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Author(s): Iain Gordon and Dmitriy Rumynin
Article Title: Subregular Representations of ${\frak s} {\frak l}_{n}$ and
Simple Singularities of Type A
n−1Year of publication: 2003
Subregular Representations of
sl
n
and Simple
Singularities of Type
A
n
1
IAIN GORDON1and DMITRIY RUMYNIN2
1
Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, U.K. e-mail: [email protected]
2
Mathematics Department, University of Warwick, Coventry, CV4 7AL, U.K. e-mail: [email protected]
(Received: 19 March 2001; accepted in final form: 26 June 2002)
Abstract. Alexander Premet has stated the following problem: what is a relation between subregular nilpotent representations of a classical semisimple restricted Lie algebra and non-commutative deformations of the corresponding singularities? We solve this problem for typeA.
Mathematics Subject Classifications (1991). Primary: 17B50; Secondary: 14J17, 16G10, 16S80.
1. Introduction
1.1. The last few years have seen interest grow in the representation theory of reduc-tive Lie algebras in posireduc-tive characteristic. Classical results of Kac and Weisfeiler show that the nilpotent coadjoint orbits of the Lie algebra comprise a natural para-meter set for the representation theory, [25]. It is thus a basic problem to understand how the geometry of the nilpotent coadjoint orbits influences representation theory. A milestone in this direction is Premet’s theorem which asserts that the dimension of any simple module associated to the orbit O has dimension divisible byp1=2ðdimOÞ,
wherepis the characteristic of the field, [21]. Building on Premet’s theorem, Jantzen studied the subregular nilpotent coadjoint orbit in detail, obtaining a great deal of information on not only the simple modules, but also baby Verma modules, [14].
1.2. More recent work of Lusztig suggests a relationship between the representation theory associated to a nilpotent coadjoint orbit Oand the geometry of a transverse slice to O, together with its desingularisation in the Springer resolution, [18]. The relationship would tie the simple and projective modules associated to Oto certain elements in theK-theory of the transverse slice and its desingularisation. In the sub-regular case, the transverse slice is a Kleinian singularity and the Springer resolution provides its minimal resolution.
1.3. Quantisations of Kleinian singularities were introduced by Hodges [11] for type
Aand later by Crawley-Boevey and Holland [7] for all types. Premet has suggested a possible relationship between these quantisations and subregular representations of a
simple Lie algebra. In [22] Premet examines this in characteristic zero. This paper is concerned with exploring Premet’s suggestion in the modular typeAcase. In a sub-sequent paper we consider Lusztig’s conjectures in the subregular case. It is the quan-tisations which build a bridge between the representation theory and the geometry, allowing us to connect the two sides.
1.4. Let us give some more detail. We refer the reader to later sections of the paper for unexplained definitions and notation. Fix an integern>0. LetK(respectivelyL) be an algebraically closed field of characteristicp(respectively arbitrary characteris-tic). We will assume throughout thatpandnare coprime. LetFbe the prime subfield of K. Given an integerm2Zwe denote its reduction modulop bymm 2F.
1.5. LetUbe the enveloping algebra ofslnðKÞand forv2K½tletTðvÞbe Hodges’
non-commutative deformation of a Kleinian singularity of type Aover K. Let Uw be the reduced enveloping algebra for a subregular nilpotent functional w and let
Uw;lbe the central reduction ofUwdetermined by the weightl. We show in Theorem 7.2 that, for p>n, there exists a polynomial vl2K½t such that Uw;lffiMatðtðvlÞÞ, where tðvlÞis a central reduction of TðvlÞ. Moreover, this isomorphism respects a natural Z-grading on the algebras Uw;l and tðvlÞ and so induces an equivalence between the category of Uw;l-T0-modules and a category of suitably graded tðvlÞ -modules.
1.6. A crucial step in the above theorem is the comparison of Uw;l andtðvlÞwith a basic algebra we call the no-cycle algebra.This algebra is defined over any fieldL, depends on a positive integer k, and is denoted NLðkÞ. The no-cycle algebra is a string algebra and hence its indecomposable modules can be described by a simple combinatorial procedure. As an application of this comparison we give a sufficient condition for the indecomposability of a baby Verma module belonging to the block ofUwdetermined by a regular weightl. This is formulated in terms of the geometry of the Springer fibre Bw.
1.7. There are several clear directions for future work arising from this paper. Firstly, the hypothesis p>n should be weakened to p and n being coprime. It would also be highly desirable to extend the results to other types. At the end of Section 7 we sketch how to extend our results to type B. It seems likely, how-ever, that the techniques used here are not sufficient for this in general. Further-more, we expect NCðnÞ-modules to correspond to a central reduction of the subregular representations of a quantum group of type A at a root of unity. We have not, however, checked this here.
Section 4 we study subregular representations of slnðKÞand relate them to the
no-cycle algebra. Section 5 takes a brief look at Gro¨bner–Shirshov bases for non-commutative algebras and their representations. In Section 6 we study Hodges’ quan-tisation using, in particular, the results of the previous section. In Section 7 we prove Theorem 7.2, whilst in Section 8 we present the application to baby Verma modules.
2.
Z
-Categories
2.1. AZ-category is an AbelianL-category equipped with exactshift functors,½ifor eachi2Z, together with natural isomorphisms½i ½ j ! ½iþj. IfCandDare both Z-categories, we say that anL-linear functorF:C!Dis aZ-functor if the functors
F ½iand½i Fare naturally isomorphic for everyi2Z. AZ-functorF:C!Dis a Z-equivalence if there exists aZ-functorC:D!Csuch thatCFffi1CandFCffi1D. We sayCandDare equivalentZ-categories. Note that an equivalence betweenCandD
need not be aZ-equivalence [10, Section 5].
2.2. Let R¼Li2ZRi be a Z-graded (noetherian) algebra, that is RiRjRiþj.
A Z-graded R-module is an R-module, M, together with a L-space decomposi-tion M¼Li2ZMi satisfying RiMjMiþj. The category of Z-graded (finitely
generated)R-modules, denotedR-Grmod (R-grmod) is an example of aZ-category. By definition, we haveðM½iÞj¼Mji:for alli;j2Z.
2.3. Given X;Y2Candi2Z, set HomðX;YÞi¼HomCðX½i;YÞ. We define
HomðX;YÞ ¼M
i2Z
HomðX;YÞi;
a Z-graded vector space. The identification HomCðX½i;YÞ ffiHomCðX½iþj;Y½jÞ yields a composition law for X;Y;Z2C: HomðY;ZÞjHomðX;YÞi!
HomðX;ZÞiþj. Then, thanks to 2.2, the space EndðXÞ ¼HomðX;XÞ becomes a
Z-graded L-algebra and HomðX;YÞ a Z-graded EndðXÞ-module. The functor HomðX;Þ is aZ-functor fromCto EndðXÞ-Grmod.
3. The No-cycle Algebra
3.1. Letk2Nbe a fixed integer. LetQbe the directed graph withkvertices and 2k
edges labelledaiandbifori2Z=kZ, see Figure 1. LetLQbe the path algebra ofQ. Theno-cycle algebra,NLðkÞ, is the quotient ofLQby the two sided ideal generated by all non-trivial paths in Qwhich start and end at the same vertex. By inspection,
NLðkÞis akð2k1Þ-dimensional algebra.
3.2. If L admits a primitive nth root of unity z then the no-cycle algebra can be described as a ‘skew coinvariant algebra’. LetGbe a cyclic group of ordern acting on L½X;Ybygm:ðX;YÞ 7! ðzmX;zmYÞ. There exists an isomorphism
NLðkÞ ffi
L½X;Y
which sendsXtoPnk¼1bk,Yto
Pn
k¼1akandgto
Pn k¼1z
nk
ek. This realization is of fundamental importance since the ‘skew coinvariant algebra’ controls much of the geometry surrounding the resolution of the Kleinian singularity of type Ak1.
3.3. The algebraNLðkÞisZ-graded: a vertex idempotenteihas degree 0,aihas degree
1 and bi has degree 1. Following 2.2 the category of Z-graded modules will be
denoted byNLðkÞ-grmod.
3.4. We need some notation before describing several NLðkÞ-modules. For
i2Z=kZ, introduce formal inverses of the arrows ai (respectively bi), written ai
(respectivelyb
i). The head (respectively tail) of an arrow,c, is denotedhðcÞ
(respec-tively tðcÞ), and we define hðcÞ ¼tðcÞ (respectively tðcÞ ¼hðcÞ). We form formal
paths of length t, c1;. . .;ct, where each cj is of the form c or c for some arrow c and tðcjÞ ¼hðcjþ1Þ. Given a formal path c1;. . .;ct, we define its inverse to be c
t;. . .;c1, where ðcÞ
equals
c.
3.5. For t4k, letSt be the set of formal pathsc1. . .ct such that ifcj¼ai
(respec-tively bi) then cjþ1 is either ai1 or bi1 (respectively biþ1 or aiþ1), and similarly if cj¼a
i (respectivelybi). Furthermore, ift¼kthen exclude fromSkthe formal paths
consisting entirely ofa’s or entirely ofb’s and also the inverses of such formal paths. Fort<k(respectivelyt¼k) letrtbe the equivalence relation onSt which identifies
a formal path with its inverse (respectively its cyclic permutations and their inverses). Let Wt be a set of equivalence class representatives inSt forrt.
3.6. Fort<k, an elementC¼c1. . .ct 2Wtdefines atþ1-dimensional
indecompo-sablestring NLðkÞ-moduleStðCÞ. A basis is given byfz0;. . .;ztgwhere, for 14j4t,
the element zj is concentrated at vertexhðcjÞ, andz0 is concentrated at vertextðc1Þ.
For 14j4tifcj¼c, an arrow, definecðzjÞ ¼zj1, whilst ifcj¼c, definecðzj1Þ ¼ zj. All other arrows inQact as zero.
3.7. Fort¼k, and elementC¼c1;. . .;ck2Wktogether with a scalarl2L, define
an indecomposable band NLðkÞ-module BdlðCÞ. A basis is given by fz0;. . .;zk1g
where, for 04j4k1, the element zj is concentrated at vertex hðcjÞ. If ck¼c,
an arrow, define cðz0Þ ¼lzk1, whilst if ck¼c, define cðzk1Þ ¼l1z0. For
14j4k1, if cj¼c, an arrow, then cðzjÞ ¼zj1, whilst if cj¼c define cðzj1Þ ¼zj. All other arrows of Qact as zero.
3.8. The algebraNLðkÞbelongs to a family of tame algebras called string algebras. Any non-zero indecomposable NLðkÞ-module of dimension no greater than k is isomorphic to either StðCÞor BdlðCÞfor some uniqueCandl[6, Section 3] (see also [2]). Observe that the modules StðCÞadmit aZ-grading compatible with theZ-grading on NLðkÞintroduced in 3.3.
4. The Reduced Enveloping Algebra
4.1. Let w2slnðLÞ be the functional vanishing on upper triangular matrices and
defined as follows on the strictly lower triangular matrices
wðEi;jÞ ¼ 1 ifi
¼jþ1 and 14j4n2;
0 otherwise:
Let P (respectively Pþ) be the weight lattice (respectively dominant weights) of
SLnðLÞ with respect to the standard choice of torus and Borel subgroup. Let
f$1;. . .; $n1g be the fundamental weights and let r¼$1þ þ$n1. The Weyl
group W is the symmetric group Sn acting on P. The W-orbit through l2P
contains a unique representative in Pþ. We also need a dot action given by
wl¼wðlþrÞ r; w2W;l2P:
4.2. LetBbe the flag variety. As a set this consists of all Borel subalgebras ofslnðLÞ,
that is those subalgebras which are conjugate under SLnðLÞto the upper triangular
matrices. The cotangent bundle of Bis naturally identified with the variety
f
N
N ¼ fðx;bÞ:xðbÞ ¼0g slnðLÞB
where the first projectionp1 becomes the moment map. The Springer fibreBwis the subvariety p1
1 ðwÞof B.
There is a simple way to parametriseBw, [24, Section 6.3]. Given a basisuiofLn, let Fðu1;. . .;unÞ be a flag with the span of u1;. . .;uk as the k-dimensional space.
Let vi be an element of the standard basis ofLn so that Ei;jvj¼vi. We introduce
the flag
for all ðk;aÞ 2 ðf1;. . .;n1g LÞ [ fð0;0Þg. The irreducible components ofBw are projective lines Pk, 14k4n1 where
Pk¼ fFnk;aja2Lg [ fFnk1;0g:
For 24k4n1 the components Pk1 and Pk intersect transversally at a point pk1;k¼Fnk;0. ComponentsPi andPj withjijj>1 do not intersect.
Consider the following one parameter subgroup of the diagonal matrices in SLnðLÞ
T0¼ fnðtÞ ¼tE1;1þtE2;2þ þtEn1;n1þt1nEn;n:t2Lg:
Notice that T0 stabilisesBw, sincenðtÞ Fi;a¼Fi;tna.
4.3. Let us further assume that L¼C. By the Jacobson-Morozov theorem there exists an sl2-triple e;h;f2slnðCÞ such that TrðexÞ ¼wðxÞfor each x2slnðCÞ. Let N be the variety of nilpotent elements inslnðCÞ. Let
Vw¼ fm2slnðCÞ j 8x2slnðCÞmð½x;fÞ ¼wð½x;fÞg:
By [24, Theorem 6.4 and Section 7.4] Vw is a Kleinian singularity of typeAn1 and
Lw¼p11ðVwÞis its minimal desingularisation with the exceptional fibreBw.
4.4. For the rest of this section we will assume that p>n. We expect, however, all results to hold under the weaker condition that p andn are coprime. The proof of Proposition 4.13 is the crucial point where we require the restriction p>nand all other results of this section, up to 4.17 inclusive, will continue to hold if this pro-position can be proved under the weaker hypothesis.
4.5. We are going to work with the Lie algebra slnðKÞ now. For any element X2slnðKÞletX½p2slnðKÞdenote thepth power ofX.
Let UðslnðKÞÞbe the universal enveloping algebra of slnðKÞ. For anyX2slnðKÞ
the element XpX½p2Uðsl
nðKÞÞ is central. We will study representations of the
following subregular reduced enveloping algebra
Uw:¼
UðslnðKÞÞ
ðXpX½pwðXÞp:X2sl
nðKÞÞ
:
LetL¼P=pP, anF-vector space, and letc:P!Lbe the quotient map. Both the usual and the dot actions ofWonPpass to L. Let WðlÞbe the stabiliser of l2L
under the dot action. Set
C0¼ l2P:lþr¼
X
ri$iwithri50 andp5ðr1þ þrn1Þ
n o
:
4.6. BLOCKS
By [5, Theorem 3.18] we have a block decomposition
Uw¼
M
l
Bw;l;
wherel runs over a set of representatives of theW-orbits onL.
Let hbe the diagonal Cartan subalgebra ofslnðKÞ. The Weyl group acts by alge-bra automorphisms on the polynomial ringSðhÞ. Forl2Lthe partial coinvariants give a local, graded algebra
Cl¼
SðhÞWðlÞ ðSðhÞWþÞ:
Thanks to ([19], Theorem 10) and ([22], Theorem 8.2) there is an injective algebra homomorphism Cl!Bw;l, whose image is central in Bw;l;. Henceforth, we will
identifyCl with its image inBw;l.
4.7. We will be concerned with the category of finite-dimensional Uw-modules,Uw -mod, or more specifically the subcategory of Bw;l-modules. These categories have graded analogues which we introduce now.
By construction wðtXt1Þ ¼wðXÞ for all X2sl
nðKÞ and t2T0. As a result the
action ofT0 on UðslnðKÞÞpasses to an action on the quotientUw.
Following Jantzen [12], aUw-T0-module is a finite-dimensional vector spaceVover
Kthat has a structure both as a Uw-module and as a rationalT0-module such that
the following compatibility conditions hold:
(1) We have tðXvÞ ¼ ðtXt1Þtvfor allX2sl
nðKÞ;t2T0 andv2V;
(2) The restriction of theslnðKÞ-action onVto LieðT0Þis equal to the derivative of
theT0-action onV.
We obtain the categoryUw-T0-mod, whose objects are theUw-T0-modules and whose
morphisms are the T0-equivariantUw-module homomorphisms.
For i2Z, there are shift functors ½i:Uw-T0-mod!Uw-T0-mod. These send a
givenUw-T0-moduleVto the object having the sameUw-module structure but with
T0 acting by nðtÞv¼tipnðtÞvfor nðtÞ 2T0 and all v2V. This makes Uw-T0-mod a
Z-category.
By [9, 9.3] the full Z-subcategory Bw;l-T0-mod of Uw-T0-mod is well-defined.
Its objects are Bw;l-modules with a compatible rational T0-action. The projection
functor prl:Uw-T0-mod!Bw;l-T0-mod is aZ-functor.
4.8. Let F:Uw-T0-mod!Uw-mod denote the functor which forgets the T0
simple inUw-T0-mod and any lift of a projective indecomposableUw-module is pro-jective indecomposable in Uw-T0-mod. SupposeMis gradable, that is there exists a Uw-T0-module V such that FðVÞ ¼M. Then, by ([15], Remark 1.5), we have FðsocVÞ ¼socMandFðradVÞ ¼rad M.
For any M;N2Uw-T0-mod, using the notation of 2.3, we have
HomUwðFðMÞ;FðNÞÞ ¼ iHomUw-T0ðM½i;NÞ, ([10], Section 2).
4.9. The category Uw-T0-mod admits a contravariant self-equivalence, D whose
square is canonically isomorphic to the identity functor, ([13], Sections 1.13 and 1.14). MoreoverDfixes the simple modules inBw;l-T0-mod, ([13], Proposition 2.16).
4.10 TRANSLATION FUNCTORS
Using the mapc:P!L, we will abuse notation by writingBw;l-T0-mod and prlfor l2P(we should really take the image of l underc). Given l;m2C0 we define a
translation functor
Tlm:Bw;l-T0-mod!Bw;m-T0-mod
by TlmðVÞ ¼PrmðEVÞ, where E is the simple SLnðKÞ-module with the highest
weight wðmlÞ 2Pþ for somew2W.
Note that we get (in general) more than one functorBw;l-T0-mod!Bw;m-T0-mod
for fixedlandm: ifmandm0are two distinct weights inC
0withcðmÞandcðm0Þin the
same W-orbit thenTlm andTlm0 will be two (in general) distinct functors fromBw;l
-T0-mod to Bw;m-T0-mod¼Bw;m0-T0-mod.
4.11. BABY VERMA MODULES
Givenb2Bwandl2Pwe have a one dimensional representation ofU0ðbÞ, the
sub-algebra ofUwgenerated by the elements ofb, described as follows. Letg2SLnðKÞbe
such thatgconjugatesbto the upper triangular matrices inslnðKÞ, saybþ. There is a
one dimensional U0ðbþÞ-module which is annihilated by strictly upper triangular
matrices and on which diagonal matrices act via cðlÞ. Conjugation by g provides an isomorphism betweenU0ðbÞandU0ðbþÞ, and so gives a one dimensionalU0ðbÞ
-module, say Kl. It can be checked that this module is independent of the choice of g2SLnðKÞ. Induction yields ababy Verma module
Vðb;lÞ ¼UwU0ðbÞKl
This is a Bw;l-module on whichCl acts by scalar multiplication.
4.12. The moduleVðbþ;lÞcan be given the structure of aBw;l-T0-module whereT0
acts on 11 throughl. Set Vðbþ;lÞ0¼DðVðbþ;lÞÞ(it follows from [12, 11.16(1)]
PROPOSITION [13, Theorem 2.6]. Suppose l2C0 with lþr¼Pri$i and let r0¼p ðr1þ þrn1Þ.
ðiÞ The category Bw;l-T0-mod has simple modules L0;L1;. . .;Ln1 ðup to isomor-phism and shiftÞwhere the module Lihas dimension pðn2n2Þ=2
rn1iðif rn1i¼0 then Lishould be omitted from the list of simple modulesÞ.
ðiiÞ For each i between0 and n1there exists a uniserial module Vi2Bw;l-T0-mod whose Loewy layers are Li;Liþ1½1;. . .;Li1½1n ðcount the subscripts modulo n, and omit Viand Liwhenever rn1i¼0Þ.
ðiiiÞ For each i between0and n1there exists a uniserial module V0
i 2Bw;l-T0-mod whose Loewy layers are Li;Li1½1;. . .;Liþ1½n1 ðcount the subscripts modulo n, and omit V0
i and Liwhenever rn1i¼0Þ.
ðivÞ For any m2P such thatcðmÞand cðlÞare in the same W-orbit there exists a unique i and j2Zðrespectively i0;j0Þ such that Vðb
þ;mÞ ffiVi½ j ðrespectively
Vðbþ;mÞ0ffiVi00½j0Þ.
4.13 ENDOMORPHISMS
Letl2Pand letZlbe the centre ofBw;l. RecallClZl. LetMbe inBw;l-T0-mod.
There is a homomorphism
yM:Zl!EndBw;lðFðMÞÞ;
sendingz2Zl to the endomorphismðm7 ! zmÞ.
PROPOSITION. Let l2C0 and Q be a projective indecomposable module in Bw;l-T0-mod. The homomorphism
yQ:Zl!EndBw;lðFðQÞÞ
is surjective.
Proof. To make the paper as self-contained as possible, we will give a direct proof of this for regular weightsl, since the general case relies on an unpublished result of Jantzen and So¨rgel, [16].
It is enough to prove this in the ungraded case. LetP0;. . .;Prbe all distinct (up to
isomorphism) projective indecomposableBw;l-modules. The algebraClhas a simple socle [8, Corollary 3.9]. Therefore if AnnClðPiÞis non-zero it must contain the socle
of Cl. The equality 0¼AnnClðBw;lÞ ¼ \iAnnClðPiÞ, implies that Cl acts faithfully
on at least one projective indecomposable, sayP0. By Proposition 4.12 and a result
of Jantzen ([12], Proposition 10.11) the dimension of EndBw;lðP0Þ ¼dimCl, so Zl
generates EndBw;lðP0Þ.
Now assume l is regular. It follows from ([13], Section 2.3) that we can find a translation functor T such that TðP0Þ ffiPi and, by ([12], Section 11.21), that T is
auto-morphism of Zl, sayTT~. It follows that EndBw;lðPiÞis generated by TT~ðClÞ Zl, as
claimed.
Iflis not regular the above argument fails since it need no longer be true that we can find a translation functorTwhich is a self-equivalence and sendsP0toPi. In this
situation we use the following fact, ([16], C.6 Claim 2): if the highest weight ofPi(in
the graded category) belongs toC0and does not lie on the affine wall, then the
cano-nical mapCl!EndBw;lðPiÞis an isomorphism. Hence, it is enough to show that we
can find a representative of the isomorphism class ofPiwhose highest weight belongs to C0 and does not lie on the affine wall. A straightforward calculation shows that
this follows from ([14], Section 2.3). &
4.14. It is possible toslightlyweaken the hypothesisp>nwhen using the results of [16], and hence the hypothesis of this whole section. The proof of Jantzen’s results, however, do not apparently generalise to the best case where p andn are coprime. We leave these details to the interested reader.
4.15. Let Jbe the unique maximal ideal ofZl.
LEMMA. Let Pibe the projective cover of Liin Bw;l-T0-mod. Then½Pi=JPi:Li½j ¼dj0. Proof. It is enough to prove this for ungraded Bw;l-modules. Suppose that
FðLiÞ is a composition factor of FðPiÞ=JFðPiÞ. Thus FðLiÞ appears as a direct summand of radmFðPiÞ=radmþ1FðPiÞfor somem2N. Hence we have a commutative
diagram
Thanks to Lemma 4.13 there exists z2Zl such that the above endomorphism of
FðPiÞis multiplication byz. Thenz2=J since, by hypothesis, the composition factor FðLiÞdoes not lie inJFðPiÞ. SinceZlis local it follows that the endomorphism is an isomorphism and so FðLiÞlies in the head ofFðPiÞas required. &
4.16. TWO CENTRAL REDUCTIONS
Let JJ~be the unique maximal ideal of Cl and recall that J is the unique maximal ideal ofZl. We introduce two central reductions
~ U Uw;l¼
Bw;l
~ J JBw;l
; Uw;l¼
Bw;l
JBw;l
:
SinceJJ~andJare homogeneous bothUU~w;landUw;linheritZ-gradings fromBw;l. The category of graded modules Uw;l-T0-mod is thus a full subcategory ofBw;l-T0-mod
4.17. The main result of this section follows.
PROPOSITION. Suppose l2C0 with lþr¼Pri$i, and let r0¼p ðr1þ þ rn1Þ. Let k be the number of non-zero ri’s. Then Uw;l is Morita equivalent to the
no-cycle algebra NKðkÞ. Moreover, iflis regular there is aZ-equivalence of categories
Uw;l-T0-mod!NKðnÞ-grmod:
Proof. LetAbe a finite dimensional algebra with simple modulesS1;. . .;Sr. The
Gabriel quiver of A is the directed graph with vertices labelled from 1 to r and dim Ext1
AðSi;SjÞedges fromitoj. By ([3], Proposition 4.17)Ais Morita equivalent to
the the path algebra of its Gabriel quiver factored by some admissible ideal, that is an ideal generated by linear combinations of paths of length at least two.
Recall the notation of Proposition 4.12. Let 04i1 <i2< <ik4n1 be such
that Lit 6¼0, or, equivalently, rn1it 6¼0. Set st¼itþ1it for 14t<k and set sk¼nþi1ik. Since Li appears only once as a composition factor of Vi, ZðBw;lÞ acts by scalars on Vi, making Vi a Uw;l-module. By [13, Proposition 2.19], for
t6¼t0 modulokandj2Z
Ext1Uw;lT0ðLit½j;Lit0Þ ¼
K; ift0¼tþ1;j¼s
t ort0¼t1;j¼ st1;
0; otherwise.
(
Thus the Gabriel quiver ofUw;l is of the form 3.1, possibly with loops added at the vertices. LetBbe the quotient of this quiver which is Morita equivalent toUw;l. We will showBis isomorphic toNKðkÞ.
The projective covers of the simpleB-modules are spanned by the paths ending in a fixed vertex. Hence, Lemma 4.15 shows that there can be no loops at vertices and further, thatBis therefore a quotient ofNKðkÞ. In particular, its dimension is at most
kð2k1Þ.
LetTit be the kernel of the sum of two projectionsVit V
0
it !Lit. By Proposition
4.12
½FðTitÞ:FðLit0Þ ¼
2; if t6¼t0
1; if t¼t0:
SinceFðTitÞis a quotient module ofFðPitÞ, the dimension ofBcan be estimated by
dimB¼End M
k
t¼1 FðPitÞ
!
¼X
t;t0
½FðPitÞ:FðLit0Þ5
X
t;t0
½FðTitÞ:FðLit0Þ ¼kð2k1Þ:
We deduce that BffiNKðkÞ, proving the first statement of the theorem. This also proves thatTit is the projective cover ofLit inUw;l-T0-mod.
Letlbe regular, so thatk¼n, and letT¼ Tit. Thanks to 2.3 and 4.8 the algebra E¼EndUw;lðFðTÞÞ
op has a Z-grading. By ([10], Theorem 5.4) U
But, up to a choice of scalars,bi corresponds to aUw;l-T0-module homomorphism
sending Ti½1to Tiþ1, andai corresponds to the aUw;l-T0-module homomorphism
sendingTiþ1½1toTi. This proves the second claim. &
4.18. TWO CENTRAL REDUCTIONS (II)
In general the inclusion ClZl is strict, ([8], Corollary 3.9). However, the natural homomorphism UU~w;l!Uw;l is an isomorphism. This follows from the fact in the proof of Proposition 4.3 that the map
y1:Cl!EndBw;lðFðPÞÞ
is an isomorphism for any projective indecomposable module P. The arguments of 4.15 and 4.17 are then valid for UU~w;l, from which the isomorphism follows. We expect this continues to hold under the weaker hypothesis that p and n are coprime.
5. Gro¨bner-Shirshov Bases
5.1. We are going to use Gro¨bner-Shirshov bases for associative algebras (see [4] for two-sided ideals and [17] for one-sided ideals). In this section we quickly explain the technique to make our paper self-contained. Although the version for one-sided ideals [17] is sufficient for our ends, we generalise to arbitrary mod-ules to avoid repetitions.
5.2. LetR¼LhX1;X2;. . .;Xlibe a free associative algebra. LetFbe a freeR
-mod-ule with generatorsY1;. . .;Yk. Althoughlandkare natural numbers here, one can
use, with certain care, the technique for transfinite ordinals.
Let Rbe the set of monomials inR,F the set of monomials inF. The setR[F
admits a partial multiplication with a two-sided unit 1R(one agrees thatm1¼mfor m2F). We always make the assumption that a product is defined when we write the product. We start with a linear order onR[F such that
. 8z2R[F z1R;
. z1z2 ¼)wz1vwz2v;
. 8z2R[F the setfw2R[Fjzwgis finite.
A degree lexicographical order is most practical but there are different orders. For a nonzero elementfofR[Fwe denote the highest term offbyff.
5.3. A pair of subsets SR and TF determine an algebra A¼R=RSR and a left A-module M¼ARðF=RTÞ ¼F=ðRTþRSFÞ. The technique of Gro¨bner–
Each f2S[T gives a rewriting rule ff!fff. We write a?b if b can be obtained fromaby using rewriting rules. Note that one cannot rewrite a monomial
ff t unlesst¼1 or f2S.
5.4. COMPOSITION
For certain f;g2R[F, w2R[F, we can form a composition ðf;gÞw. If
w¼ff V ¼Wgg¼WZVfor someW;Z;V2R[FwithZ6¼1 then the composition is
ðf;gÞw¼f VWg:
Ifw¼Wff V ¼ggfor someW;V2R[F then the composition is
ðf;gÞw¼Wf Vg:
These two cases are mutually exclusive.
A pairðS;TÞis a Gro¨bner–Shirshov pair ifðf;gÞw?0 for all possiblef;g2S[T andw2R[F.
The following version of Shirshov’s composition lemma can be proved by stan-dard methods [4]. Ifk¼1 and T¼ ;then the statement is the standard version of Shirshov’s composition lemma [4]. If k¼1 andTarbitrary then it is a version for left ideals [17].
5.5. SHIRSHOV’S COMPOSITION LEMMA
ForS andTas above, we consider the set of monomials
B¼ fZ2F j 8f2S[T8W;V Z6¼Wff V g:
If ðS;TÞ is a Gro¨bner–Shirshov pair then the image of B is a basis of M as an L-vector space.
Moreover, for everyðS;TÞthere exits a Gro¨bner–Shirshov pairðS0;T0Þsuch that
RSR¼RS0RandRSFþRT¼RS0FþRT0.
5.6. BUCHBERGER’S ALGORITHM
A proof of existence of a Gro¨bner–Shirshov pair uses transfinite recursion, called Buchberger’s algorithm. It proceeds as follows. One starts with ðS0;T0Þ ¼ ðS;TÞ.
Given ðSm;TmÞwe produce the next pair ðSmþ1;Tmþ1Þ such that RSmR¼RSmþ1R
and RSmFþRTm¼RSmþ1FþRTmþ1. Consider all possible compositions ðf;gÞw
with f;g2Sm[Tm. To each such composition, apply a sequence of rewriting rules
vv!vvvwithv2Sm[Tm so thatðf;gÞw?½f;gw and½f;gw cannot be rewritten
of rewriting rules to use. Another sequence can give a different answer. Define the following sets
Im¼ f½f;gwj f;g2Sm[Tm; ½f;gw6¼0g;
Jm¼ fgj f;g2Sm[Tm; ðf;gÞw¼Wf Vg; ½f;gw6¼0g:
Now we can make the recursion step,
Smþ1¼ ðSm[ ðIm\RÞÞ nJm;Tmþ1¼ ðTm[ ðIm\FÞÞ nJm:
5.7. TERMINATION OF BUCHBERGER’S ALGORITHM
If ðS;TÞ is a Gro¨bner-Shirshov pair then S1¼S0, T1¼T0, and the procedure
terminates immediately. If S is a Gro¨bner-Shirshov basis (equivalently ðS;;Þ is a Gro¨bner–Shirshov pair), and R=RSR is noetherian then the procedure terminates after finitely many steps.
6. Hodges’ Quantisation
6.1. KLEINIAN SINGULARITIES
Let z2Kbe a primitive root of unity of degreen. Set
G¼ gi:g¼ z 0
0 z1
;
a subgroup of SL2ðKÞ. The natural action of G on K2 induces an action on
K½X;Y:gX¼zX;gY¼z1Y. The invariants ofK½X;Yunder this action are generated byXn;XYandYn. Thus, the orbit spaceK2
=Ghas co-ordinate ring
OðK2=GÞ ¼K½Xn;XY;Yn ffiK½A;B;H ðABHnÞ:
The varietyK2=Ghas an isolated singularity at 0, aKleinian singularity of type An1.
6.2. Let vðzÞ 2K½z be a polynomial of degree n, whose roots lie in F. Following [11,1], we define an associative algebra,TðvÞ, overKwith generatorsa;bandh satis-fying the relations
ha¼aðhþ1Þ;hb¼bðh1Þ;ba¼vðhÞ;ab¼vðh1Þ: ð1Þ
There exists a filtration onTðvÞsuch that grðTðvÞÞ ffiK½A;B;H=ðABHnÞ:In other
words,TðvÞis a deformation of a Kleinian singularity of typeAn1.
6.3. AZ-CATEGORY
There is also aZ-grading onTðvÞ: we assignadegreen,bdegreenandhdegree 0. We want to study a category of gradedTðvÞ-modules similar in spirit to the construc-tion ofBw;l-T0-mod in Section 4.8.
Sincepandnare coprime, we can find an inverse ofnninF, saycc. We will consider the full subcategory of finitely generated Z-graded TðvÞ-modules consisting of objects
M¼M
j2Z
Mj;
such that hacts onMj through scalar multiplication byjjcc. Note thataMjMjþn
(respectivelybMjMjn) showing that this definition is compatible with the relation ha¼aðhþ1Þ(respectivelyhb¼bðh1Þ). Denote this category by TðvÞ-grmod.
For i2Z, there is a shift functor ½i:TðvÞ-grmod!TðvÞ-grmod: given M2
TðvÞ-grmod, setðM½iÞj¼Mjpifor all j2Z. This makesTðvÞ-grmod aZ-category.
We let F:TðvÞ-grmod!TðVÞ-mod denote the functor which forgets the Z-structure.
6.4. We will be interested in a finite dimensional central quotient ofTðvÞ.
LEMMA 6.1. The centre of TðvÞis generated by ap;bpand hph. It is isomorphic to the algebra of functions on a type An1 Kleinian singularity.
Proof. It is straightforward to check thatap,bp andhphare central elements.
By constructionTðvÞis a freeK½h-module with basisfai;bj:i;j50g. It follows from the relations in TðvÞthat the degrees of the homogeneous components of any non-zero central elements must be a multiple ofpn. Sinceapandbp are central we must
find which polynomials in h are central. Let qðhÞ be such a polynomial. Since
aqðhÞ ¼qðhþ1Þawe deduce that the roots ofqare invariant under integer addition. It follows thatqðhÞis a polynomial inhph as required.
Using the defining relations once more we have
apbp¼vðhÞvðhþ1Þ. . .vðhþp1Þ:
Since vðhÞhas degree n inh, it follows that apbp¼ ðhphÞn. Hence, the centre of
TðvÞ is a quotient of the ring of functions of a Kleinian singularity of type An1.
Any proper quotient of the ring of functions on a Kleinian singularity has dimension 0 or 1. Thus sinceTðvÞis finitely generated over its centre and has Gelfand–Kirillov
dimension 2, the centre must be the entire ring of functions. &
6.5. Now we can introduce the protagonist of this section:
tðvÞ ¼ TðvÞ
Since the idealðap;bp;hphÞis homogeneous,tðvÞinherits aZ-grading fromTðvÞ. We denote the full subcategory of TðvÞ-grmod consisting of Z-gradedtðvÞ-modules bytðvÞ-grmod.
6.6. In order to studytðvÞwe introduce an intermediate algebra
TðvÞ ¼TðvÞ=ðap;bpÞ: ð2Þ
Let us use a degree lexicographical order on non-commutative associative mono-mials ina of degree 1,bof degree 2n1, andh of degree 1 withh>b>a.
The relations of (1) already form a Gro¨bner–Shirshov basis. It follows that mono-mials not containingha,hb,ba, orabas submonomials form a basis ofTðvÞ.
For any polynomialfðzÞ 2K½zand a positive integeri, we denote
fðiÞðzÞ ¼ Yi1
k¼0
fðzþkkÞ; fðiÞðzÞ ¼ Yi1
k¼0
fðzkkÞ:
LEMMA 6.2. The following relations, together with those in ð1Þ and ð2Þ, form a Gro¨bner-Shirshov basis ofTðvÞ,
apiv
ðiÞðh1Þ for i¼1;. . .;p; ð3Þ
bpivðiÞðhÞ for i¼1;. . .;p1: ð4Þ
Proof. Let us obtain all relations in (6.6) recursively. Fori¼1;. . .;p,
ðbavðhÞ;apiv
ðiÞðh1ÞÞbapihni¼bapivðiÞðh1Þ ðbavðhÞÞapi1hni
¼bapiðvðiÞðh1Þ hniÞ þvðhÞapi1hni?vðhÞapi1ðvðiÞðh1Þ hniÞþ
þvðhÞapi1hni¼vðhÞapi1vðiÞðh1Þ?avðhþ1Þapi2vðiÞðh1Þ?
?api1vðhþpi1Þv
ðiÞðh1Þ ¼api1vði1Þðh1Þ:
Similarly, fori¼1;. . .;p1;ðbpiv
ðiÞðhÞ;abvðh1ÞÞabpihni?bpi1vðiþ1ÞðhÞ. Now
we need to show that all remaining compositions are trivial. The highest terms of defining relations areha,hb,ba,ab,apihni, andbpihni. Let us make certain that all
compositions are trivial. Firstly,
ðbavðhÞ;abvðh1ÞÞbab¼bvðh1Þ vðhÞb?bvðh1Þ bvðh1Þ ¼0:
Similarly,ðabvðh1Þ;bavðhÞÞaba?0:Then
ðhbbhþb;bpivðiÞðhÞÞhbpihni ¼hbpiðhnivðiÞðhÞÞ bhbpi1hniþbpihni?
Analogously, the compositions ðapiv
ðiÞðhÞ;abvðh1ÞÞapb and ðbpivðiÞðhÞ;
abvðh1ÞÞabp are trivial. Then
ðhaaha;abvðh1ÞÞhab
¼hvðh1Þ ahbab?hvðh1Þ aðbhbÞ ab
¼hvðh1Þ abh?0:
Another possible composition to consider is
ðhaaha;apiv
ðiÞðhÞÞhapihni
¼hapiðvðiÞðhÞ hniÞ þahapi1hniþapihni?
?apiððhiÞðv
ðiÞðhÞ hniÞ þ ðhi1ÞhniþhniÞ
¼apiv
ðiÞðhÞðhiÞ?0:
The remaining compositions ðhaaha;apihniÞ
apihnia,ðhbbhþb;apihniÞapihnib,
ðhaaha;bpihniÞ
bpihnia, and ðhb;bpihniÞbpihnib are trivial by a similar
argument. &
COROLLARY 6.7. The dimension of TðvÞ is np2. Moreover, there is a direct sum decomposition
TðvÞ ¼ M
p
i¼1
apiK½h=ðv
ðiÞðh1ÞÞ
" #
M
p1
j¼1
bjK½h=ðvðpjÞðhÞÞ
" #
: ð5Þ
Proof. The direct sum decomposition (5) follows at once from the description of the Gro¨bner–Shirshov basis ofTðvÞ. Adding dimensions of summands, we arrive at the dimension ofTðvÞ, that is 2ðnþ2nþ ðp1ÞnÞ þpn¼np2. &
6.8. If we write gcdðfðZÞ;gðZÞÞfor the greatest common divisor of two polynomials then decomposition (5) is inherited:
tðvÞ ¼M
p
i¼1
apiK½h
ðgcdðvðiÞðh1Þ;hphÞÞ
M
p1
j¼1
bjK½h
ðgcdðvðpjÞðhÞ;hphÞÞ
: ð6Þ
This decomposition allows one to compute the dimension oftðvÞ. Let r1;. . .;rn12Zbe such thatri50 and p5r1þ þrn1 and
vðzÞ ¼Y
n1
i¼0
ðz ðr1þ þriÞÞ:
Setr0¼p ðr1þ þrn1Þ.
Proof. Iffðz;iÞ ¼z ðr1þ þriÞthen the roots offðjÞðz;iÞarer1þ þri;. . .; r1þ þrijj. If j5ri then r1;. . .;ri1;. . .;r1þ þrijjare already roots of fðjÞðz;iþ1Þ. Thus, the dimension of the first summand in (6) is
p2X
n1
i¼0
ð1þ2þ þ ðri1Þ þriþriþ riÞ;
where the number of summands in the parenthesis is p. This sum equals
p2þ ðpPin¼01r2iÞ=2. Similarly, the second summand has dimension p2 ðpþ
Pn1
i¼0 r2iÞ=2, so that the total dimension is 2p2
Pn1
i¼0 r2i. &
6.9. BABY VERMA MODULES (II)
Letu(respectivelyu0) be the subalgebra oftðvÞgenerated byaandh(respectively,b
andh). Similarly letU(respectively,U0) be the subalgebra ofTðvÞgenerated byaand
h (respectivelybandh).
We introduce two sets of baby Verma modules. Forl2ZletKl(respectively,K0l) be the one-dimensional U-module (respectively, U0-module) with basisj0i
(respec-tivelyj1i) and
hj0i ¼llj0i; aj0i ¼0; hj1i ¼llj1i; bj1i ¼0:
The baby Verma moduleVðlÞ(respectively VðlÞ0) is
VðlÞ ¼TðvÞ UKl ðrespectivelyVðlÞ0¼TðvÞ U0K0 lÞ:
LEMMA 6.10. ðiÞ If ll is not a root of vðzÞ then VðlÞ ¼0, whilst if ll is a root of
vðzÞthen VðlÞhas a basis of p elementsj0i;bj0i;. . .;bp1j0i. We have abkj0i ¼vðllkkÞbk1j0i; hbkj0i ¼ ðllkkÞbkj0i:
ðiiÞIfll1is not a root ofvðzÞthen VðlÞ0¼0, whilst ifll1is a root ofvðzÞthen VðlÞ0has a basis of p elementsj1i;aj1i;. . .;ap1j1i. We have
bakj1i ¼vðllþkk1Þak1j1i; hakj1i ¼ ðllþkkÞakj1i:
Proof. (i) For the generatorj0i 2VðlÞone observes that
0¼baj0i ðbavðhÞÞj0i ¼vðhÞj0i ¼vðllÞj0i:
Thus if ll is not a root of vðzÞthenj0i ¼0 andVðlÞ ¼0.
Letllbe a root ofvðzÞ. LetSbe the Gro¨bner–Shirshov basis ofTðvÞconstructed in Lemma 6.6. The module VðlÞ is determined by the pair ðS;fðhllÞj0i; aj0igÞand this turns out to be a Gro¨bner–Shirshov pair. Indeed, there are three elements in
S whose leading monomials end witha:
ðaj0i;bavðhÞÞa ¼baj0i ðbavðhÞÞj0i ¼vðhÞj0i?vðllÞj0i ¼0;
ðap;aj0iÞa¼apj0i ap1aj0i ¼0:
Elements ofSwhose leading monomials end withhfall into two types:
ðapivðiÞðh1Þj0i;ðhllÞj0iÞh ¼ ðllap
ihni1þapiðv
ðiÞðh1Þ hniÞÞj0i?
vðiÞðll1Þapij0i?0;
ðbpiv
ðiÞðhÞ;ðhllÞj0iÞh¼ ðllbpihni1þbpiðvðiÞðhÞ hniÞÞj0i?bpivðiÞðllÞj0i ¼0:
Direct computation now yields the formulas for the action.
(ii) The proof is analogous. &
Thanks to the lemma we can consider VðlÞand VðlÞ0 as objects in TðvÞ-grmod. Indeed if VðlÞ (respectively VðlÞ0) is nonzero we let bkj0i (respectively akj1i) span
theðlkÞn(respectivelyðlþkÞn) homogeneous component.
6.11. Under our assumptions, we have ðllkkÞp¼llkk. It follows that
ðhphÞVðlÞ ¼0 (respectivelyðhphÞVðlÞ0¼0), and soVðlÞandVðlÞ0 are objects in tðvÞ grmod.
PROPOSITION. Let vðzÞ ¼Qin¼01ðz ðr1þ þriÞÞ be as in 6:8 and let r0¼ p ðr1þ þrn1Þ.
ðiÞ The category tðvÞ-grmodhas simple modules L0;. . .;Ln1ðup to isomorphism and shiftÞwhere the dimension of Liis rn1iðif rn1i¼0then Li should be omitted from the list of simple modulesÞ.
ðiiÞ For each i lying between 0 and n1 there exists a uniserial module Vi2tðvÞ-grmodwhose Loewy layers are Li;Liþ1½1;. . .;Li1½1n ðcount sub-scripts modulo n, and omit Vi and Li whenever rn1i¼0Þ.
ðiiiÞ For each i lying between 0 and n1 there exists a uniserial module V0
i 2tðvÞ-grmod whose Loewy layers are Li;Li1½1;. . .;Liþ1½n1 ðcount sub-scripts modulo n, and omit V0
i and Li whenever rn1i¼0Þ.
ðivÞ For any l2Z such that ll is a root of vðzÞ there exists a unique i and j2Z
ðrespectively i0;j0Þsuch that VðlÞ ffiVi½j ðrespectively Vðlþ1Þ0
ffiV0
i0½j0Þ.
Proof. SetV0 ¼Vð0ÞandVi¼Vðr1þ þrn1iÞ½i. Note that ifrn1i¼0 then ViffiViþ1½1. Since every tðvÞ-module has a u-fixed point we see that any simple tðvÞ-module is a quotient of FðViÞ for some i (where F is the forgetful functor to ungraded modules). By Lemma 6.10(i) FðViÞ is isomorphic to K½b=ðbpÞ as a
K½b=ðbpÞ-module. SinceK½b=ðbpÞis a local algebra it follows thatV
i has a simple
In Vi the element brn1ij0i is annihilated by a and belongs to the nðr1þ þrn1ðiþ1ÞÞ þpicomponent, so there is a gradedtðvÞhomomorphism
yi:Viþ1½1 !Vi:
The cokernel ofyi, sayLi, has dimensionrn1iand is simple ifrn1i6¼0 since it is
generated by any basis vectorbij0iit contains. Ifr
n1i6¼0 it follows from Lemma
6.10 that Li has a uniqueu-fixed point, namelyj0i. Hence, if rn1i andrn1j are
non-zeroLi andLjare isomorphic if and only if i¼j. This proves (i).
Since Vi is simple-headed, dimVi¼p and Pin¼01ri¼p the chain of homomor-phisms
Vi1½1n ! yi2
Vi2½2n ! yi3
!yiþ1 Viþ1½1 ! yi
Vi
proves (ii). The proof of (iii) is similar. Part (iv) is clear. &
PROPOSITION 6.12. Let vðzÞ ¼Qin¼01ðz ðr1þ þriÞÞ be as in 6:8 and set r0¼p ðr1þ þrn1Þ. Let k be the number of nonzero ri’s. Then tðvÞ is Morita equivalent to NKðkÞ. Moreover, if k¼n, there is aZ-equivalence of categories
tðvÞ-grmod!NKðnÞ-grmod:
Proof. Let 04i14 4ik4n1 be such that rn1it 6¼0. Let Qit be the
projective cover of Lit intðvÞ-grmod. Recall the general formula, [3, Section 1.7]
dimtðvÞ ¼X
k
t¼1
dimQitdimLit:
LetTitbe the kernel of the sum of two projectionsVitV
0
it !Lit. ThenTit has head
isomorphic to Lit so is a quotient of Qit. By Lemma 6.9 and Proposition 6.11
dimTit ¼2prn1it. Using Lemma 6.8 we find
Xk
t¼1
dimQitdimLit5
Xk
t¼1
dimTitdimLit ¼
Xk
t¼1
ð2prn1itÞrn1it
¼2p2X
k
t¼1
r2n1it ¼dimtðvÞ;
proving that Tit ffiQit.
LetT¼ Tit. The basic algebra oftðvÞis EndtðvÞðFðTÞÞ
op
. Letbt(respectivelyat) be the homomorphism FðTitÞ !FðTitþ1Þ (respectively FðTitþ1Þ !FðTitÞ) associated to
the composition factor FðLitÞ of FðTitþ1Þ (respectively FðTitÞ) lying in the second
Loewy layer of FðVitþ1Þ (respectively FðV
0
itÞ). It is straightforward to check that at
and bt, together with the idempotents arising from the projections in
EndtðvÞðFðTÞÞ, generate the basic algebra and satisfy the relations of the no-cycle
algebra. Since
the first statement of the proposition follows. The second statement is proved in the
same manner as Proposition 4.17. &
7. Proof of Premet’s Conjecture
7.1. We requirep>nfor the following theorem. Thanks to 4.18 we can replaceUw;l with UU~w;l throughout, if we wish.
THEOREM 7.2. Let p>n. Suppose l2C0 with lþr¼Pri$i and let vðzÞ ¼
Qn1
i¼0 ðz ðr1þ þriÞÞ. There is an isomorphism Uw;lffiMatpðn2n2Þ=2ðtðvÞÞ:
Moreover, there is aZ-equivalence of categories
Uw;l-T0-mod!tðvÞ-grmod:
Proof. Setr0¼p ðr1þ þrn1Þand letkbe the number of nonzerori’s. Let
04i14 4ik4n1 be such that rn1it 6¼0. Let L1;. . .;Lk (respectively M1;. . .;Mk) be the simple Uw;l-T0-modules (respectively graded tðvÞ-modules)
appearing in Proposition 4.12 (respectively Proposition 6.11) and let P1;. . .;Pk
(respectivelyQ1;. . .;Qk) be their projective covers. We have
Uw;lffiEndUw;lðFðPtÞ
pðn2n2Þ=2r
n1itÞ ffiMat
pðn2n2Þ=2ðEndUw;lðFðPtÞ rn1itÞÞ
and
tðvÞ ffiEndtðvÞðFðQtÞrn1itÞ:
Thanks to our construction ofPtin Proposition 4.17 andQt in Proposition 6.12 we have a graded isomorphism
EndUw;lðFðPtÞ
rn1itÞ ffiEnd
tðvÞðFðQtÞrn1itÞ;
proving the first statement of the theorem, together with an equivalence. The
equiva-lence is a Z-equivalence by ([10], Theorem 5.4). &
7.3. EXTENSION TO TYPE B
For subregular representations of Lie algebras of type B, Jantzen has proved an analogue of Proposition 4.12 and calculated several extension groups, ([13], Section 3). Then,if we can prove an analogue of Proposition4:13, it follows formally from the arguments of Sections 4, and 6 and the above that the central reduction of a block of a subregular reduced enveloping algebra of type Bis a matrix ring over a central reduction of Hodges’ deformation of a Kleinian singularity of type A.
Unfortunately, the proof of Proposition 4.13 does not immediately generalise to type B since it is no longer true that we can find all highest weights in the funda-mental alcove, C0. For regular weights, however, this can be remedied as follows.
associated to a regular weight l. Rickard’s results in ([23], Section 3) remain valid in this situation. To check this requires the specific information on the wall-cross-ing functors, denoted gi for 14i4n, provided by Jantzen in ([14], Section H). Thus, for 14i4n, there are derived self-equivalences Fi on the bounded derived categories of Bw;l-modules. In particular, given any Bw;l-module M, FiðMÞ is the
complex giðMÞ !M, the map being given by the counit of gi. Combining the
known behaviour of baby Verma modules under wall-crossing, [12, 11.20], with existing results on filtrations of the projective indecomposable modules by baby Verma modules, ([12], Proposition 10.11), it can be shown that, given two projective indecomposable Bw;l-modules Q and Q0, there are integers 14i1;. . .;it4n such
that Fi1;. . .;FitðQÞ is quasi-isomorphic to Q
0. Since it is known that the centre
of Bw;l is a derived-invariant, the proof of Proposition 4.13 can be generalised, using the derived category, to deal with Bw;l. As a result we find the central reduc-tion of Bw;l is a matrix ring over the central reduction of Hodges’ deformation of a Kleinian singularity of type A2n1.
8. More on Baby Verma Modules
8.1. We want to study baby Verma modules in ‘general position’. To do so, we need a general lemma.
LEMMA 8.1. Let A be a finite dimensional L-algebra and Y a connected algebraic variety over L. Suppose Ma, a2Y, is a flat family of finite-dimensional A-modules
over Y. Then the Grothendieck group element ½Ma 2K0ðAÞis independent ofa. Proof. LetBbe the basic algebra ofA. There exists aðB;AÞ-bimoduleN, flat over
A, such that the functor NAinduces an equivalence between the categories of
finite-dimensionalA-modules and finite-dimensionalB-modules. LetKa ¼NAMa, a flat family ofB-modules overY. Given a primitive idempotente2B, it suffices to show that the dimension of eKa is independent of a2U, a Zariski open neigh-bourhood of a point. Without loss of generality we can trivialise the family locally on
U, givingKU. Thenedefines an algebraic family of projection operators on the finite dimensional vector spaceK,
ea:k7!prKðe ðk;aÞÞ:
Since the dimension ofeKais equal to the rank ofea, and the latter is constant since
Yis connected. &
8.2. Givenl2X, it is an interesting problem to describe the isomorphism classes of all baby Verma modulesVðb;lÞasbruns overBw. Iflis regular then the description of Bw, Proposition 4.12 and Lemma 8.1 show that, for everyb2Bw,
½Vðb;lÞ ¼X
n1 i¼0
Therefore, on passing to the no-cycle algebra, we see that suchVðb;lÞcorresponds to a NKðnÞ-module MðbÞ, such thateiMðbÞ is one-dimensional for all i. Thanks to Section 2, all modules of this dimension are known.
Letðk;aÞ 2 ðf1;. . .;n1g KgÞ [ ð0;0Þand letbk;a be the stabiliser ofFk;a. Sup-pose first thata¼0. Since the torusT0stabilisesbk;0for anyk, twists by elements of T0 provide a grading ofVðbk;0;lÞ. ThereforeMðbk;aÞis gradable and, by Section 3, must be a direct sum of string modules ifnis odd (note that some band modules are gradable for even n). Whenlis regular, we expect that gradable band modules are not baby Verma modules, and that baby Verma modules are indecomposable.
The generic case is dealt with in the following proposition.
PROPOSITION 8.3. Keep the above notation and let l2X be a regular weight,
14k4n1anda6¼0. Then the baby Verma module Vðbk;a;lÞis indecomposable.
Proof. It follows from Section 3 that ifMðbk;aÞis not gradable it is necessarily a band module and, hence, indecomposable. Thus, by Proposition 4.17, it suffices to show thatVðbk;a;lÞdoes not admit aT0-grading.
Suppose for a contradiction that Vðbk;a;lÞ admits a T0-grading. Let Lk be the
sl2ðKÞ-subalgebra generated by Enk;n;Enk;nkEn;n and En;nk. Any Borel
sub-algebra belonging to Pk is uniquely determined by its intersection with Lk. Let
lk¼lðEnk;nkEn;nÞ. A straightforward calculation shows that the restriction of Vðbk;a;lÞ to Lk has a direct summand isomorphic to the baby Verma module for Lk induced frombk;a\Lkwith highest weightlk.
Fort2T0 the elementtð11Þ 2Vðbk;a;lÞis a highest weight vector for the Borel subalgebratbk;a, yielding an isomorphismVðbk;a;lÞ ffiVðtbk;a;lÞ. Sincelis regular lk6¼ 1, and so, by ([20], Main Theorem), the baby Verma modules forLkinduced from
different Borel subalgebras ofLkwith highest weightlkare not isomorphic. Hence, by
the last paragraph, the isomorphismVðbk;a;lÞ ffiVðtbk;a;lÞforcesVðbk;a;lÞto have
infinitely many nonisomorphic direct summands, a contradiction. &
Acknowledgements
Both authors were visiting and partially supported by MSRI while this research was begun and extend their thanks to that institution. Much of the research for this paper was undertaken while the first author was supported by TMR grant ERB FMRX-CT97-0100 at the University of Bielefeld. The second author was partially supported by EPSRC grants GR/M68886 and GR/M75037.
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