Weight Modules over Generalized Weyl Algebras*

ARTICLE NO. 0270

Weight Modules over Generalized Weyl Algebras*

Yuri A. Drozd,^†Boris L. Guzner, and Sergei A. Ovsienko

Faculty of Mechanics and Mathematics, Kie_¨Taras She_¨chenko Uni_¨ersity, Vladimirskaya 64, 252 017 Kie¨, Ukraine

Communicated by Kent R. Fuller Received April 5, 1995

This work originated in an attempt to comprehend a striking likeness between representations and cohomology theories of some algebras, such

Ž . Ž .

as sl 2, C and its non-degenerated quantizations, modular sl 2 and

Ž .

degenerated quantizations of sl 2, C , Weyl algebra A₁, and others. As a result, the notion of generalized Weyl algebras Žof which all those men- tioned above turned out to be examples and weight modules have arisen.. Fortunately, almost all the techniques necessary to elaborate the theory w are present in the literature. The most important tools can be found in 1,

x

3 . So we were able to give a nearly complete description of weight modules over generalized Weyl algebras via reduction to a class of linear

Ž .

categories called chain and circle categories resembling those arising in w x1 . We could emphasize that, though categories seem to be more compli- cated than algebras, as one needs to deal with many objects, this multitude itself provides a certain convenience, and sometimes a decisive one, at least for calculation, but often for comprehension as well.

In Sections 1 and 2 the subjects of the paper}generalized Weyl algebras, chain and circle categories, and weight modules}are introduced and the main correlations established between them. In Sections 3 and 4, a description of weight modules over chain and circle categories is given, while in Section 5 it is translated back to the generalized Weyl algebras.

*This work was partially supported by the International Science Foundation under Grant RKJ000, and the Foundation for Fundamental Research of Ukraine under Grant 1 1 3Ž . r50.

†E-mail: Drozd@uni-alg.kiev.ua.

491

0021-8693r96 $18.00

CopyrightQ1996 by Academic Press, Inc.

All rights of reproduction in any form reserved.

1. GW-ALGEBRAS AND WEIGHT MODULES

1.1. Let R be a ring. A category CC is said to be an R-category if its

Ž .

morphism sets CC i, j for the objects i, j are equipped with R-bimodule structure, the multiplication of its morphisms is R-linear with respect to

Ž . Ž .

both left and right R-module structure, and ar bsa rb for any possible a, bgMorCC, rgR. Remark that, even if R is commutative, we do not suppose that arsrafor agMorCC, rgRŽin the latter case C is said to

. Ž .

be R-linear . If CC contains only one object i.e., is simply a ring , we call it an R-ring.

1.2. From now on suppose the ring R is commutative. Let CC be an R-category and M be a CC-module, i.e., an additive functor from CC to the categoryAb of Abelian groups. Then, for each igObCC, the group M iŽ .

Ž .

becomes an R-module if we put r_¨s r1_{i ¨}. We write, as usual, a_¨

Ž . Ž .

instead of M a_¨ for _¨gM i , a: iªj in CC, and call elements of

"

_igObCC M i the elements of the module M.Ž .

1.3. Denote by Max R the set of all maximal ideals m;R. For

< 4

mgMax Rand an R-module M, put M_ms ¨gM m_¨s0 . A CC-mod- ule M is said to be a weight module if MsÝ_mgMaxR M_m. Denote by

Ž . < 4

W

W CC the category of all weight CC-modules. Put SuppMs m M_m/0 Ž .

and, for each subset v:Max R, denote by W_v CC the full subcategory of Ž .

W

W CC consisting of all modules M with Supp M:v.

1.4. Suppose we are given an automorphism s of the ring R and an

Ž .

element tgR. Then define the generalized Weyl algebra GW-algebra

Ž .

AsA R, s, t as the R-ring generated over R by two elements X, Y subject to the following relations:

v XrssŽ .r X and rYsYsŽ .r for any rgR;

v YXst and XYssŽ .t .

One can easily check that both as a left and as a right R-moduleAis free

^{n n} < 4

with a basis 1, X ,Y ngN.

Ž .

Remark. In most cases cf. below , R is an algebra over some field K ands is an algebra automorphism; thusAis really a K-algebra. Neverthe- less, we retain the term ‘‘GW-algebra’’ even if it contains no ground field at all.

² :

1.5. The cyclic group s generated by s acts on the set Max R.

Denote by Vthe corresponding orbit set. It follows from the definition of

Ž . y1

A R, s, t that XM_m:M_sŽm. and YM_m:M_{s Žm.} for any A-module M and mgMax R. Hence, the following statement is obvious.

Ž . Ž .

PROPOSITION. WW A s@_vgVWW_v A, i.e., any weight A-module M de- composes into a direct sum: Ms

[

_vgV^{M with}_v ^{Supp M}_v:v.

Ž . w x

1.6. EXAMPLES. i Let RsK T , the polynomial ring over some

Ž . 4

field K, tsT, and s T sqTq1 for some qgK_ 0 . Then A is a

Ž .

quantization of the usual Weyl algebra A₁overK if qs1, thenAsA₁ . Ž .ii Let RsK H,w Tx, the polynomial ring in two variables, tsT,

Ž . Ž . Ž .

s H sHy1, and s T sTqH. Then A,Ug, the universal en- veloping algebra of the 3-dimensional simple split Lie algebra g over K ŽslŽ2, K., if char K/2 ..

Žiii. Now let RsK T,w Z, Z^y1x Žpolynomials in T and Laurent

. Ž . ^y1 Ž . Ž ^y1.^y1Ž

polynomials in Z ,tsT,s Z sq Z, and s T sTq qyq Z

y1

. 4

yZ for some qgK_ 0, "1 . Then we obtain as A a quantization U_qŽ .g of the algebra of the preceding example.

2. CHAIN AND CIRCLE CATEGORIES

2.1. Let L be a local commutative ring, m its only maximal ideal, and

< 4

KsLrm its residue field. Suppose we are given a family t_i igZ of

Ž .

elements of L. Then we define the chain category CC L, t_i as the

Ž .

L-category with the object set Z, generated over L by the set of

< 4

morphisms X_i,Y_i igZ, where X_i: iªiq1 and Y_i: iq1ªi, subject to the relations:

v X r_i srX_i and Y r_i srY_i for each rgL and each igZ;

v Y X_{i i}st_{i i}1 and X Y_{i i}st_{i i}1_q1 for each igZ.

Ž .

2.2. PROPOSITION. Let AsA R, s, t be a GW-algebra, vgV an

Ž . Ž . Ž .

infinite orbit cf. 1.5 , and mgv. Then WW_vA ,WW CC for the chain Ž ^yiŽ ..

category CCsCC R_m, s t .

Ž . Ž . i ⁱ

Proof. Given a weight module MgWW A, put N i sMs Žm.. As s

; i Ž .

induces an isomorphism R_mªR_sŽm., we can consider N i as an R_m-

Ž . Ž

module. For _¨gN i, put Xi_¨sX_¨ and Yiy1_¨sY_¨. Then Xi_¨gN iq

. Ž . Ž .

1 and Yiy1_¨gN iy1 cf. 1.5 . Moreover, if rgRm, then X ri _¨s Ž . ^iq1Ž .

Xs r _¨ss r X_¨srXi_¨. Analogously, Y ri _¨srYi_¨. Further, Y Xi i_¨s

yiŽ . ^yiŽ .

YX_¨ss t _¨ and X Y_{i i¨}ss t _¨. Hence, N is a CC-module and we

Ž . Ž .

obtain a functor F:WW_v A ªWW CC .

Ž . Ž .

Conversely, if NgWW CC , put Ms

[

i^{N i and supply M with the} yiŽ .

A-module structure putting r_¨ss r _¨, X_¨sX_i¨, and Y_¨sY_iy1¨ for

Ž . Ž . Ž .

¨gN i and rgR. It gives us a functor F9: WW CC ªWW_v A obviously inverse to F.

2.3. Now suppose we are given a natural pgNŽnot necessarily prime,

. < 4

perhaps even ps1 , an automorphismt of L, and a family t_i igZ_p of

Ž .

elements of L. Then we define the circle category CC_p L, t_i, t as the L-category with the object set Z_psZrpZ, generated by the set of mor-

< 4

phisms X_i,Y_i igZ_p, where X_i: iªiq1 and Y_i: iq1ªi, subject to the relations:

v X r_i srX_i, Y r_i srY_i, and X Y_{i i}st_{i i}1_q1 for each rgL and each igZ_p except for py1;

v X_py1r stŽ .r X_py1, rY_py1 sY_py1tŽ .r for each r gL and

Ž .

X_py1Y_py1st t_py1 1 ;₀

v Y X_{i i}st_{i i}1 for each igZ_p.

Ž .

2.4. PROPOSITION. Let AsA R, s, t be a GW-algebra, vgV an

Ž . Ž .

orbit with p elements, and mgv. Then WW_v A ,WW CC for the circle Ž ^yiŽ . ^p.

category CCsCC_p R_m, s t , s .

The proof is analogous to that of 2.2 and hence is omitted.

3. WEIGHT MODULES OVER CHAIN CATEGORIES 3.1. To describe weight modules, we need some simple and known results. Recall that a full subcategory SS:CC is called a skeleton of CC provided the objects of SS are pairwise non-isomorphic and any object of C

C is isomorphic to some object of SS. It is evident that in this case the categories of CC- and of SS-modules are equivalent.

Ž w x. Ž . Ž .

3.2. LEMMA cf., e.g., 2 . i Let M be a simple CC-module and M i /

Ž . Ž .

0 for some igObCC.Then M i is a simple CC i, i-module.

Ž .ii Con_¨ersely, for any simple CCŽi, i.-module N there exists a unique Žup to isomorphism. simple CC-module M such that M iŽ .,N as CCŽi, i.-modules.

Ž .

3.3. From now on in this section CC denotes a chain category CC L, t_i Žcf. 2.1 ..

Ž .

Call igZa break for CC if t_igm. Denote by B the set of all breaks.

As B:Z, it inherits the natural order. The following statement is evident.

PROPOSITION. A skeleton SS of the chain category CC can be chosen as follows:

1. If BsB, thenOb SSs 40 .

2. If B/B and has no maximal element, thenObSSsB.

3. If B/B and m is its maximal element, thenOb SSsBjmq 1 .4

3.4. Surely, the objects of SS can be numbered by integers belonging to

Ž .

some interval I:Z perhaps left or right or two-sided unbounded . Denote by jthe object of SS corresponding to jgI.

Ž .

It follows from the definition 2.1 that mCC is a two-sided ideal in CC.

Hence, we can form the factor SSsSSrmSS and, of course, weight SS- modules are just SS-modules. But the same definition implies the following description of SS.

Ž .

3.5. PROPOSITION. If BsB, then SS 0, 0 sK for the only object 0 in SS.Otherwise, SS is the K-category generated by the set of morphisms

<

X ,Y JgI,non-maximal , where X: jªjq1,Y: jq1ªj,

½

j j

5

j j

commuting with the elements of K and subject to the relations X Y_{j j}s0 and Y X_{j j}s0.

Ž .

In particular, SS j, j sK for any jgI.

3.6. COROLLARY. Isomorphism classes of simple weight CC-modules are in 1]1 correspondence with the elements of I. Namely, for each jgI, the corresponding simple CC-module M is defined as follows:_j

Ž . Ž

1. If j is a break,then M ij sK for jy1-iFj we put jy1s y`

. Ž . Ž .

if jy1fI, i.e., j is the minimal break, M i_j s0 otherwise; M Y_{j i} s1

Ž . Ž . Ž

and M X_{j i} st_i the multiplication by t in K_i for all jy1-i-j other- wise these mappings are automatically zeros..

2. If jsmq1 for the maximal break m, then M i_jŽ .sK for i)m,

Ž . Ž . Ž .

M i_j s0 otherwise; M Y_{j i} s1and M X_{j i} st for all i_i )m.

Ž . Ž . Ž .

3. Last, if BsB,then M i₀ sK, M Y_{0 i} s1, and M X_{0 i} st for_i all igZ.

Here, as before, KsLrm.

3.7. In the case considered one can also determine all SS-modules, hence, all weight CC-modules. Namely, let J:I be a non-empty subinter- val of I and J^X:J be any subset not containing the maximal element of J Žif the latter exists . Then define an. SS-module NsN J,Ž JX.by the rules:

v N jŽ .sK if jgJ and N jŽ .s0 otherwise;

v N XŽ _j.s1 if jgJX and N XŽ _j.s0 otherwise;

v N YŽ _j.s1 if jfJX and N YŽ _j.s0 otherwise.

Now an easy exercise in linear algebra leads to the following result.

Ž X.

3.8. THEOREM. All SS-modules N J, J are indecomposable and any SS-module decomposes uniquely into a direct sum of modules isomorphic to

Ž ^X. some of the N J, J .

3.9. COROLLARY. For each pair J^X:J as in 3.7, define a CC-module

Ž X. Ž

MsM J, J as follows. Denote by l the maximal break preceding J or ls y` if such breaks do not exist. and by m the maximal element of J pro<sub>¨</sub>ided it is a break or ms` otherwise. Put:

v M iŽ .sK for l-iFm and M iŽ .s0 otherwise;

v if l-i-m and i is not a break, then M YŽ _i.s1 and M XŽ _i.st_i;

v if igJ and is a break, then

M XŽ i.s1 if igJ and M XX Ž i.s0 otherwise;

X 4

M YŽ i.s1 if ifJ j m and M YŽ i.s0otherwise.

ŽIn other cases M XŽ _i.s0 and M YŽ _i.s0 automatically..

Ž ^X.

Then all M J, J are indecomposable weight CC-modules and any weight CC-module decomposes uniquely into a direct sum of modules isomorphic to

Ž ^X. some of the M J, J .

4. WEIGHT MODULES OVER CIRCLE CATEGORIES

Ž . Ž .

4.1. In this section CC denotes a circle category CC_p L, t_i, t cf. 2.3 .

Ž .

Again call igZ_p a break for CC if t_igmand denote by B the set of all breaks.

PROPOSITION. A skeleton SS of the circle category CC can be chosen as follows:

1. If BsB,then ObSSs 40 . 2. If B/B,then ObSSsB.

4.2. Of course, there is no natural order on Z . Instead we can define a_p

Ž .

‘‘circular order’’: i-j-k means that 0-jyi-kyi-p in Z . If SS contains m objects, put them in 1]1 correspondence with Z_m in such way that i-j-k in Z_m implies i-j-k in Z_p Žsurely, it means nothing if ms1 . Here, again,. j denotes the element of ObSS corresponding to jgZ_m.

4.3. The definition 2.3 again implies that mCC is an ideal inCC, so we can form the factor SSsSSrmSS and give the following description.

y1

Ž . Ž . w x

PROPOSITION. i If BsB, then SS 0, 0 ,PsK x, x , t , the skew Laurent polynomial ring o_¨er K with the automorphismt induced byt.

Ž .ii If B/B, then SS is the K-category generated by the set of

< 4

morphisms X_j: jªjq1 and Y_j: jq1ªj jgZ_m subject to the rela- tions:

v X and Y commute with the elements of K if j_{j j} /my1;

v X_my1astŽ .a and aY_my1sY_my1tŽ .a for all agK;

v X Y_{j j}s0 and Y X_{j j}s0 for all jgZ_m. In particular,

² <

S

S

Ž

j, j

.

,DsK x, y xysyxs0, xastŽ .a x and aysytŽ .a for all agK:.

4.4. PROPOSITION. Ž .i All simple P-modules are of the form PrfP for irreducible elements fgP.

Ž .ii All simpleD-modules are the following:

v N₀sDrŽx, y.;

v N_1,fsDrŽ Žf x., y., where f/x is an irreducible element of the w x

skew polynomial ring P₀sK x,t ;

Ž Ž ..

v N_2,fsDr x, f y , where f/y is an irreducible element of the

T w y1x

skew polynomial ring P₀sK y,t .

Ž . w x

Proof. The assertion i is well known, asP is a principal ideal ring 4 . To prove ii , we need to remark that bothŽ . xN and yN are submodules in anyD-module N. Hence, if N is simple, each of them coincides either with N or with 0. But xNsN implies yNs0 and vice versa. Thus, either

Ž .

xNsyNs0 i.e., N,N₀ , or xNsN, yNs0, or xNs0, yNsN.

Obviously, DryD,P₀, DrxD,P^T₀, and both of them are principal ideal rings, which implies the assertion ii .Ž .

4.5. Remark that the ringsP₀ andP^T₀ are anti-isomorphic; hence, there is a natural 1]1 correspondence between their irreducible elements.

Namely, to a polynomial fsa₁qa x₂ q???qa x_d ^dy1qx^d of P₀ there corresponds a polynomial f^Tsa₁qya₂q???qy^dy1a_dqy^dgP^T₀. Using Lemma 3.2, we are able to reconstruct all simple weight CC-modules.

If BsB, take an irreducible element fsa₁qa x₂ q???qa x_d ^dy1qx^d Ža₁/0 in. P. The corresponding simple module can be defined as follows:

v M i_fŽ .sKd for all igZ_p;

v M Y_fŽ _i.s1 and M X_fŽ _i.st_ifor all i/py1;

v M X_fŽ _py1._¨stŽF t_{f py1¨}., where

0 0 0 ??? 0 ya₁ 1 0 0 ??? 0 ya₂

0 1 0 ??? 0 ya

Ffs ³

. . . .

. . . .

. . . .

^{0 0 0 ??? 1 yad}

0

y1y1

v M Y_fŽ _py1._¨sF_f t Ž .¨ .

< <

If B sm)0, define first the simple modules M_j for jgZ_m:

v M i_jŽ .sK if jy1-iFjand M i_jŽ .s0 otherwise;

v M Y_jŽ _i.s1 and M X_jŽ _i.st_iif jy1-i-jand i/py1;

y1

v M Y_jŽ _py1.st and M X_jŽ _py1.stt_py1if jy1-py1-j.

Now let fsa₁qa x₂ q???qa x_d ^dy1qx^d/x be an irreducible ele- ment ofP₀. Define the modules M_1,f and M_2,f in the following way:

v M_1,fŽ .i sM_2,fŽ .i sKd for all i;

v M_1,fŽY_i.s0 and M_2,fŽX_i.s0 for all i;

v M_1,fŽX_i.st_iand M_2,fŽY_i.s1 for i/py1;

v M_1,fŽX_py1. is the semi-linear mapping with automorphism t de-

Ž .

fined by the matrix F_f cf. above ;

y1

v M2,fŽYpy1. is the semi-linear mapping with automorphism t defined by the matrix F^y_f ¹.

4.6. Recall that two elements f, g of a ring L are said to be similar if LrfL,LrgL.

COROLLARY. Ž .i If BsB,then, for each irreducible polynomial fgP, the module M is a simple weight_f CC-module,each simple weight CC-module is isomorphic to one of them, and M_f,M if and only if f and g are similar_g in P.

Ž .ii If B/B, then all modules M_j, M_1,f, and M_2,f, for irreducible

Ž .

fgP₀ f/x , are simple weight CC-modules, each simple CC-module is isomorphic to one of them,and the only isomorphisms between these modules are M_1,f,M_1,g and M_2,f,M_2,g if f and g are similar inP₀.

4.7. To classify non-simple CC-modules, we need to suppose them to be

Ž Ž .

finite-dimensional i.e., all M i to be finite-dimensional vector spaces over

. Ž .

K : otherwise we must at least classify all linear or semi-linear mappings in infinite-dimensional vector spaces}the problem, which seems hopeless.

The case BsBis rather simple: we must determine finite-dimensional P-modules. But P is a principal ideal domain; thus the answer is well

w x

known 4 : the only indecomposable modules are N_fsPrfP with f indecomposable in P Žit means, by definition, that PrfP is indecompos- able and. N_f,N_g if and only if f and g are similar inP.

4.8. If B/B, the morphisms X_j,Y_j satisfy the relations X Y_{j j}s0 and Y X_{j j}s0. Thus the problem almost coincides with that of classification of

w x

pairs of mutually annihilating linear mappings solved in 3 . We need only w x

take into account the graduation, different from that in 3 , and the fact that our mappings are not linear but semi-linear. However, we encounter no significant trouble and, accurately following the quoted work, we arrive at a description of finite-dimensional SS-modules.

< <

Denote by D the free monoid generated by two letters x, y. Let w be the length of the word wgD. For any such word wsz z_{1 2} ??? z_n, where

4 Ž

z_kg x, y, and any jgZ_m, define the SS-module NsN_j,w module of the w x.

first kindin the terminology of 3 as follows.

Consider nq1 symbols e₁,e₂, . . . ,e_n. As a K-basis of N lŽ .take the set

Ž .

of all e_k with jqksl in Z_m . If there are no such indices, then N lŽ .s0. Define the action of X_l andY_l on this basis by the rules

e_kq1 if z_kq1sx X el ks

½

0 otherwise;

e_ky1 if z_ksy Yly1eks

½

0 otherwise.

In particular, X_{jqn n}e s0 and Y_{jy1 0}e s0. For instance, if ws« Žthe

< < .

empty word, « s0 , then N_j,« is the simple SS-module corresponding to j.

Of course, defining X_my1 and Y_my1 on other elements, we must take into account the fact that they are not linear but semi-linear mappings

y1 Ž

with automorphismst andt , respectively the next paragraph, as well as 5.5 and 5.6, is concerned with the same remark ..

4.9. Now call a word w an m-word if its length n is a multiple of m w x and non-periodic if it is not a power of another m-word. LetP0sK x,t be the skew polynomial ring with automorphism t and fsa₁qa x₂ q???qa x_d ^dy1qx^d/x^d be an indecomposable polynomial in P₀. Put

T ry1Ž . Ž .

a_r st a_r rs1, . . . ,d . For any non-periodic m-word w and any

Ž w x.

jgZ_m define an SS-module NsN_w,f of the second kind 3 as follows.

Ž .

Considerdnelements e_{k s} ks1, . . . ,n,ss1, . . . ,d. Take as a basis of

Ž . Ž .

N l the set of all e_{k s} with k'l modm. Define the action of X_l and

Y_ly1 on this basis by the rules

e if k/nand z sx

¡

^{kq1 ,s kq1}

e_{1 ,sq1} if ksn, z₁sx, and s/d

~

X el k ss

¢

y₀ Ýd_rs1a e_{r 1r} if_otherwise;ksn, z₁sx, and ssd

e if k/1 and z sy

¡

^{ky1 ,s k}

e_n,sq1 if ks1, z₁sy, ands/d

~

Yly1ek ss

¢

y₀ Ýd_rs1a eT_{r n r} if_otherwise.ks1, z₁sy, andssd

4.10. THEOREM. Ž .i If B/B,then the modules N_j,w and N_w,f built in 4.8 and 4.9 are indecomposable SS-modules and any indecomposable finite- dimensional SS-module is isomorphic to one of them.

Ž .ii The only isomorphisms between these modules are N_w,f,N_wŽk.,g,

Ž . Ž .

where k'0 mod m , f and g are similar in P₀, and w k sz_kq1 ??? z z_{n 1}

??? z_k, i.e., is a cyclic permutation of w.

4.11. We can reconstruct weight CC-modules M_f, M_j,w, and M_w,f corre- sponding to the SS-modules N_f, N_j,w, and N_w,f, respectively. Namely, M_f is defined as in 4.5. If NsN_j,w orN_w,f, as defined in 4.8 and 4.9, then the corresponding M is defined by the rules

v M iŽ .sN lŽ . ifly1-iFl;

v X_l andY_l acts as X_l andY_l;

v if i is not a break and i/py1, then M XŽ _i.st_iand M YŽ _i.s1;

v last, if py1 is not a break, then M XŽ _py1.stt_py1 and M YŽ _py1. sty1.

Ž .

In particular, the simple modules M_j, M_1,f, and M_2,f cf. 4.5 coincide, respectively, with M_j,«, M_x^m_,f, and M_y^m_,f.

4.12. COROLLARY. Ž .i If BsB,any indecomposable weight CC-module is isomorphic to M for some indecomposable element f_f gP and M_f,M if_g and only if f and g are similar inP.

Ž .ii If < <B sm)0, any indecomposable weight CC-module is isomor- phic either to M_j,w for some jgZ_m and wgD or to M_w,f for some non-periodic m-word wgD and an indecomposable element fgP₀, f/x^d. The only isomorphisms between these modules are M_w,f,M_wŽk.,g with f and

Ž .

g similar inP₀ and k'0 mod m .

4.13. Remark. Corollaries 4.6 and 4.12 ‘‘almost’’ solve the problem of describing weight CC-modules. We still have the problem of classifying indecomposable elements inPandP₀up to similarity, which may be highly non-trivial. However, even the ‘‘usual’’ problem of describing irreducible polynomials over an arbitrary field is far from being solved, though no one doubts that the Frobenius normal form is a ‘‘good classification’’ of linear mappings in finite-dimensional vector spaces.

5. BACK TO GW-ALGEBRAS

5.1. Now return to the definitions and notations of 1.4 and 1.5. So A

Ž .

denotes a GW-algebra A R, s, t . We are able to restore the weight A-modules according to Propositions 2.2 and 2.4 and the description of the weight modules over chain and circle categories.

For mgMax R, put K_msKrmandt_mstqmgK_m. Call m a break ŽforA.if t_ms0. Let B be the set of all breaks. If vgVsMax Rr²s:, denote B_vsBlv.

Ž . For each finite orbit v of p elements, fix a maximal ideal m v gv

Ž . w

supposing that it is a break provided B_v/B. Put K_vsK_mŽv.,P_vsK_v x,

y1 x w x

x ,t_v andP_0vsK_v x,t_v , where t_v denotes the automorphism of K_v

p Ž ^pŽ . .

induced bys note that s m sm for mgv .

² :

The maximal ideals m;R belonging to finite s -orbits will be called

pŽ .

periodic. Really, it means that s m sm for some p)0.

5.2. Now construct the full set of representatives of isomorphism classes of indecomposable weight CC-modules M with Supp M:v.

< < Ž .

Suppose first that v s`and B_vsB. Let V v s

[

_mgv^Km^{. Supply}

Ž . Ž . ^y1Ž .

V v with A-module structure putting X_¨ss t_m¨ and Y_¨ss ¨ for

¨gK_m.

< <

5.3. Suppose v sp-`and B_vsB. For any indecomposable polyno-

dy1 d Ž . Ž .

mial fsa1qa x2 q???qa xd qx gPv with a1/0 , letV v, f s

[

_mgv^K_m^d. Supply it withA-module structure putting, for _¨gK_m^d:

v X_¨ssŽt_m¨.if m/mŽv.;

v X_¨ssŽF t_{f m¨}.if msmŽv.;

v Y_¨ssy1Ž .¨ if sy1Žm./Žm.Žv.;

v Y_¨sFy_f 1sy1Ž .¨ if sy1Žm.smŽv., where the matrix F_f was defined in 4.5.

< <

5.4. Suppose that v s` and B_v/B. Consider the natural order on

Ž . ^X Ž .4

v: m-s m. If B possesses a maximal element m, put B_{v v}sBj s m ,

X X Ž ^{X Y}

otherwise B_vsB_v. Let J:B_v be an interval i.e., m-m-m in B_v

X Y . ^X

and m , m gJ implies mgJ and J :J be any subset not containing

the maximal element of J if the latter exists. Denote by m the maximal₀ break in v preceding all elements of J or y`if it does not exist; by m<sub>1</sub> the maximal element of J if it exists and is a break orq`otherwise. For each mgv, putV_msK_m if m₀-mFm₁andV_ms0 otherwise. Supply

Ž ^X.

V v, J, J s

[

_mgv ^V_m ^withA-module structure putting, for _¨gV_m,

¡

^sŽ^t_m^¨. if m is not a break

~

X

X_¨s

¢

s₀Ž .¨ if m_otherwise;gJ

y1 X

4

0 if s Žm.gJ j m1

Y_¨s

½

s_y1Ž .¨ otherwise.

< < < <

5.5. Suppose v sp-`and B_v sm)0. Consider the natural ‘‘cir-

X X iŽ . ^{Y k}Ž .

cular order’’ on v: m-m -m0 means that m ss m and m ss m for some 0-i-k-p. Define the 1]1 correspondence Z_mªB_v, jªm_j in such a way that i-j-k in Z_m implies m_i-m_j-m_k in v and

Ž . Ž . Ž .

m₀sm v cf. 5.1 . Denote by jm the only jgZ_m such that m_jy1-m Fm_j.

Ž .

For each jgZ_m and each word wsz z_{1 2} ??? z_ngD cf. 4.8 , consider nq1 symbols e₁,e₂, . . . ,e_n. If mgv, denote by V_m the vector space over

Ž . Ž . Ž

K_m with a basis consisting of all pairs m, e_k such that jqksjm in

. Ž .

Z_m . PutV v, j,w s

[

_mgv ^V_m and supply it withA-module structure by the rules

¡

^tm

Ž

^sŽ^m.^,ek

.

if m is not a break

~

XŽm,e^k.s

¢

₀

Ž

_sŽm.,ekq1

.

if m_otherwise;gBand zkq1sx

¡ Ž

^sy1Ž^m.^,ek

.

if m is not a break

~

^y1

YŽm,ek.s

¢

₀

Ž

_s Ž_m._,eky1

.

_{if motherwisegBand z}k_sy

Žtaking into account, of course, that X andY are semi-linear ..

Ž .

5.6. Now let wgD be a non-periodic m-word cf. 4.9 and fsa1q a x₂ q???qa x_d ^dy1qx^d/x^d be an indecomposable polynomial fromP_0v.

Ž .

Consider dnsymbols e_{k s} ks1, . . . ,n, ss1, . . . ,d. If mgv, denote by

Ž .

V_m the vector space over K_m with a basis consisting of all pairs m, e_{k s}

Ž . Ž .

such that k'j mod m . Put V v,w, f s

[

_mgv ^V_m and supply it with

A-module structure by the rules

¡

^tm

Ž

^sŽ^m.^,ek s

.

if m is not a break

sŽm.,e if mgB,k/nand z sx

Ž

kq1 ,s

.

kq1

~

XŽm,ek s.s

Ž

_sŽm.,e1 ,sq1

.

if mgB,ksn, z1sx, s/d

dy1

yÝrs1 ar

Ž

sŽm.,e1r

.

if mgB,ksn, z1sx, ssd

¢

_{0 otherwise;}

YŽm,ek s.

¡ Ž

^sy1Ž^m.^,ek s

.

^ifsy1mis not a break

y1 y1

s Žm.,e ifs Žm.gB,k/1 and z sy

Ž

ky1 ,s

.

k

~

^{y1 y1}

s

Ž

s Žm.,en,sq1

.

ifs Žm.gB,ks1,z1sy,s/d

dy1 T y1 y1

yÝrs1 ar

Ž

s Žm.,en r

.

ifs Žm.gB,ks1,z1sy,ssd

¢

_{0 otherwise}

Žcf. 4.9 for the definition of a^T_r..

Ž . Ž . Ž . Ž ^X. Ž

5.7. THEOREM. i TheA-modules V v ,V v, f ,V v, J, J ,V v, j,

. Ž .

w , and V v, w, f defined, respecti_¨ely, in 5.2]5.6 are indecomposable weight A-modules.

Ž .ii E_¨ery weight A-module M such that all MŽm.with periodic m are finite-dimensional decomposes uniquely into a direct sum of modules isomor- phic to those listed inŽ .i .

Žiii. The only isomorphisms between the listed modules are:

v VŽv, f.,VŽv, g if f and g are similar in P. _v;

v VŽv,w, f.,VŽv,w kŽ ., g if f and g are similar in. P_0v and k'0 Žmod B< <_v..

Ž . Ž .

5.8. THEOREM. The weight A-modules V v , V v, f for irreducible

Ž . Ž . Ž .

fgP_v,V v, j, 0 ,V v, j,« , and V v,w, f for f irreducible inP_0v and

p p < <

wsx or y ,where ps v, are simple and each simple weightA-module is isomorphic to one from this list.

5.9. The construction of indecomposable weight modules implies di- rectly the description of their supports. Here is their list:

v If VsVŽv.,VŽv, f., orVŽv,w, f., then SuppVsv.

v Let m be the maximal element of₁ J provided it exists and is a break orq`otherwise; m be the maximal break preceding₀ J provided it

Ž ^X. < 4

exists ory`otherwise. Then SuppV J, J s mgv m₀-mFm₁.

v SuppVŽv, j,w.sv if ls< <w G< <B_v y1, otherwise SuppVŽv, j,

. ^kŽ Ž ..< 4

w s s m j 0GkGl.

5.10. Call a weight A-module M R-finite if it has finite length as an Ž .

R-module. It means that M m s0 for all but a finite number of maximal Ž .

ideals m and dim_K M m -`for each m. Remark that if R is a finitely

m

generated algebra over some field K, it means that the module M is finite-dimensional over K.

COROLLARY. Ž .i A has R-finite weight modules if and only if there exists

< < < <

such orbitv that either v -` or B_v G2.

Ž .ii All indecomposable R-finite weight A-modules are isomorphic to

Ž . Ž . Ž . Ž

either V v, f ,V v, j,w ,V v,w, f for some finite orbit v or to V v, J,

X.

J , where J has both a minimal and a maximal element and the latter is a break.

Žiii. All R-finite weight A-modules are semi-simple if and only if all orbits are infinite and ha_¨e at most two breaks.

Ž . Ž . Ž .

5.11. Remark. For the algebrasAsUg orU_qg from Example 1.6 ii Ž .

or iii our notion of ‘‘weight modules’’ seems rather unusual. As a rule, in these cases anA-module M is said to be a weight module if it possesses an eigenbasis for the only operatorHor, respectively, Z. We demand also the same for TsXY, which one can replace by the central element CsH²

Ž . Ž .Ž ^y1.

qHq2T if char K/2 or, respectively,Cs qy1 qyq qqZq

y1 Ž .

Z the ‘‘Casimir operator’’ .

This, of course, does not imply the description of simple modules but has a great influence on indecomposable ones. Namely, under this weaker condition, the classification of indecomposable modules with finite- dimensional eigenspaces for Hor Zcan be given if orbits are infinite, i.e., if char Ks0 or, respectively, q is not a root of unity. This is due essentially to the fact that there can be no more than two breaks in an orbit. But whenever orbits become finite, there is no chance, in some

Ž . w x

sense, to give a good classification. For details in the case of Ug, cf. 1 ; forU_qŽ .g the calculations are similar.

REFERENCES

1. Yu. A. Drozd, On representations of Lie algebra sl 2 ,Ž . Visnik Kie¨.Uni¨.Ser.Mat.Mekh.

Ž .

25 1983 , 70]77.

2. Yu. A. Drozd, S. A. Ovsienko, and V. M. Futorny, Harish-Chandra subalgebras and Gelfand]Zetlin modules,in‘‘Finite Dimensional Algebras and Related Topics,’’ Kluwer Academic, Dordrecht, 1994.

3. I. M. Gelfand and V. A. Ponomarev, Indecomposable representations of the Lorenz

Ž . w Ž .

group,Uspekhi Mat.Nauk23, No. 2 1968 , 3]40. Russian Math.Sur_¨eys23, No. 2 1968 , x

1]58

4. N. Jacobson, ‘‘The Theory of Rings,’’ Amer. Math. Soc., Providence, RI, 1943.