/a ℓ instead). Define a Z 2 -grading on I by setting I0 = {i ∈ I | |α i | ∈ 2Z} and I1 = I\I0. Then, (C6) is satisfied. Among the diagrams obtained from this construction, we will work only with those satisfying (C4).
Then, A is a GCM as in §2.1, and Γ is its Dynkin diagram. We additionally have the data: Bilinear pairing: (·, ·) : Q × Q −→ Z.
(2.6) Assume further that Γ is a quiver. That is, we have a pair of maps s : H → I and t : H → I (the source and the target). We assume that a is compatible with the quiver structure in the sense that it is equivariant with respect to the source and target maps: s(a(h)) = a(s(h)) and t(a(h)) = a(t(h)) for all h ∈ H. Set and let m(i, j) = lcm{(α i , α i ), (α j , α j )} = 2s i s j / gcd(s i , s j ). As noted in [Ro1], The data above defines a matrix Q = (Q ij (u, v)) i,j∈I , see [Ro1]. Each Q ij (u, v) ∈ K ij {u, v}, and the (skew-)polynomial entries in Q are defined by Q ii (u, v) = 0, and for i = j, In particular, the entries in Q satisfy the properties in [KKT,(3.1)].
To prove (d), first assume j ∈ I0. Note that by assumption (C6) s i is odd and s j is even. Therefore, m(i, j)/(α i , α i ) = s j / gcd(s i , s j ) is even, proving the lemma in this case.
Next, assume j ∈ I1. In this case, both m(i, j)/(α i , α i ) = s j / gcd(s i , s j ) and m(i, j)/(α j , α j ) = s i / gcd(s i , s j ) are odd by assumption (C6). However, −2(α i , α j )/m(i, j) = a ij gcd(s i , s j )/s j is even, since a ij is even by (C4) and both s i and s j are odd (again, by (C6)). Since uv = −vu ∈ K ij {u, v}, whenever a and b are both odd, the result follows.

Quantum superalgebras and bilinear forms
3.1. Kac-Moody superalgebras. Associated to a GCM A as in §2.1 and §2.2 is a Kac-Moody superalgebra g = g(A) (see [Kac]), and a quantized enveloping superalgebra (see [BKM]). Let f − be the super analogue of Lusztig's algebra f , generated over Q(q) by {θ i |i ∈ I} and subject to a signed analogue of Serre relations (cf. (3.2) below). The θ i parameterized by i ∈ I0 (respectively, i ∈ I1) are even (respectively, odd). For k ≥ 0 and i ∈ I, we shall denote by θ Besides the standard relation among divided powers, the signed Serre relations in A f − can be written as: Definition 3.1. Assume a GCM A satisfies (C1)-(C6). We define a bar-involution¯: Note that all the divided powers θ (k) i are bar-invariant under the assumption (C6), and hence we have¯: Remark 3.2. With respect to this apparently new bar-involution, we will develop a theory of canonical basis in a forthcoming work with Sean Clark.
3.2. Kac-Moody algebras. If we forget the parity on I in 2.1, we shall write the corresponding GCM matrix by A + . Associated to the GCM A + is the usual Kac-Moody algebra g + = g(A + ), and write f + for Luszig's algebra f . For k ≥ 0 and i ∈ I, we shall abuse notation slightly and denote by θ i , for k ≥ 1 and i ∈ I, subject to the usual Serre relations, which we suggestively write as: (3.6) 3.3. Covering Kac-Moody algebras. Finally, we present a common framework to describe the presentations in §3.1 and §3.2, and in the process justify our seemingly inconsistent use of notation therein. To this end, fix an indeterminant π, and for a ring R, we introduce a new ring R π = R[π]/(π 2 − 1).
Denote by ′ f π = ′ f π 0 ⊕ ′ f π 1 the free associative algebra over Q(q) π generated by even generators θ i for i ∈ I0 and odd generators θ i for i ∈ I1. We have parity p(x) = 0 for We denote by θ The algebra structure on ′ f π ⊗ ′ f π is given by The following is a super analogue of [Lu2,Proposition 1.2.3]. Note that the v in Lusztig corresponds to our q −1 . Though the identities below look almost identical to those in loc. cit., we give a detailed proof where some super signs show up.
Clearly (x, y) = 0 unless (homogeneous) x, y have the same weight in Q, which implies they must have the same parity. All elements involved below will be assumed to be homogeneous.
This proves (b). It remains to prove (c). The cases when y ′′ is 1 or θ j can be checked directly. Assume that (c) is known for y ′′ replaced by y or y ′ and for any x, x ′ . We then prove that (c) holds for y ′′ = yy ′ . Write On the other hand, For a summand to make nonzero contribution, we must have |x ′ 1 | = |y 2 | and |x 2 | = |y ′ 1 | in Q and, therefore, both p(x ′ 1 ) = p(y 2 ) and p(x 2 ) = p(y ′ 1 ). It follows that the powers of q and π in (3.12) and (3.13) match perfectly. Hence the two sums in (3.12) and (3.13) are equal, proving (c).
We then define (half of) the covering Kac-Moody algebra to be the quotient algebra ′ f π by the radical as f π = ′ f π /Rad(·, ·). The bilinear form (·, ·) on ′ f descends to a non-degenerate bilinear form (·, ·) on f π satisfying the same properties as in Proposition 3.3, and we also have an induced algebra homomorphism r : Recall from [Lu2,1.3.5] the quantum binomial formula: for any a ≥ 0 and x, y in a Q(v)-algebra such that xy = v 2 yx, (3.14) Now we convert (3.14) to notations in our super setting.
Lemma 3.4. For any a ≥ 0 and x, y in a Q(q)-algebra such that xy = π i q −2 i yx, Proof. For i even, p(i) = 0 and so (3.15) is simply (3.14) with v = q −1 i . Now assume i is odd, i.e, p(i) = 1, and so π i = π. Fix a square root √ π of π once for all. We have xy = v 2 yx, if we introduce a new indeterminate v by letting Hence (3.14) is applicable, and we have This identity can then converted to (3.15) by using the following identities: for a ≥ t ≥ 0, ( 3.17) Since π = π −1 , this proves the lemma.
3.5. Bilinear form. We now study the divided powers in relation to the homomorphism r and the bilinear form (·, ·).
Lemma 3.5. For any a ∈ Z ≥0 and any i ∈ I, we have Proof. When i ∈ I0, this is simply [Lu2,Lemma 1.4.2], which was proved directly using (3.14). Now assume i ∈ I1. Thanks to the identity (3.9), the assumption of Lemma 3.4 is satisfied with x = 1 ⊗ θ i and y = θ i ⊗ 1. Hence this lemma follows directly from (3.15) by the definition of the divided power based on (3.7).
Lemma 3.6. For any a ∈ Z ≥0 and any i ∈ I, we have We will only prove the first identity, as the second identity is elementary.
The argument is similar to the proof of [Lu2,Lemma 1.4.4], which corresponds to the case for i ∈ I0. We proceed by induction. Note that the lemma holds for a = 0 or 1. Assume the lemma holds for a and also for a ′ . Using Lemma 3.5 and (3.10) we have , this gives the result.
3.6. Quantum Serre relations. We first formulate the following super analogue of the last formula in the proof of [Lu2,Lemma 1.4.5].
Lemma 3.7. Assume i ∈ I is odd. Let n ∈ Z ≥0 and let a, where the sum is taken over all t, t ′ , s, s ′ in Z ≥0 such that

19)
and we have denoted Proof. We first compute by Lemma 3.5 that It follows by Lemma 3.5 that (3.23) (3.24) The right-hand side of (3.24) can then be converted to (3.18) using Lemma 3.6 repeatedly and noting that This proves the lemma. Now we are ready to state and prove the following fundamental result, called the quantum Serre relations. Recall p(k; i, j) from (3.3).
Theorem 3.8. Let A be a GCM satisfying (C1)-(C5). For any i = j in I, the following identities hold in f π : (3.25) Proof. The strategy is to show the element on the left-hand side of (3.25) orthogonal to f π with respect to (·, ·). For i even, the same proof for [Lu2,Proposition 1.4.3] applies here without any change, regardless of the parity of j. Now assume that i is odd. Hence p(i) = 1 and π i = π. We still proceed as in [Lu2] and keep track of the super signs carefully in the meantime.
Using (3.19) to get rid of t and s ′ , we can rewrite ♣ in Lemma 3.7 as where the sum is taken as in Lemma 3.7. Here we have where ♠ arises from the conversion (3.17). Recall p(a ′ ; i, j) = a ′ p(j) + 1 2 a ′ (a ′ − 1), and note that a ij = 1 − n is an even integer by (C4). Some elementary and lengthy manipulation using (3.19) allows us to rewrite ♦ in (3.28) (in terms of s, t ′ , b, b ′ ) as Recalling π 2 = 1, it follows that √ π ♦ is independent of s, s ′ , t, t ′ and can be moved to the front of in (3.27). To prove (3.25), it suffices to show that a+a To that end, recalling (3.27) and noting a ij + n − 1 = 0, it remains to verify the identity where the sum is taken as in Lemma 3.7 again. We can factorize the sum on the left hand side as Since at least one of b and b ′ is positive, one of the two factors must be zero thanks to a classical binomial identity (cf. [Lu2,1.3.4(a)]).
3.7. The bar involution. It is shown in [BKM] that the integrable modules of the quantum Kac-Moody superalgebras have the same characters as their classical counterparts, generalizing the earlier work of [Lu1]. Now based on this fact and proceeding as in [Lu2,33.1], we can establish Theorem 3.9(a) below as a super analogue of [Lu2,Theorem 33.1.3] (which is reformulated as Theorem 3.9(b) below). Recall the algebras f − , f + , f π were introduced in earlier subsections.
Remark 3.11. Theorem 3.9(a) identifies half of the quantum Kac-Moody superalgebra associated to the generalized Cartan matrix A in §2.1 with the quotient of a free algebra by the radical of an analogue of Lusztig's bilinear form. A similar result for quantum osp(1|2n) was also obtained in [Ya] using a different normalization of bilinear form such that (θ i , θ i ) = 1 for i odd and with additional signs appearing in the definition of the form on f − ⊗ f − (see also [Gr]).
We will write A f π for the A π -subalgebra of f π generated by the divided powers θ , subject to the quantum Serre relation (3.25).
( 3.29) We have that q i = π s i q −1 i , and a calculation gives It now follows from (C6) that the quantum integers [k] i are bar-invariant, and so the divided powers θ (k) i are bar-invariant as well. Thus, this induces a bar-involution¯: 4. Spin quiver Hecke algebras 4.1. Generators and relations. Fix an ℓ × ℓ GCM A as in §2.1, and continue to assume (C6) as usual. Let n ∈ Z ≥0 , and assume for ν ∈ Q + (see (2.5)) that ν = n 1 α 1 + · · · + n ℓ α ℓ and n 1 + · · · + n ℓ = n (i.e. ν has height ht(ν) = n). Let I ν ⊂ I n be the S n -orbit of the element (1, . . . , 1 n 1 , . . . , ℓ, . . . , ℓ where S n acts on I n by place permutation: w · (i 1 , . . . , i n ) = (i w(1) , . . . , i w(n) ). Equivalently, We now define an algebra based on the data above in terms of generators and relations. When I1 = ∅, the algebra is nothing but the quiver Hecke algebra of Khovanov-Lauda and Rouquier [KL1,Ro1]. In the general case I1 = ∅ we are considering, these algebras were recently defined in [KKT]. We refer to them as spin quiver Hecke algebras as explained in the introduction. In the special case when I = I1 is a singleton, the algebra is the spin nilHecke algebra, a nil version of the spin Hecke algebra first introduced in [Wa,3.3]. The spin quiver Hecke algebra is defined to be where H − (ν) is the unital K-algebra, with identity 1 ν , given by generators and relations as described below.

4.3.
The rank 1 case. Note that the diagram of a single node • corresponds to Lie superalgebra osp(1|2). In this section, we will also consider the diagram • which corresponds to sl(2). These are quivers with compatible automorphism (the trivial one) and the corresponding GCM is A = (2), and there is only one polynomial Q(u, v) ≡ 0.
In the even case •, the quiver Hecke algebra is the nilHecke algebra NH n generated by subalgebras K[Y ] for Y = {y 1 , . . . , y n } and the nil-Coxeter algebra NC n . The nil-Coxeter algebra is generated by the divided difference operators ∂ + r , 1 ≤ r < n, which are subject to the relations

The nil-Coxeter algebra act faithfully on K[Y ] via
where the simple transposition acts on polynomials by permuting the variables as usual.
For the odd case, the corresponding spin (or odd) nilHecke algebra NH − n is a nil version of the spin Hecke algebra in [Wa,3.3] and we follow the presentation in [EKL,2.2] below. Let K[Y ] − be a skew-polynomial ring in n variables (that is, the quotient of the free algebra on Y = {y 1 , . . . , y n } by the relation y r y s = −y s y r for r = s). Then, NH − n is generated by subalgebras K[Y ] − and the spin (or odd) nil-Coxeter algebra NC − n . The spin nil-Coxeter algebra is generated by an odd analogue of divided difference operators ∂ − r , 1 ≤ r < n, which are subject to the relations Here the action of S n on K[Y ] − is given by s r (y k ) = −y sr(k) .

A basis theorem.
We now show that H − (ν) satisfies the PBW property. More precisely, we have the following.
Note that by (4.12) this depends on the choice of reduced expression. However, we have the following proposition, the proof of which is almost identical to [BKW,Proposition 2.5].
Proposition 4.4. Let i ∈ I ν and w ∈ S n . Assume that w = s k 1 · · · s kt is a presentation of w as a product of simple transpositions.
(a) If the presentation w = s k 1 · · · s kt is reduced, and w = s ℓ 1 · · · s ℓt is another reduced presentation of w, then where the sum is over u < w in the Bruhat order, f u (y) is a polynomial in y 1 , . . . , y n , and deg τ u f u (y)e(i) = deg τ w e(i) for all u. (b) If the presentation w = s k 1 · · · s kt is not reduced, then τ k 1 · · · τ kt e(i) can be written as a linear combination of words of the form τ kr 1 · · · τ kr s f (y)e(i), such that 1 ≤ r 1 < · · · < r s ≤ t, s < t, w = s kr 1 · · · s kr s is reduced, f (y) is a polynomial in y 1 , . . . y n , and deg τ k 1 · · · τ kt e(i) = deg τ kr 1 · · · τ kr s f (y)e(i).
We now turn to the proof of Theorem 4.3. To this end, let P − (ν) be the subalgebra of H − (ν) generated by the elements y 1 , . . . , y n and e(i), i ∈ I ν . Fix a total ordering on I, and define a family of polynomials P = (P ij (u, v)) i,j∈I by the formula (4.21) Then, Theorem 4.3 follows immediately from the following proposition.
Proposition 4.5. There is a faithful action of the algebra H − (ν) on P − (ν), where y s and e(i) act by left multiplication (s = 1, . . . , n, i ∈ I ν ), and where S n acts on P − (ν) according to s k (y r e(i)) = (−1) p(i k )p(i k+1 )p(ir) y s k (r) e(s k · i). Note that H − (ν) is almost positively Z-graded, cf. [KL1,p.24]. Indeed, it is nontrivial only in degrees greater than, or equal to, − i n i (n i −1), for ν = i∈I n i α i . In particular, for any M ∈ Mod − (ν), we may define its (q, π)-dimension The specialization π → 1 recovers the usual graded dimension of the module M , while π → −1 produces the graded super dimension. Additionally, we define the graded character of M ∈ Mod − (ν) by This is an element of the free Z((q)) π -module with basis labelled by I ν . Define ch + q M = ch π q M | π=1 and ch − q M = ch π q M | π=−1 . Then, the mapping which sends e(i) to the natural embedding {P i ֒→ H − (ν)} defines an isomorphism be the Z-linear involution: π = π, q = πq −1 , and There is a bilinear form given by Note that (P i , P j ) = dim π q e(j)H − (ν)e(i). For future reference, we also note that natural form on [Proj H − (µ) ⊗ H − (ν)] is given by Proof. For P, Q ∈ Proj − (ν), we compute The simple objects in Mod − (ν) belong to the category Rep − (ν). Let S − 0 (ν) be the unique maximal graded ideal in S − (ν), spanned by S n -invariant polynomials without constant term. We have the following. Proof. Recall the commuting family of elements Y ν from (4.23). These elements are positively graded, and therefore act nilpotently on any finite dimensional module. Given a simple module, we can find a simultaneous eigenvector for the action of Y ν (necessarily with eigenvalue 0). By definition, S − 0 (ν) acts as 0 on this vector. But, by Proposition 4.6, S − 0 (ν) is contained in the center of H − (ν) and so this determines its action on the entire simple module. It now follows that every simple module factors through the 2 n1 (n!) 2dimensional algebra Define the A π -sesquilinear (i.e., antilinear in first variable) Cartan pairing given by For each simple module L ∈ Rep − (ν), there exists a (unique up to (Z×Z 2 )-homogeneous isomorphism) projective indecomposable cover P L ∈ Proj − (ν). The modules L and P L are dual with respect to (5.14). Moreover, we have

5.2.
More rank 1 calculations. In this subsection, we will consider the algebra H − (nα i ), which is isomorphic to either the nilHecke algebra or spin nilHecke algebra, depending on whether i is even or odd, respectively. Accordingly, we will revert to the notation of §4.3 to better facilitate comparison with [La,Section 3] and [EKL,Section 2]. Indeed, this section is essentially a review of results from [La, EKL] which we need, adapted for computations with left modules (cf. (5.28) below) and enhanced by the insertion of π to keep track of the parity. This amounts to reading their diagrams from top to bottom, as opposed to bottom to top (alternatively, applying the antiautomorphism ψ). Additional signs also appear, caused by (4.9) which differs from the corresponding formula in loc. cit., but is commonly used in the literature, cf. [BK2].
Proposition 5.3. The natural action of NH ± n on K[Y ] ± defines an isomorphism Define the Demazure operator Then, using (4.9), (∂ ± r ) 2 = (−∂ ± r y r )(−∂ ± r y r ) = ∂ ± r (±∂ ± r y r+1 − 1)y r = −∂ ± r y r = ∂ ± r . Moreover, it is straightforward to verify that these elements satisfy the type A braid relations: In particular, for each w ∈ S n , the element ∂ ± w is well-defined in terms of any reduced expression of w. Set Since the spin nil-Coxeter algebra satisfies the braid relations for the spin symmetric group as opposed to the standard braid relations, the elements ∂ − w , w ∈ S n are only welldefined up to sign and, therefore, we will fix reduced expressions w = s r 1 · · · s r k for each element w ∈ S n and define ∂ − w = ∂ − r 1 · · · ∂ − r k to remove this ambiguity. For the moment, let w 0 n denote the longest word in S n . We define the fixed reduced expression for w 0 n inductively by w 0 1 = 1, and w 0 n = s 1 s 2 · · · s n−1 w 0 n − 1 , for n > 1. Then, we have a well-defined element For k < n, let † S k ≤ S n be the subgroup of permutations of {n − k + 1, . . . , n} and, for w ∈ S k , let † w ∈ † S k denote the corresponding element. The following useful fact is proved in [EKL].
Proof. The proof follows by induction using (4.9) and the fact that ∂ ± r e ± n = 0.
5.3. Categorical Serre relations. From now on, we write e i,n = e ± n since the ± notation can be recovered from the parity of i, and translate NH ± n to the notation of H − (nα i ). This defines the unique projective indecomposable H − (nα i )-module (5.28) and .

In particular, there is an isomorphism
Proof. Assume i is even, so p(k; i, j) ≡ 0. Then, this result is proved in [KL2] (all the relevant maps being even). The main technical tools needed are Lemma 5.6 and Lemma 5.8. From now on, we assume i is odd. The necessary technical facts carry over and, once we keep careful track of the parity, essentially the same proof as in [KL2] works.
In the special case a ij = 0, the theorem states that there is an isomorphism P ji ∼ = Π p(j) P ij . The relevant map is given by right multiplication by τ 1 e(ij) ∈ H − (α i + α j ). In order to be a morphism in Proj − (α i + α j ), this map must preserve the Z 2 -grading. To see this, note that τ 1 e(ij) is odd when j is odd, and even when j is even. In particular, the corresponding morphism is a parity preserving isomorphism P ji → Π p(j) P ij , as required.
The maps α k+1,k are going to be maps in a chain complex. Therefore, we prove by induction that, for any N ≥ 2, α k+1,k α k,k−1 = 0 in H − (N α i + α j ), for 1 ≤ k ≤ N − 1. We want to emphasize that this particular statement holds independently of the meaning of N given in the statement of the theorem (as will be necessary when making the inductive step). To that end, let (5.34) Then, we have e(k) = e(i k )e(k). When k = 1, we calculate as above that α 2,1 α 1,0 = τ 1 τ 2 e(1)τ 1 e(0) = τ 1 τ 2 τ 1 e(0).
6. Categorification of quantum superalgebras 6.1. Induction and restriction. Let µ, ν ∈ Q + . Assume throughout the section that ht µ = m and ht ν = n, and let D m,n be the set of minimal left S m × S n -coset representatives in S m+n . The natural embedding H − (µ) ⊗ H − (ν) ֒→ H − (µ + ν) maps e(i) ⊗ e(j) to e(ij) for all i ∈ I µ and j ∈ I ν , and ij ∈ I µ+ν is obtained by concatenation. The image of the identity element 1 µ ⊗ 1 ν is the idempotent Define the functor Proof. The set {1 µ,ν τ w |w ∈ D m,n } is a basis for 1 µ,ν H − (µ + ν) as a free graded left H − (µ) ⊗ H − (ν)-module.

We have
Ind µ+ν µ,ν (P i ⊗ P j ) = P ij . Passing to direct summands, we deduce that the same holds if we replace P i and P j by P i (k) and P j (l) , respectively.
The following formula for restriction follows from Theorem 6.3.
Property (a) follows from Proposition 6.5 and then checking that the homomorphisms r and [Res] are compatible on the generators θ i and [P i ] under γ π . The bar-involution on [Proj − ] fixes each P i and satisfies (5.6), and (b) follows. for every M ∈ Proj − (ν). Recall the algebra A f + and A f − from Section 3.

Let [Proj
Corollary 6.9. There exist injective homomorphisms γ + : A f + → [Proj − ] π=1 and γ − : 6.4. The isomorphisms γ − and γ π . The arguments in [Kle1,Chapter 5] are insufficient to derive the surjectivity of γ − directly. In particular, the proof of the independence of characters [Kle1,Theorem 5.3.1] fails since we have ch − q L = 0 if ch π q L = ch π q ΠL (see (5.2) and (5.3)). We now give an alternative argument based on representation theory of quantum Kac-Moody superalgebras to show this never happens.
Denote by ChM = µ dim M µ e µ the (formal) character of an algebra or an module M = ⊕ µ M µ which is graded by Q + and free over the base ring A or Q(q). We have a natural partial order ≥ on the collection of characters: for characters g, h, we have g ≥ h if and only if g − h is a nonnegative integer linear combination of e µ for µ ∈ Q + . Lemma 6.11. We have Ch A f + = Ch A f − . Proof. By [Kac], the super Weyl-Kac character formula for integrable modules holds and therefore so does the super Weyl-Kac denominator formula. According to [BKM], the integrable modules of quantum Kac-Moody superalgebras have the same characters as for their classical counterpart at q → 1 (generalizing Lusztig's work for quantum ). Hence the super Weyl-Kac denominator formula holds, and so where ρ and the Weyl group W for the Kac-Moody superalgebra associated to the GCM A [Kac] coincide with the counterparts for the usual Kac-Moody algebra associated to the GCM A + . On the other hand, it is well known that Ch A f + is given by exactly the same formula (by a combination of Weyl-Kac denominator formula and Lusztig's result for quantum Kac-Moody). Hence the lemma follows.
Recall a finite-dimensional simple module S of an associative superalgebra A is of type M if it remains to be simple with the Z 2 -grading forgotten, and is of type Q otherwise. Recall the parity-shift functor Π. It is known (cf. for example [Kle1,Chapter 12]) that a simple A-module S is of type Q if and only if there exists an even isomorphism of A-modules: ΠS ∼ = S. Proposition 6.12. We have Proof. Using the Cartan pairing (5.14), [Proj − ] π=1 has a basis of projective indecomposable modules P L labeled by all simple modules L of the spin quiver Hecke algebra (of both type M and type Q). On the other hand, since a type Q simple module L always admits an even isomorphism ΠL ∼ = L, therefore [Proj − ] π=−1 has a basis labeled by the type M simple modules. Hence, we have Ch[Proj − ] π=1 ≥ Ch[Proj − ] π=−1 . By Theorem 6.10, the map γ + is an isomorphism. Combining this with the injection γ − gives us a sequence of inequalities of formal characters: All inequalities much be equalities by Lemma 6.11.
The equality Ch A f − = Ch[Proj − ] π=−1 above implies the following. Arguing as in [KL1,Proposition 3.20] and using Corollary 6.13, we now deduce the part (a) of the following theorem. Part (b) then follows from (a), Proposition 6.8 and Theorem 6.10.
(a) The map γ − : holds, and it implies (and is indeed equivalent to) the following property of simple modules of a spin quiver Hecke algebra (which was conjectured in [KKT,page 3]).
Proposition 6.15. Each simple module of a spin quiver Hecke algebra is of type M.