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SVV algebras.

Introduction. Following a conjecture of Enomoto and Kashiwara in [EK06] concerning categories of modules over affine Hecke algebras of type B, proved in general by Varagnolo and Vasserot [VV11], Kashiwara and Miemietz conjectured analogous results for type D affine Hecke algebras, see [KM07]. They predicted that categories of modules over affine Hecke algebras of type D categorify a highest weight module over a certain quantum group. This conjecture was confirmed by results of Shan, Varagnolo and Vasserot in [SVV11], in which they introduce and use a family of graded algebras, by showing that categories of modules over these algebras are equivalent to categories of modules over affine Hecke algebras of type D. These algebras are similar to KLR algebras and to VV algebras, the latter of which were studied in [Wal]. This note is a study of these algebras, which we call SVV algebras. We obtain a Morita equivalence between SVV algebras and a direct product of VV algebras. We use KLR algebras to show that, in certain settings, SVV algebras are graded affine quasi-hereditary and graded affine cellular.

1. Preliminaries and notation. Throughout this paper we will denote by k a field with char(k) [nor equal to] 2 and by a grading we will always mean a Z-grading. We write q to denote both a formal variable and a degree shift functor which shifts the degree by 1. So qM is a graded A-module with kth graded component [(qM).sub.k] = [M.sub.k-1], where M = [[direct sum].sub.n2Z] [M.sub.n] is a graded A-module.

The Weyl group of type D, WB has generators [s.sub.0], [s.sub.1]; ..., [s.sub.m-1] which are subject to the relations [s.sup.2.sub.i] [for all]i , [s.sub.i][s.sub.i]+1[s.sub.i] = [s.sub.i]+1[s.sub.i][s.sub.i+1] for 1 [less than or equal to] i [less than or equal to] m - 2, [s.sub.0][s.sub.2] = [s.sub.i][s.sub.0] [for all]i [not equal to] 2, [s.sub.i][s.sub.j] = [s.sub.j][s.sub.i] for [absolute value of (i - j)] > i and i [less than or equal to] i,j [less than or equal to] m - 1, [s.sub.0][s.sub.2][s.sub.0] = [s.sub.2][s.sub.0][s.sub.2]. Now let [[tau].sub.0], [[tau].sub.i], ..., [[tau].sub.m-1] be the generators of the type B Weyl group, [W.sup.B.sub.]. We can consider [W.sup.D.sub.m] a subgroup of [W.sup.B.sub.m] via the injection [W.sup.B.sub.m] [right arrow] [W.sup.B.sub.m] given by [s.sub.0] [right arrow] [[tau].sup.0][[tau].sup.1] [[tau].sup.0], [s.sub.k] [right arrow] [[tau].sup.k] for i [less than or equal to] k [less than or equal to] m - 1.

There are exactly two parabolic subgroups of [W.sup.D.sub.m] which are both isomorphic to the symmetric group on m letters. We label these subgroups [G.sub.m] and [S.sub.m]; they are the subgroups generated by [s.sub.1], [s.sub.2], ..., [s.sub.m-1] and So,[s.sub.2], ...,[s.sub.m-1], respectively.

1.1. SVV algebras. In this section we recall a family of graded algebras which were introduced by Shan, Varagnolo and Vasserot [SVV11]. We will call them SVV algebras.

We start by fixing an element p [member of] [k.sub.x]. Consider the action of Z [??] [Z.sub.2] on [k.sub.x] given by (n, [+ or -] 1) x [lambda] = [p.sub.2n][[lambda].sup.[+ or -]1]. Fix a Z [??] [Z.sub.2]-orbit [I.sub.[lambda]]. So I = [I.sub.[lambda]] = {[p.sub.2n][[lambda].sup.[+ or -]1] | n [member of] Z} is the Z [??] [Z.sub.2]-orbit of [lambda]. To I we associate a quiver [GAMMA] = [[GAMMA].sub.I]. The vertices of [GAMMA] are the elements i [member of] I and we have arrows [p.sup.2]i [right arrow] i for every i [member of] I. We always assume that [+ or -]i [??] I and that p [not equal to] [+ or -] 1. If p [member of] [I.sub.[lambda]] then we can write [I.sub.[lambda]] = [I.sup.+.sub.[lambda]] [??] [I.sup.-.sub.[lambda]], where [I.sup.[+ or -].sub.[lambda]] = {[p.sub.2n][[lambda].sup.[+ or -]1] | n [member of] Z}. Similarly, when p [member of] [I.sub.p] and p is not a root of unity then we can write [I.sub.p] = [I.sup.+.sub.p] [??] [I.sup.- .sub.p], where [mathematical expression not reproducible]. Let [mathematical expression not reproducible] has finite support, [v.sub.i] [member of] [Z.sub.[greater than or equal to]0] [for all]i}. Recall the height of v [member of] is denoted by [absolute value of (v)] and is defined as [absolute value of (v)] = [[SIGMA].sub.i[member of]I] [v.sub.i]. We also recall that i [member of] I has multiplicity one in v if [v.sub.i] = 1.

Throughout this work we assume that if p [member of] I then p is not a root of unity, so that we always have [I.sub.[lambda]] = [I.sup.+.sub.[lambda]] [??] [I.sup.-.sub.[lambda]].

For any v [member of] [sup.[theta]]NI we have [v.sub.i] = [v.sub.i-1] for all i [member of] I which means that we can always write v = [v.sup.+] + [v.sup.-], where [mathematical expression not reproducible]. It now makes sense to talk about the KLR algebras associated to [v.sup.+] and to [v.sup.-]. Denote these algebras by [Rv.sup.+] and [Rv.sup.-], respectively. Recall that the KLR algebras are a family of graded algebras that have been introduced in [KL09] and [Rou] in order to categorify quantum groups.

For v [member of] [sup.[theta]]NI with [absolute value of (v)] = 2m, define

[mathematical expression not reproducible].

For v [member of] [sup.[theta]]NI with [absolute value of (v)] = 2m, m > 1, the SVV algebra, denoted by [degrees]R[([GAMMA]).sub.v], is the graded k-algebra generated by elements

[mathematical expression not reproducible]

which are subject to the following relations.

(a) [mathematical expression not reproducible].

(b) The [x.sub.l]'s commute.

(c) For 1 [less than or equal to] k [less than or equal to] m - 1,

[mathematical expression not reproducible].

For 1 [less than or equal to] k < m - 1;

[mathematical expression not reproducible]:

(d) For [mathematical expression not reproducible].

(e) For [mathematical expression not reproducible]:

The grading on [degrees]R[([GAMMA]).sub.v] is given as follows:

[mathematical expression not reproducible]

where [absolute value of (i [right arrow] j)] denotes the number of arrows from i to j in the quiver [GAMMA].

If v = 0 we set [degrees]R[([GAMMA]).sub.v] = k [direct sum] k and if v = i + [i.sup.-1], for some i [member of] I then,

[degrees]R[([GAMMA]).sub.v] = k[x]e(i) [direct sum] k[x]e([i.sup.-1]):

The action of [W.sup.D.sub.m] on i =([i.sub.1], ..., [i.sub.m]) [member of] [sup.[theta]][I.sup.v] is given via

[mathematical expression not reproducible]

for 1 [less than or equal to] k < m.

Take w [member of] [W.sup.D.sub.m] and fix a reduced expression [mathematical expression not reproducible]. We then set [mathematical expression not reproducible] and, for the identity element 1 [member of] [W.sup.D.sub.m], we have [[sigma].sub.e](i) = e(i). From the relations we can see that [[sigma].sub.w]e(i) is dependent upon the choice of reduced expression of w. Therefore whenever we write [[sigma].sub.e](i) it should be understood that, although not always specified, we are fixing a choice of reduced expression of w.

We can visualise the algebra [degrees]R[([GAMMA]).sub.v] as a quiver with the vertices given by the idempotents e(i) and the arrows labelled by generators [x.sub.1], ..., [x.sub.m], [[sigma].sub.0], ..., [[sigma].sub.m-1] and determined by the relationship between idempotents. It is always the case that this quiver has two connected components so that we always have [degrees]R[([GAMMA]).sub.v] [congruent to] [e.sub.1][degrees]R[([GAMMA]).sub.v][e.sub.1] x [e.sub.2][degrees]R[([GAMMA]).sub.v][e.sub.2], where [e.sub.1], [e.sub.2] are certain idempotents in [degrees]R[([GAMMA]).sub.v]. In addition, we will see in Lemma 2.4 that, as algebras, these two components are isomorphic. Therefore it suffices to study one of these components in order to understand the algebra. For example, to show that [degrees]R[([GAMMA]).sub.v] is graded affine cellular and affine quasi-hereditary it is enough to show that one of the components, say [e.sub.2][degrees]R[([GAMMA]).sub.v][e.sub.2], has these properties (using Remark 1.4 and Proposition 1.6).

Given the data in the definition, together with a fixed v [member of] [sup.[theta]]NI, we have a VV algebra [W.sub.v] (see [Wal] for the definition). In [SVV11], the authors note that there is a canonical inclusion of algebras [degrees]R[([GAMMA]).sub.v] [??] [W.sub.v] given by e(i), [[sigma].sub.k], [x.sub.i] [??] e(i), [[sigma].sub.k], [x.sub.i] for k = 1, ... m - 1, i = i, ..., m, [[sigma].sub.0] [??] [pi][[sigma].sub.1][pi] so that the SVV algebras are unital subalgebras of the VV algebras.

Lemma 1.1 (Basis theorem for SVV algebras). Take v [member of] [sup.[theta]]NI with [absolute value of (v)] = 2m. For each element w [member of] [W.sup.B.sub.m] fix a reduced expression. The set of elements

[mathematical expression not reproducible]

forms a k-basis for [degrees]R[([GAMMA]).sub.v].

Proof. The proof is similar to that of [KL09, Theorem 2.5]. In particular we show this set is a spanning set in exactly the same way as for KLR algebras. For linear independence we use the polynomial representation of [degrees]R[([GAMMA]).sub.v] (see [SVV11]) and show that the elements in the set act by linearly independent operators.

Remark 1.2. Suppose we take v [member of] [sup.[theta]]NI such that supp([v.sup.+]) consists of two connected components. In other words, v = [v.sub.1] + [v.sub.2] where supp([v.sup.+]) and supp([v.sup.+]) are both connected and there are no arrows between any i [member of] supp([v.sup.+]) and j [member of] supp([v.sup.+]). It can be shown, using a similar proof as in [Wal, Proposition 2.6], that there is a Morita equivalence [mathematical expression not reproducible]. So we may assume that v is chosen in such a way that supp([v.sup.+]) is connected.

1.2. Affine cellularity. We now recall the definition of an affine cellular algebra [KX12]. Here we use the basis definition which is analogous to the way that Graham and Lehrer defined cellular algebras in [GL96]. The definition we recall is taken from [Cui], where it is shown to be equivalent to the basis-free definition in [KX12]. Let k be a noetherian domain and let A a unitary k-algebra. By an affine algebra, we mean a commutative k-algebra of the form B = k[[x.sub.1], ..., [x.sub.t]]/I, for some ideal I and some positive integer t.

Definition 1.3. We say that ([LAMBDA], M, B, C, *) is an affine cell datum for A, where ([LAMBDA], [less than or equal to]) is a finite poset, M([lambda]) is a finite set for each [lambda] [member of] [LAMBDA], [B.sub.[lambda]] is an affine k-algebra with an anti-involution [[sigma].sub.[lambda]], C = {[C.sup.[lambda].sub.s,t] | [lambda] [member of] [LAMBDA] and s, t [member of] M([lambda])} is a subset of A, and * is a k-linear anti-involution on A, if the following are satisfied.

(a) For each [lambda] [member of] [LAMBDA], let [[??].sup.[lambda]] be the right [B.sub.[lambda]]-span of [mathematical expression not reproducible]. Then [mathematical expression not reproducible] is a [B.sub.[lambda]]-basis of the right [B.sub.[lambda]]-module [[??].sub.[lambda]], and A = [[direct sum].sub.[lambda][member of][LAMBDA]] [[??].sup.[lambda]] as k- modules.

(b) For each [lambda] [member of] [LAMBDA], let [mathematical expression not reproducible]. For [lambda] [member of] [LAMBDA], s [member of] M([lambda]) and a [member of] A, b [member of] [B.sub.[lambda]], there exist coefficients [r.sup.s.sub.v](a) [member of] [B.sub.[lambda]] such that for all t [member of] M([lambda]),

[mathematical expression not reproducible],

and the coefficients [r.sup.s.sub.v](a) [member of] [B.sub.[lambda]] are independent of t.

(c) For all [lambda] [member of] [LAMBDA], s, t [member of] M([lambda]), and for any b [member of] [B.sub.[lambda]], [([C.sup.[lambda].sub.s,t] x b).sup.*] = [C.sup.[lambda].sub.s,t] x [[sigma].sub.[lambda]] (b).

The algebra A is said to be affine cellular if such an affine cell datum exists.

Remark 1.4. Let [A.sub.1] and [A.sub.2] be affine cellular algebras with affine cell data ([[LAMBDA].sub.1], [M.sub.1], [B.sub.1], [C.sub.1], [*.sub.1]) and ([[LAMBDA].sub.2], [M.sub.2], [B.sub.2], [C.sub.2], [*.sub.2]), respectively. Then [A.sub.3] := [A.sub.1] x [A.sub.2] has an affine cell datum ([[LAMBDA].sub.3], [M.sub.3], [B.sub.3], [C.sub.3], [*.sub.3]) where [[lambda].sub.3] = [[lambda].sub.1] [union] [[lambda].sub.2] with partial ordering given as follows: [lambda] [less than or equal to] [mu] if and only [lambda] and [mu] both lie in [[lambda].sub.i], for i [member of] {1, 2}, and moreover [LAMBDA] [less than or equal to] [mu] in [[LAMBDA].sub.i]. We also have, for any [lambda] [member of] [[LAMBDA].sub.3],

[mathematical expression not reproducible]

with the obvious anti-involutions. Furthermore, [C.sub.3] = [C.sub.1] [union] [C.sub.2] and [*.sub.3] is the anti-involution on [A.sub.3] defined by, [*.sub.3]([a.sub.1], [a.sub.2]) = ([*.sub.1]([a.sub.1]), [*.sub.2]([a.sub.2])). One can quickly check that this really does define an affine cell datum for [A.sub.3] so that [A.sub.3] = [A.sub.1] x [A.sub.2] is affine cellular.

Since [degrees]R[([GAMMA]).sub.v] [congruent to] [e.sub.1][degrees]R[([GAMMA]).sub.v][e.sub.1] x [e.sub.2][degrees]R[([GAMMA]).sub.v][e.sub.2], in order to show that [degrees]R[([GAMMA]).sub.v] is affine cellular it suffices to show that each component is affine cellular.

1.3. Affine quasi-heredity. We also recall the notions of affine quasi-heredity, for left Noetherian Laurentian algebras, and affine highest weight categories, introduced by Kleshchev [Kle15] as a graded analogue of the theory of Cline, Parshall and Scott. KLR algebras of finite Lie type are graded affine quasi-hereditary, as are certain classes of VV algebras. Let B be the class of all positively graded polynomial algebras. Recall from [Kle15] that a two-sided ideal J C A is an affine heredity ideal if; (SI1): [Hom.sub.A](J, A/J) = 0, (SI2): as a left module J [congruent to] m(q)P([pi]) for some graded multiplicity m(q) [member of] Z[q, [q.sup.-1]] and some [pi] [member of] [PI] such that [B.sub.[pi]] := [End.sub.A][(P([pi])).sup.op] [member of] B, and (PSI): as a right [B.sub.[pi]]-module P([pi]) is free finite rank.

By Lemma 6.5 in [Kle15], if J is an ideal in A which is projective as a left A-module, then (SI1) is equivalent J being an idempotent ideal, i.e. J = AeA, for an idempotent e 2 A.

Definition 1.5. An algebra A is affine quasi-hereditary if there exists a finite chain of ideals

[mathematical expression not reproducible]

with [J.sub.i+1]/[J.sub.i] an affine heredity ideal in A/[J.sub.i], for all 0 [less than or equal to] i < n. Such a chain of ideals is called an affine heredity chain.

Proposition 1.6. If A, B are affine quasi-hereditary algebras then the direct product A x B is an affine quasi-hereditary algebra.

Proof. Suppose A and B have affine heredity chains

[mathematical expression not reproducible]

respectively. Then one shows that

[mathematical expression not reproducible]

is an affine heredity chain for A x B.

2. Results. We remind the reader that if p [member of] I then we assume that p is not a root of unity.

Lemma 2.1. For v [member of] let e = igI,+ e(i). Then e[degrees]R[([GAMMA]).sub.v]e = [Rv.sup.+], i.e. every SVV algebra has a distinguished idempotent subalgebra isomorphic to a KLR algebra.

Proof. The proof is similar to [Wal, Proposition 1.17].

Remark 2.2. More generally, we can always express any v [member of] in the form [mathematical expression not reproducible], for some [mathematical expression not reproducible], where [mathematical expression not reproducible]. Let [mathematical expression not reproducible]. Provided i + [i.sup.-1] is not a summand of [??], for any i [member of] I, one can use the same argument in Lemma 2.1 show that e[degrees]R[([GAMMA]).sub.v]e [congruent to] [R.sub.[??]], the KLR algebra associated to [??].

Let [S.sub.m] := ([s.sub.0], [s.sub.2], [s.sub.3], ..., [s.sub.m-1]) be the parabolic subgroup of [W.sup.D.sub.m] generated by [s.sub.i], i = 1. We have already noted that [S.sub.m] is isomorphic to the symmetric group [G.sub.m].

Take v [member of] [sup.[theta]]NI, [absolute value of (v)] = 2m, and let [i.sub.min] be the summand [p.sup.2k][lambda] of [v.sup.+] such that k is minimal. Let [mu] = [v.sup.+] - [i.sub.min] + [i.sub.min] and let j [member of] [I.sup.[mu]] = {([j.sub.1], ..., [j.sub.m]) [member of] [I.sup.m] | [[SIGMA].sup.m.sub.k=1] [j.sub.k] = [mu]} be such that j = [i.sup.-.sub.min] and [j.sub.2], ..., [j.sub.m] are ordered by power of p. For example, when v = m[lambda] + m[[lambda].sup.-1] for some m > 1 and some [lambda] [member of] I, we have [i.sub.min] = [lambda] and j = ([[lambda].sup.- 1], [lambda], [lambda], ..., [lambda]). Let J be the following subset of [sup.[theta]][I.sup.v].

J := {w x j | w [member of] [S.sub.m] [subset] [W.sup.D.sub.m]}

where the action of elements w [member of] [S.sub.m], considered elements of [W.sup.D.sub.m], should be the obvious one.

Example 2.3. Take [mathematical expression not reproducible]. Then [i.sub.min] = [lambda], j = ([[lambda].sup.-1], [p.sup.2][lambda], [p.sup.4][lambda]) and

[mathematical expression not reproducible].

Let D = D([W.sup.D.sub.m]/[G.sub.m]) and D = D([W.sup.D.sub.m]/[G.sub.m]) denote the minimal length left coset representatives of [G.sub.m] and [S.sub.m] in [W.sup.D.sub.m], respectively. Note that D consists of elements w' which are obtained from w [member of] D by replacing every occurrence of [s.sub.0] and every occurrence of [s.sub.1] in a reduced expression of w with [s.sub.1] and with [s.sub.0], respectively. Then put [mathematical expression not reproducible] and [mathematical expression not reproducible]. By considering the way in which the elements w [member of] D, w' [member of] D' act on these tuples i we see that we have [e.sub.1] + [e.sub.2] = 1 in [degrees]R[([GAMMA]).sub.v] and therefore a decomposition [degrees]R[([GAMMA]).sub.v] = [e.sub.1][degrees]R[([GAMMA]).sub.v][e.sub.1] x [e.sub.2][degrees]R[([GAMMA]).sub.v][e.sub.2].

Lemma 2.4. There is a k-algebra isomorphism [e.sub.2][degrees] R[([GAMMA]).sub.v] [e.sub.2] = [e.sub.1][degrees] R[([GAMMA]).sub.v][e.sub.1].

Proof. Define the map

[mathematical expression not reproducible]

and extend k-linearly and multiplicatively. Examining the defining relations of [degrees] R[([GAMMA]).sub.v] we see that this is well-defined and is therefore an algebra homomorphism. In fact, it is an isomorphism of k-vector spaces and hence an isomorphism of algebras as there is an obvious inverse map.

As previously mentioned, the quiver which represents the algebra [degrees]R[([GAMMA]).sub.v] has two connected components. Lemma 2.4 tells us that the algebras which are associated to these connected components are isomorphic.

Corollary 2.5. There is an algebra isomorphism [e.sub.-][degrees]R[([GAMMA]).sub.v][e.sub.-] = [Rv.sup.+], where [e.sub.-] := [[SIGMA].sub.i[member of]J] e(i).

Proof. Under the isomorphism in Lemma 2.4 [mathematical expression not reproducible] is mapped to [e.sub.-] = [[SIGMA].sub.i[member of]J] e(i). It follows that e[degrees]R[([GAMMA]).sub.v]e is isomorphic to [e.sub.-][degrees]R[([GAMMA]).sub.v][e.sub.-]. Now we use Lemma 2.1.

Fix any element q [member of] [k.sub.x] in such a way that q [??] I. We can then define a VV algebra [W.sub.v] for any given v [member of] [sup.[theta]]NI, see [Wal] for details.

Theorem 2.6. There is a Morita equivalence [degrees]R[([GAMMA]).sub.v] ~ [W.sub.v] x [W.sub.v]. In particular, this demonstrates that any irreducible type D module arises from an irreducible type B module.

Proof. Using Lemma 2.4, it suffices to show there is a Morita equivalence [e.sub.2][degrees]R[([GAMMA]).sub.v][e.sub.2] ~ [W.sub.v]. Recall that we set [mathematical expression not reproducible].

We can also consider [e.sub.2] an element of [W.sub.v]. Proving that [e.sub.2][W.sub.v][e.sub.2] [congruent to] [e.sub.2][degrees]R[([GAMMA]).sub.v][e.sub.2] as k-algebras together with the fact that [e.sub.2] is full in [W.sub.v] implies that [W.sub.v][e.sub.2] is a progenerator in [W.sub.v]-Mod such that [mathematical expression not reproducible]. Then, by standard Morita theory, Morita equivalence between [e.sub.2][degrees]R[([GAMMA]).sub.v][e.sub.2] and [W.sub.v] follows.

Remark 2.7. In order to define VV algebras one must fix p, q [member of] [k.sub.x]. We remark here that when q [member of] I we do not have Morita equivalence between VV algebras and SVV algebras.

2.1. Setting: p [not member of] I. For the following two corollaries we will assume p [not member of] I.

Corollary 2.8. Suppose p [not member of] I. For v [member of] [sup.[theta]]NI, there is a Morita equivalence [mathematical expression not reproducible]. In particular, this demonstrates that any irreducible type D module arises from an irreducible type A module when p [not member of] I.

Proof. Fix an element q [member of] [k.sup.x] in such a way that q [??] I. Then we have, by Theorem 2.10 in [Wal], Morita equivalence [mathematical expression not reproducible]. Now use Theorem 2.6.

Corollary 2.9. When p [not member of] I, with the additional constraint of p not a root of unity, the algebras [degrees]R[([GAMMA]).sub.v] are affine quasi-hereditary and affine cellular.

Proof. We first note that affine quasi-heredity is a Morita invariant between unital algebras. Then by Proposition 1.6 and Corollary 2.8 it suffices to show that the algebras [mathematical expression not reproducible] are affine quasi-hereditary. This is true when p is not a root of unity (see [Kle15, Section 10.1] and results throughout [BKM14]). This proves affine quasi-heredity. It was shown in [KLM13] that KLR algebras of type [A.sub.[infinity]] are affine cellular. This, together with Remark 1.4, shows that the algebras [mathematical expression not reproducible] are affine cellular in this case. The proof is completed by applying Lemma 3.4 in [Yan14].

2.2. Setting: p [member of] I. Now suppose p [member of] I and assume p is not a root of unity. Then we can write [I.sub.p] = [I.sup.+.sub.p] [??] [I.sup.-.sub.p], where [mathematical expression not reproducible] and [[GAMMA].sub.I] is of the form

[mathematical expression not reproducible].

Remark 2.10. If we take v [member of] [sup.[theta]]NI such that the number of summands of v equal to p is less than 2 (i.e. [v.sub.p] < 2), then all the results of Subsection 2.1 apply to [degrees]R[([GAMMA]).sub.v]. Namely, we are again reduced to type A via the Morita equivalence [mathematical expression not reproducible]. Therefore, if p [member of] I we may assume that p has multiplicity at least two in v, i.e. [v.sub.p] [greater than or equal to] 2.

Let A be the path algebra A = k([a.sub.1] [??] [a.sub.2]) where the arrow from [a.sub.1] to [a.sub.2] is labelled [u.sub.1] and the arrow from [a.sub.2] to [a.sub.1] is labelled [u.sub.2]. We consider A a left k[z]-module with the action defined by; z x [a.sub.1] = u2 uiai and z * [a.sub.2] = [u.sub.1][u.sub.2][a.sub.2]. Suppose p [member of] I, p is not a root of unity and we take v [member of] [sup.[theta]]NI. Recall that in [Wal, Section 2.6] we define the structure of a right k[z]-module on [mathematical expression not reproducible].

Corollary 2.11. When p is not a root of unity and [v.sub.p] = 2 there is a Morita equivalence [mathematical expression not reproducible].

Proof. Use Theorem 2.6 and [Wal, Theorem 2.46].

doi: 10.3792/pjaa.94.7

Acknowledgements. I thank Dr. Vanessa Miemietz for reading a draft of this work and for her helpful comments and feedback. This work was completed at Universite Paris Diderot (Paris VII) and was supported in part by the European Research Council in the framework of the The European Union H2020 with the Grant ERC 647353 QAffine.

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Ruari Donald WALKER

Bureau 627, Universite Paris Diderot-Paris VII, Batiment Sophie Germain, 75205

Paris Cedex 13, France

(Communicated by Masaki KASHIWARA, M.J.A., Dec. 12, 2017)

2010 Mathematics Subject Classification. Primary 16D90, 17D99.
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Title Annotation:Shan, Varagnolo and Vasserot
Author:Walker, Ruari Donald
Publication:Japan Academy Proceedings Series A: Mathematical Sciences
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2018
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