This article was downloaded by: [North Carolina State University]On: 17 October 2014, At: 05:29Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20

Projective functors for quantized envelopingalgebrasChen Liu aa Department of Mathematics , University of Oregon , Eugene, OR, 97403, USAPublished online: 27 Jun 2007.

To cite this article: Chen Liu (1994) Projective functors for quantized enveloping algebras, Communications in Algebra,22:6, 2125-2172, DOI: 10.1080/00927879408824959

To link to this article: http://dx.doi.org/10.1080/00927879408824959

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purposeof the Content. Any opinions and views expressed in this publication are the opinions and views of theauthors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should notbe relied upon and should be independently verified with primary sources of information. Taylor and Francisshall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, andother liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relationto or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

COMMUNICATIONS IN ALGEBRA, 22(6), 2125-2172 (1994)

PROJECTIVE FUNCTORS

FOR QUANTIZED ENVELOPING ALGEBRAS

Chen Liu

Department of Mathelnatics University of Oregon

Eugene, OR 97403 USA

o. Introduction

Let g be a semisimple Lie algebra, U(g) its uiiiversal enveloping algebra

and Z = Z(U(g)) t,lle ceutcr of U ( g ) Let M be the category of all U(g)-

modules, M z f the subcategory consisting of all 2-finite modules and let V

be a finite dilnensiollal U(g)-module. Then

where F is the ground field, defines a functor: M ----t M. The restrictions

FVIMz, : M Z f --' M Z f for various V and their direct suininands are

called by Bernstein and Gelfaid (see (21) projective functors. They have

proved in (21 that each projective fiulctor can be written as a direct sum of

i~idecolnposable ones. arid they have classified all indeconlposable projective

functors. Tliis is what we refer to as the theory of projective fuilctors in the

classical case. Tliis theory has many important applications. To inention a

few of them, Bernstein and Gelfalid have applied it to prove the cquivalence

2125

Copyright 0 1994 by Marcel Dekker, Inc

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

2126 LIU

of certain categories of U(g)-modules arid to prove the isoniorpl~ism between

tlic lat,tices of two-sided ideals in U ( g ) and the subinotlule latt,ices of Verriia

iilodulcs. And, perhaps nnost interesting of all is the applicat,ioi~ to Harish-

Cliaiidra ~i~odules . For illore details. see [2].

Aftkr the work of Lusztig aiitl lliaily ot,liers. the theory of qi~ai~tized en-

veloping algebras, cspec.ially t,liat of tlieir finite tliilici~sio~ial represclntatiolls.

looks very siil~ilar to t,llat of 111iiversa1 ei~veloping. algebras of Lie algebras. So,

it is natural to ask whetlier the Bcri~st,eiii and Gelfalid's work geiicralizes t,o

quantiactl ciivelopi~ig algebras. Bn6. t,lierc are essential diff(:rerict:s bct,wceii

the t,wo wlneii we look beyond tlnc fiinite tlil~iensiolial reprcscut,at,ions. For

exa~nplc. the adjoint rc~)rcscint~;tt,ioii of Uq(g) t,urns out to be not scunisiinple

(see [13] 4.14). wlnile t,lic adjoirit rcl)rcscntation of IJ(g) i s . a fact wliic:l~ llas

played a rolc ill tlic: Beri~stci~i ant1 Gelfalid's work (see [2] 3.6). Mucln lias yet

to be done hefore we call discuss projective func't,ors for qi~mt,izctl eliveloping

algebras.

Iii this paper. we t,reat the simply coiirlected qnaiitizcd enveloping al-

gebra U = U(,(g) (For the defiiiitioii. see l.l).whicli ~ o i i t ~ a i ~ l s t,lic adjoint

~p~aii t ized ciivt:l~j~iiig algci~ra U' = Ui(g) as a Hopf snbalgcbra.

Tlir crucial instr~ulient Bcrr~steiii ant1 Gclfaild used i ~ i their t8reatincut of

the classical case is a Duflo tlleorenl wl~ich stat,cs that t,lle a~mihilator of any

Vcrnia module is generated as an ideal by the keriiel of the corresponding

~ c n t r a l clnaracter (see [2] 1.13 (13)). Wlietlier the sainc theorem holds for U

is not yet kilown (cf. [13] 5.6). So. wc turii to a subdgebra F ( u ) (see 2.1).

w l~ ic l~ forms t h esseiit~ial ~ i ~ 1 . t of U ( see Tlieorem 2.1.1) a ~ ~ d was first studied

by Joscpln and Letzt>er ill [GI. It is for this subalgebra, tlnat we develop the

projective fiuictor theory. Based on tlieir work in [GI and [7], we are able

to show that the above nientioned Duflo tlleorem lmlds for ?(u) (Theorem

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS 2127

2.2.1), which enables us to reduce the study of projective functors to that

of their effect on Verrna modules (Theorem 3.3.1) as Bernstein and Gelfand

have done in the classical case ([2] 3.5).

Parameterizations of the indecoinposable projective functors may be

bizarre if q is an indeterminate. But when q E C * which is not a root of

1, we have a nice classification (Theorem 5.2.1), which largely resembles the

classical case.

The sanle theory has also been developed for tlle adjoiut quantized en-

veloping algebras Ui(g) in [13]. But in the adjoint case, there are some

exceptions (see 1131 chapter 5). In the simply connected case dealt with in

this paper. however, there are no exceptions.

In order to concentrate on the proof of the main theorem (Theorem

3.3.1) and on that of the classification theorem (Theorem 5.2.1), we leave

out the proofs for sonle of the results introduced in section 1 and 2 and

the proof for Lenma 3.1.2. We just briefly lnerltioll tlle ideas of the proofs

and give the related references. Anyway, for anyone who wants to know the

detail, there are complete proofs for the adjoint case in [13], and generalizing

them to the simply connected case is straight forward.

1. Some General Results

1.1 Let A = (aij)nXn be an indecoinposable Cartan matrix of finite

type; dl , d2, . . . , d, relatively prime integers such that the matrix (diaij),,,

is symnletric and positive definite; q an indeterminate or any non-zero com-

plex number which is not a root of 1. Set q; = q d ~ .

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

2128 LIU

Let g be the Lie algebra over C with Cartan matrix A = (aij),,,:

b a Cartail suhalgebra of g: n = (al. 02.. . . ,a,) the siniple roots: I? =

C:=l Zcu, t,he root lattice: w l , w2,. . . . w, the furldaumital weights: P =

P(I1) = Cr=l Zwi the weight lattice and P+ = P(IT)+ = xr=l Nu, t,lie

dominant integral weights: W tlie Wcyl group of p. Let ( , ) be the W -

invariant scalar product on b* with (a , , a, ) = d,n,, for all i , j .

Let F > C ( q ) b e an extension field of C ( q ) . If q is a coiiiplex ~n~ i i ihe r

tliat is not a root of 1. we call take F to be C = C(q) . In this case.

we clioose a fixed log q and writ,c q" = e" 1°gq for any (z E C. If q is an

indet,erminate, we require that F contains all q" with n E Q: by fixiiig a

group l~omoniorpliism: a H q" frolii ( C , +) to (F*. .). Such a field exists.

For example. G = {q" 1 n E C) is an abelian group isolnorpl~ic t80 ( C . +).

let C [GI he the group algebra, it is ail int,egral domain, wc can take F to be

the fraction field of Q: [GI. For simplicity we assurrie that F is algebraically

closed.

Assigii to X E P ( n ) a symbol r(X) = KA. Tlieii K = { K ~ I X E ~ ( n ) ) is an abeliaii gronp a i d T is a group ison~orpliisni from P i n ) to K . Let U(,(p)

or siiiiply U be the ;tssociat,ive algebra wit,ll identity 1 over F generat,ed by

E,. F i d 1 ( I < i < 1 2 ) and satisfying the followil~g relations:

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS

with

and

U,(g) is a Hopf algebra, called tlie simply connected quantized envelop-

ing algebra associated to g. with comultiplication A, antipode S and counit

E defined by:

Let K. be tlie F-algebra anti-automorphism defined by (cf. [G] 4.8):

Let S' be the C-algebra antiautomorphisnl defined by (for q an indeter-

minate):

1.2. Let U;(g) or simply U' be the subalgebra generated by Ei, F, and

K : i = 1 ,2 , . . . , n. It is a Hopf subalgebra, called tlie adjoint quantized

enveloping algebra associated to g. Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

2130 LIU

Let CJ+ (U-, resp.) be the subalgebra generated by E, (F, . resp.):

z = 1.2 . . . . . n: U" tlie subalgebra generated by KAaL : z = 1 . 2 . . . . .?L. Let

Uo be the subalgebra generated by K*,, : z = 1,2. . . . . 1%. We have the

following triangular tlecoiuposit~on

where (U-)+ a d (U+)+ are t,lic al~glllel~tat,iorl ideals of U - and U + , respec-

t,ively. Let Po be the projection

defined by (1.2.2). Let

A

Eleine~its ill UO will be denoted by X, f i , . . . , etc. Tlie map

wliere X, = X(K,~ ). is a bijection. Since U0 is isomorphic to tlie group algebra

of P(n). the set HomF-,l,(Uo, F) can be identified with H o m g r o l L , ( P ( ~ ) . F*)

by restricting I,!J : U0 -+ F to ?i,0 : P ( n ) + F' where $ J ~ ( X ) = $ ( K x ) . Ob-

viously Honi,,,,,,(P(~), F*) is an abelian group under pointwis~ multiplica-

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS 2131

tion. Since P(II) is a free abelian group with basis { w l w z , . . . , w,,), the map A

(1.2.3) is a group isoinorpliism. From now on we identify Uo with and

use whichever notat<ioil is convenient in our context.

For each p E b * , there is a natural extension in U o , denoted by F , via

jG(Kx) = q( ' .~ ) for all X E P(II) .

We call jG the principal extension of p. The map p - F is a group 1101110-

morpliism

f)* --+ U O . (1.2.4)

It is injective if q is an indeterminate. If q = J E C*, it is not injective since

the exponential function a ++ qu is periodic with period 2ri/ log q. However,

since we assume that q is not a root of unity, the lnap (1.2.4) induces an

elnbeddilig A

t l i = P(W @z & r UO.

Fortunately, whenever we deal with the relations anlong weights, we are only

iiivolved in a fixed weight X E (F*)" and weights in IT), and the fact that

the exponential function is periodic causes no problem. A

Any f i E U O defines an automorphisn~ of U0 via

fiKx = fi(Kx)Kx for all X E P(II).

We call

h = F o P o : U - + U 0

the Harish-Cliandra map for U . A

Let be the set of all element,^ o in Uo such that u(K,,) = kl for all A

i. One checks that is a subgroup of Uo and a(X) = a ( K x ) is a root of 1

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

2132 LIU

for all a E I?; ant1 all X E P ( n ) . Any a E I?; can be extended to ail algebra

homomorphism a : U --+ F via o ( E , ) = 0 = a(F,) . (One cliecks easily that

the relations in 1.1 are preserved.) On the other hand. the relat,ioils (1.1.2)

inlply that each algebra hoiiionlorpliism U - F maps all E, and Ij? to 0,

lirnce by (1.1.3) a11 K:t to 1. so it restricts to an eleiimlt of r; on UO. We

i.e.. r; is in one-t,o-one correspoiitlelice with the one-dinieiisional represen-

Obviously, W acts oil U0 via

cuhrx = K,(A) for all X 6 P ( n )

Olie checks that U' iiorilializes Ti on U O . It follows that W r>< ra acts 011

u'. Let

KT = ( U W ~ I a E r;).

1.3. Give11 s E IN. sct

We can extelitl the autoiuorpliisms T, ( i = 1.2.. . . . n) introduced by Lusztig

(cf. [ lo] 3.1. see also [4] 1.6) to U as follows:

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS 2133

Fix a reduced expression 7uo = s , , si, . . . s , , of the longest element w0

of W. This gives rise to an ordering of the positive roots:

Lusztig lias constructed the canoriical basis for U (see [ I l l ) . However, we

need only the followiilg weaker result.

Let Ep8 = Tz,Tl, - . . Tls-, E t a , FDa = T,, T,, . - . TZs-, Fan (= S1Ep8) . For

k = ( k ~ , . . . , h . ~ ) E z ~ , ~ ~ ~ E ~ = E ~ : E ~ ~ . . . E ~ ~ , F ~ = s ' E ~ . Tllen(cf. 141

1.7)

THEOREM 1.3.1. a) The elelneilts Ek (Fk, resp.): k E z:, form a basis of

U+ (U- , resp.) over F .

b) The elements FkKG1 . . . Km7. E', where k , r E Z:, (nzl, . . . , m,) E w7,

Zn, form a basis of U over F.

Let H be the F-bilinear map

u x u ---+ U0 (1 .3 .5)

(u, V ) ++ H(u:u) = Po(r;(u)v).

For any CY E I?+, let Pa r (a ) be the set of all k E I L ~ such that c:, k,P, = a.

Let U; be the subspace of U - spanned by F k : k E Par(a): let Ha be the

restriction of H to U;; and let det(H,) be the determinant of Ha with

respect to the basis Fk : k E Par(a) . For any /3 E R, let d p = d, and qp = q,

if p = w(a,) for some a, E ll and some ~ I J E W. One call show that det(H,)

differs only by a no11 zero scalar from the determinant of the DeConcini-

Kac-Shapovalov contravariant Hermitian form defined in [4]. Hence (cf. [4]

1.9)

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

LIU

PROPOSITI~N 1.3.2. One has rrp to a 11on zero scalar

Using the above determinant forinula and tlie inetllod in ( [4] 2.2). one

proves

THEOREM 1.3.3. The restriction to tllc cellter Z [ U ) of U of the Harish-

Chandra i m p

h : Z ( U ) + ( U O p '

is an iso~norpllisrl~ of F-algebras, called the Harish-Cllandra isomorphism for

1.4. Collsider U as a U-module via the left multipliratioll. For any

i E 5. let .Ii be tlie left ideal gellerated by (Ui)+ and K,, - (i - p?(K,, ) :

i = 1.2, . . . . n. The Verma module M ( X ) is defined to be CT/JX. Its ullique

siinple quotient is denoted by L(X) .

A U module M is called Uo-diagonalizable if M = eijEG MG. where

Mij = { m E M I K m = cj(K)m for all K E u'). (1.4.1)

be the set of all central charitcters. For each X E UO: there correspol~ds a

central character Bi sac11 t,liat each z E Z(U) acts 011 M ( X ) as the scalar

Oi(z) = X(h( z ) ) .

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QU ANTIZED ENVELOPING ALGEBRAS 2135

Using Theorem 1.3.3 and arguing similarly as in the classical case (cf.

151 7.4.7), we can prove

THEOREM 1.4.1. One has

and = 0 i 2 if and only if 1 1 = wX2 for some lu E w

Let A(0) be the set of all weights such that = 0. By the above h

theorem, A(0) consists of a single @-orbit in U O .

Let A be an associative algebra with 1 over a field F. An element v in

an A-module M is A-finite if dimF At! < m. The module M is A-finite if

every v in M is A-finite.

Let M be the category of all U modules; let Mf and Mzf be the

subcategories of M consistiilg of all finitely generated U modules a ~ l d of all

Z(U)-finite U modules, respectively.

DEFINITION 1.4.2. Let 0 be the category whose objects are all U modules

M satisfying the following conditions:

1) M is finitely generated as a U module,

2) M has a weight space decompositioll

and whose ~norphisms are U module homomorphisms.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

2136 LIU

It is an ;~beliaii category. It is clear that every Vcrnia 1i:odule M ( X ) is A

811 object, in (3, and all L(,\) : j; E UO exhanst t,lie simple ohject,s in (3.

Observe t,lmt any ~ ~ - r l i a ~ o n a l i z a b l e U-i~iod~de M is irreducible if and

ollly if it is irred~lcible as a U'-inoclule. Hence. the following theorem can he

proved in tlic sa lm wq as ill ([9] 3.2).

THEOREM 1.4.3. A finite clirncnsiond U-itiod~ilr M is irrrd~rciblc if a~ld m l y

if Ad = ~ ( i ) with = a t for n E and (1 E 11 + P*(Il) .

For each 0 E % ( I / ) . let, .JH = kwh'. and define

Use siliiilitr i~otations for O a d M z f .

For ally p E I?+. 1, E N. let,

and

T : ~ = {I E TrQ I I $ TTrL7 for ill E W.7 E Rt swli t l ~ i ~ t iJtr < rb) (1.4.3)

A - 8 For X. fi E UO, we say XZfi. if and only if ii = - 1-/3 n11d E Tfi3 for some

A

E R+ and soiue r e IN. We define n partial order on U0 by saying > f i - if and only if there exist yl, 72:. . . ,yS in R+ and il. i2. . . . ,I, in Uo silcli

Ti , YZ Ta 7 s

tliat X 2 X l > A 2 2 . . . > A , = f i . We can use this order relati011 on t,lle weights

to describe the inclusioll relati011 anlong Verma nlodules for U as it l l i ~ bee11

clone for U(g) in the classical case (cf. [ 5 ] 7.6). While doing t,liis. iiot8ice that

the deterliiinant, forrlnlla in proposition 1.3.2 makes t,lie proofs easier. The

form~l i~t io i i of the result,^ is left t,o t,lic reader.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS 2137

- 1

Let UO be the set of weights X in U^O of the fornl r, = 0 + i' for some -1

cr E and A' E fj*. Notice that for any X,@ E U0 , X 2 @ if and only if -

X = 0 + A', fi = n t 2 for the same n f I?; and A', p1 E fj* and A' 2 p' with

respect to the order relation oil fj* defined ill ( [ l l ] 1.5). Notice also that, in

the case of q E C * which is not a root of 1, we have

- -1

uo = UO .

In the case q 6 C * the decomposition of a weight in the for111 = cr + X' is

not unique in general. If lql # 1, tlleii it is unique for A' E fj&. If q is not a

root of unity (but possibly (q ( = l ) , the11 it is unique for A' E 5;.

A weight t 5 is called integral if = n t i' with o E I?; and

A' E P(II) . The deconlposition is unique for integral weights which form a

subgroup of U0 isomorphic to I?; x P(I3).

1.5. The following is the quanduln version of tlie results in the clas-

sical case proved in [3] a i d the proofs can be obtained by modifying the

correspondiug ones there.

(a) Mzf = @ MFf(B), i.e., for each M E Mzf, we can write

e~ zT)

(b) Every M f 0 has a, Jordan-Holder series.

(c) Each object in 0 is a quotient of a projective object, each siriple object

L(X) has a u n i q u ~ (up to ison~orpl~ism) projective cover P(X) and the

set of all ~(i): E U0 ~xliaust the indecomposable projcctive objects

in 0.

(d) The duality (P(X) : ~ ( 4 ) ) = ( ~ ( 4 ) : ~(i)) holds.

(e) For i, 3; t @, let diTd = ( ~ ( i ) : ~ ( i i , ) ) . Suppose that X,$ E G I , then

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

which iir turn holds if and ouly if 1 > 4 (with respect t,o the order

clefincd in 1.4). A

( f ) A weigllt i t 770 is called doilii~ia~it if tlieie is 110 weiglit, f i t U%~tcli

t,llat F - 1 = 7 for sollie 7 t I? and f i = c u i for sonre i i i E 11. The

Put Oi = [ ~ ( i ) ] . Tlirii hi : E UO for111 a hasis of K(O) ailrl for

(hi. a,) = 0 if # 4, {hi. 6,) = 1

If P is projective in 0 m d M E O, then {[PI. [ M I } = tliiir Honlv(P. M).

2. The Subalgebra F ( u )

2.1. Therc is all adjoint action of any Hopf a l g e l ~ a on itsself (see [GI

2.2). 1n particul;tr, there is an adjnii~t at:t,ion of U on U itself defined as

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QlJ ANTIZED ENVELOPING ALGEBRAS

follows:

(adu)u = u i u ~ ( u : ) . for all v, u E U with Au = ui @ u:. . . i

Following Joseph and Letzter (see [ G I ) , we define

Then, 3 ( U ) is an adU-submodule as well as a subalgebra of U. Let

Then

?(u) = KL'F(U)

is also an adU-submodule as well as a subalgebra. For the detail of the proofs

that F ( U ) and ?(u) are subalgebras, see ( [6 ] 2.3).

Remark. We may also define F ( U 1 ) = {u E U' I d i m ~ ( a d U ' ) u < ca),

Kb = T ( - 2 P t ( I I ) n I') and ?(u') = K'; '~(u ' ) . One checks that F ( U 1 ) =

3(U) n U' and Y(U') = ?(u) U'. If g is of type Es, F4 or G z , the weight

lattice P ( n ) coiiicides with the root lattice I' and we have that Uq(g) = U;(g),

F ( U ) = F ( U 1 ) and ?(u) = Y(U'). If g is of type B,, C,, D, for n even,

or E7, the weight lattice properly contains the root lattice, hence U,(g)

properly contains Ui(g ) . However, we have that for these types 2P(II) 5 I',

hence by Theorem 2.1.2 below, 3 ( U ) = F ( U t ) and ?(u) = F(u'). The

most interesting case is when g is of type A, (72. 2 2) , D, for n odd, or

Es. we have that U , F ( U ) and ?(u) properly contain U' , F(U') and ?(u'), respectively.

Modifying the argument in ([GI G.4), we can prove the following Joseph

and Letzter's theorem:

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

2 140 LIU

- THEOREM 2.1.1. The snhalgebra F(U) is generated b,y EE,, FX,, : i =

1 , 2 , . . . , n and KoK,'. Moreover, one has

Followiiig the proceeding ill ((71 !i4), one can prove

It follows froin this and from theore~ll 2.1.1 t,hat

Tlle subalgebra ?(u) is not r Hapf slibalgcbra of U . It is not closed

llnder the conlultiplicatiolr ant1 the antipode. i.e.,ill general,

Nevertheless, we still h r e the followillg

COROLLARY 2.1.4. OIIC has

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS 2141

Proof. By Theorem 2.1.1, the algebra F(u) is generated over F by

E,, FzKaz : i = 1.2. . . . , ? I and ICri KO. One easily checks that tlie corollary

holds on the generators.

Using (1121 Part C), one can easily prove that 3(U) is semisimple under

the adjoint actioil of either U or .?(u) or F(U). An argument similar to

that in ([7] 7.6) shows that

THEOREM 2.1.5. Ebr all 1 E 9 and /I E Pf (IT), one lms

where L(jZ + go is the 0-weight space of L(F t 3.

2.2. Let {yl, 7 2 , . . . , yk) be a set of coset representatives for K / K ~ K ~ ' .

The followiiig is the quantum version of the so called Duflo theorem on the

annil~ilators of Verma modules.

THEOREM 2.2.1. For al?y X E 5, one has

In particular,

All11 - M ( X ) = F(u) ker B ~ . n u )

The proof is quite lengthy and is placed in tlie Appendix.

2.3. Let ?((I)+ = Uf , ?(u)- the subalgebra generated by all FzKa,'s;

let Y(U)' = F T ( ~ P ( I I ) ) = ?(u) n U o Then the ~nultiplication niap

is an isomorphism. This is tlie triangular decolllpositioll for ?(u). We call

easily construct a basis for F(u) from Theorem 1.3.1. Take the same basis for

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

2142 LIU

xlltl order lL;N+n lcxicograpl~ically such that 61 < 62 < . . . < h 2 ~ + , , where

Let 7, be the left ideal uf f l u ) generated by .?(u)+ and K - (i - ail() for K in F(i3)O. The lilodule ?(u)/.& via left inultiplicatiotl is called a

Vc:rma 1 ~ ~ ~ d 1 1 1 c for F ( u ) . de~mt~ecl by G(i). A ~ ( ~ ) " - d i a ~ o n a l i z a t ~ l t : inodule -

?(i) is called a higlicst, weight iriodde of highest weight i - for F((r ) if -

i t is griierat,cd liy a vcctmor ox-; witall tlic property: 3 ( U ) + ~ l ~ - ; = 0 and -

1 , - = ( - ) ( for all 1 in 3 ) . Clearly. evcry highest weigllt - "

~nodnle c(X) is a quot,ient of the Veriila module M ( X ) which has a 1111iqllc . "

siulple quot,ient,. denoted by L(X). A

The rest,rictiol~ map fruln t,lle set of weights lTC1 H o ~ ~ I ~ , , , , ~ ( P ( I T ) . F*) - t,o ~ ( u ) o E Hoiilg,,,,,(2P(II), F*) is surjective since F is algebraicitlly closed;

X ( K ; Z , ) = 1 for all i. 1wlc.e X E rf. 111 other worrls, the kcriicl is equal to Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPlNG ALGEBRAS 2143

and has order 2n, since P(I l ) /2P(I l ) is elementary abelian of order 2n - Since P( I I ) f l = 0. we see that tlie restriction map is injective on -

P*). We shall identify P Z ) with its image in ?(u)o.

The restriction map takes onto

We have = Hon~,,,,,(2P(II)/2~,F*), which has order f = [P(II) : I?] =

[2P(Jl) : 2 r ] , the index of coiiuection of the root system. We call integral

weights of ?(u)' all a + X' with a E E and A' E P(II) . Then these integral

weights form a group isomorphic to x P(II ) and the restriction maps

integral weights onto integral weights (with kernel Fa0).

LEMMA 2.3.1. For d i E 5 the restriction of the simple U-module L(X) to

F ( u ) is simple.

Proof. We have a direct sum decomposition ~ ( 1 ) = $ L ( ~ ) G with

B E i + P*) c (in fact 6 E I + f) . Since tlie restriction map to - f ( U ) o is injective on P Z ) , distinct ij.s restrict to distinct weights on f ( u ) ~ .

Therefore each ?(u)-submodule of L ( X ) is a direct sum of its intersectioils

with the L ( i ) ~ ' s , hence it is Uo-stable. Now the lemma follows from Theorem

2.1.1.

DEFINITION 2.3.2. Let C? be the category whose objects are all ?(u)-modules

M satisfying the following conditions:

1) M is finitely generated as a P(U)-module,

2) M has a weight space decomposition

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

LIU

- an extcnsioii X I to UD, then by Lemma 2.3.1 ~ ( i l ) re~tr ic t~ed to 3 ( U ) is

- " also a siiliple lligliest weight inodde witli highest weight A. lience L ( X ) 2

- L ) 1 . Tlrus: Simple iiiodules in O are tlre L ( 2 ) wit11 ii

alld L ( 9 ) L(f i l ) if a i d only if 3' - f i E I?;0. Siinilarly. tile Ver~ria

iirodules and tlie highest weight modules for P(u) are the restrictiolls of the

Verma inodules and the highest weight il~odules for U . - A

Let Al he ail arbitrary niodule in 0. For each 3 E ?(u)\ consider

t,lle subspace Mi.+;. This is a subnrod~lle and M is the direct - Slllll - of subiiivdules of this type (3 running over coset reprcselltatives of jr(U)'

for u 1 , t i 2 E ?(u) and A , X I ii P(II) . It is now easy to clicck tliat tlie give11

actiom of ?(u) and U o i i ~ d ~ ~ c r . ail itctioi~ of U . (Take for u z :I weight vrctor.)

It follows from Theorein 2.1.1 that a highest weight U-moth~le V(X) is &

irreducible if and citily if it is inducible as a F(U)-1not111le. Let M be t,lw

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS 2145

obvious analogues for F(u) (see 1.4). One checks that the same decompo- - -

sition as in 1.5 (a) holds for M z f : M z f = $ M " F ~ (0); i.,, for each

@€ zF) M E Mzf, we can write

M = $ M ( 0 ) . where M (0) E G F ~ (0). (2.3.1)

@ € z T )

2.4. Recall that I'z is the set of all o E 3 such that a(K,,) = k1

for all i. Two such weights al and 02 are congruent modulo I'zO if and only

if the one dimensional U-modules with weight a1 and u2, respectively, are

isomorphic a s ?(u)-modules.

Let V be a finite dimensional irreducible Y(U)-module. Then it is a

highest weight module. By Lemma 2.3.1, it can be extended to be a U -

module. Hence, V = ~ ( i ) for some E 3. It is easy to see that X has the

form: X = a + ji for some u E I'z and some p E p + P+(II). We call 0

the type of ~ ( i ) (resp I). Let V be a finite dimensional irreducible ?(u)-

module, we say V is of congruent type 8 if it has an extension L(X) of type

a' in the congruent class (5. It is easy to see that if V is a finite dilnellsional

irreducible ?(u)-module of congruent type 8 and o' is an arbitrary element

in the class 6 , then V has an extension of type a'

Remark. The same consideration applies to arbitrary highest weight

?(u)-modules.

Example 2.4.1. Consider the case g = A2. We have wl = $(2al + a t ) ,

W2 = ! j ( ~ -t 2 ~ ~ 2 ) ; P(rI) /r = {O,L~1,221} N 253. Let JJ be a primitive third - root of 1. In the case that q E C* is not a root of 1, every weight in UtO has

the form: X = (€1, €2) + X t with ei = kl and A' E b* , where (c1,c2) denotes

the element a in UtO such that a(K,,) = ci : i = 1,2. One checks that X

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

LIU

has three distiiict exteilsions to Uu:

A

wllere (~2:;. E ~ E ; ' ) de i lo t~s tlw ~ le inn l t a in Uqideiitificd with ( F * ) ~ (see

(1.2.3)). In particular. when X' = 0, X = a = (€1, €2) 1 1 s three distinct

extensions:

0 = ( E 2 [ j , E UO.

When restricted down t,o 2P(I1), all tile exter~siolls (Q@, el(;') for a fixed

i and for different (el, r2) are equivalent. So, I I';/r;O I= 3. Hence, there -

are only 3 different coligruent types of finite dirner~siollal irreducible 3 ( U ) -

modules.

3. Projective Functors and the Main Theorem

3.1. Let V be a finite d i n ~ e l ~ s i o d ?(u)-module. Arguillg as in ([12] -

Part C). olle proves that such F(U)-i~lodules are selnisilnl,le. l~ence are direct

suins of simple highest weight modules. By the previous subscctiou. we can

extend V to bc a U-rnodule. For any M E G, the algtkra ?(u) acts on

V @F M via

- where rr E ?(u) with Au = x, u,@ IL:. By Corollary 2.1.4, this F ( U ) action

is well defined. Heucc, tlic. functor - -

FV : M -4 M

M +-+ V@FM

is well defined. However, since the ext,e~lsioll of V t,o be a U-module is not

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QU ANTIZED ENVELOPING ALGEBRAS 2147

unique, the functor Fv may depend on the extension. So it is natural that

we start with a finite dimensional U-module, instead of a a finite diniensioilal

P(u)-module, as follows.

Fix an abitrary element u E I'a. A highest weight U-module v(X) is

called of type o if = o + 1 for some X E P(ll) . A finite dimensional U -

module is said of type u if it is a direct sum of irreducible highest weight

modules of type u.

- - DEFINITION 3.1.1. (i) A functor F : M z f ---+ Mzf is called a projective

functor of type o for ?(u) if it is isomorphic to a direct suminand of the

functor Fvlfiz, for a finite diineilsional U-module V of type u. - (ii) Let 0 E Z(U) be a central character. A functor F : G($) --+ M is

called a projective 0-functor of type a for ?(u) if it is isomorpliic to a direct

sunmiand of the functor Fv(B) for a finite dimensional U-module V of type

a, where Fv(6') denotes the restriction of Fv to M(0).

The meaning of the term projective can be found in section 5 of [2].

Notice that a projective functor F preserves 6, therefore we can view (We

often do!) the image F M for any M in 6 as a U-module (see 2.3).

We list some sililple properties of projective functors of type a in the

following lemma.

L E M M A 3.1.2. Let F be a projective functor of type a. The11

(i) F is exact arid preserves direct sunis and products. Direct surnmands

of F are projective f i l i~c to~s o f type a.

(ii) I f G is a projective f1111ctor o f type a , then F $ G is a projective functor

o f type a; i f G is o f type u', then F o G a projective functor o f type

fJ f a ' .

(iii) There is a natural isomorphism o f functors:

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

LIU

and all the fmctors Pr(0') o F o Pr(0) are projective fi~nctors of type 0,

where Pr(0) is the projection M H Jf(0)) tlefined by (2.3.1). - -

(iv) F preserves 0 a i d tml~sforms projective objects of O into projective

o11es.

(v) There is a projective functor F' of type -0 left adjoi~lt to F. (Silnilark

there is a projective filnctor F" of type -a right adjoint to F ) .

One can prove the leliinla by arguing as in ( [ 2 ] 3 . 2 ) in tlie classical case

with ohviolrs modifications. However. one 113s to be careful with (v). The

functor Fv-, where the dual action oil V* is defined via the antipode S, is

both right, and left adjoint to the fwctor Fv, but tlie cailonical isolilorpliinl

doe. lmt, map Honigju) ( M V* @F N ) illt,o HolaFiui(V @F 86. N ) We

to adjust it as follows: For any p 6 H o l i i ~ ( M . V* @F N ) . k t p(p)(u @ '/7d =

x i ( I ) , if ( 7 , ) = 1 Q I . One clierks t k t t P tlius defined

induces an isomorpllis~ll from Horn- 3 ( u ) ( M , V* @ F N ) to H o l ~ i ? ( ~ ) ( V @F

M, N). For the reason of why sticking the operator K2, in the definitioll

of p. see ([I] 7.3).

3.2. Let A. B he two algebras. A module X over A @ B" is called an

( A . B)-\>in~odnle, We consider X as a left A-module ailtl right B-module: if

a E A, h E B. z E X. the11 ( a @ b ) : ~ = cmb. TO each ( A , B)-billlodlde X , we

assign a fiinctor

h ( X ) : B - mod ---+ A - lllod,

via h ( X ) ( M ) = X @o M.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPlNG ALGEBRAS 2149

In the case when A and B coincide, we use A' and A' to indicate its

different roles in the action on the modules. Similarly, if J is an ideal in A,

then ~ b n d J' are the appropriate ideals in ~ " n d AT. And, we write

for A 63 A'.

We define a ( P ( u ) , F(u))-bimodule (resp. (U, U)-bimodule) structure

on @v = V @F ?(u) (resp. V @F U ) by

for any v E V,u E ?(u) (resp. U), X E ?(u) (resp. U) with A X =

C X, @ X:. Then, - - h(@v) : M -+ M

M +-+ @V 8 ~ ( ~ ) M is naturally isomorphic to Fv.

Let Js = P ( u ) ker 6' and let Ue = U/U ker 6' and P ( u ) ~ = .?(u)/J@.

LEMMA 3.2.1. The inclusion map ?(u) - U induces an injection F ( u ) ~ -,

ue .

Proof. Suppose that u E ?(u) and u E U ker6. Write u = Xi ulzi,

where zi E ker0 and u; E U. It follows from Theoreln 2.1.1 that each -

uj t i E .F(U). So we may assume that ,u = u'a with 2 E ker 0 and u' E U.

Choose P E 2P@) such that Kpu E F ( U ) . Since 2 E Z ( U ) , (adU)(Kpu1z) =

((adU)Kpul)z. We see that (adU)Kpu1 has to be finite dimensional. Hence

Kpu' E 3 ( U ) and u' E .?(u).

Let V be as above. Consider the (?(u),?(u)~)-bimodule @ v / J g =

V 8 . F ( u ) ~ induced from av. We define an adjoint action of U on V @ U

and V @ Us, respectively, as follows:

(adX)(v 8 21) = X,V @ (adX{)u for all v E V and u E U(resp.Uo),

for any X E U with AX = X,@X:. It is related to the bimodule structure

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

LIU

because of the associativity of the coproduct A. Tlie subspace V @ ?(u)

(resp. V @ F ( u ) @ ) is a U-submodule of V @ U (resp. V @ Ue) for the adjoillt

action arid we get an induced adjoint action on v@?(u) (resp. v@?(u)@).

711

Proof. Recall that U = $ ?(U)yJ (cf. Theorem 2.1.1). We define z = 1

for a11 v E V and ,u E F ( u ) . Obviously, @ is a right U-module llomomor-

pliisn~. To sllow tJmt @ is a left U-mod111r homon1orpllisll1. it is enough to

check t l ~ a t 9 co~mnutes with tllc left actlion of ca.ch Kx with X E P ( n ) .

Since K x z = ( adKxz)Kx for all r E V @ U and since coulnlut,es with

t,lle right action of U. it is enough to show that coninlutes with the ad-

joint action of each K x . Under the adjoi~lt act,ion of U" the tensor product

@ U = + X I ) 8 U is a direct sulii of weight spaces. The weights that

occur have t,he form cr + with / L E P(I I ) . Distinct weights of t h for111 have -

distinct restrictions t,o F(U)( ' . Therefore the ~ ' - w e i ~ h t spaces of V @ U are -

also the 3(U)"-weight spaces. The same stat,e~llent lloltls for L 8 U . Since -

Ip̂ commutes with the adjoint, action of F (u ) ' it nlaps each wciglit space

( V @ U)g+; t,o (L @ U)g+;. hence coemuutes with the hiljoilrt act,ion of UO.

L E M M A 3.2.3. Let V and L be as ill Lemma 3.2.2. Then one has Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QU ANTlZED ENVELOPlNG ALGEBRAS 2151

Proof. By Lemma 3.2.1 it is enough to prove

For cp E H0111(~ (u ) , 5(U)j (@v, QL), we define

.J) = ~ ( p ) : v -+ L @ F ( u )

v H p ( u @ l ) V V E V .

Using Lemma 3.2.2 , we can show that $I E Horn- (V. Q2d) as follows: F(V)

while

= (adE*)cp(v @ 1).

Similarly, we can check for the F,K,,'s and for XoX;', a11d it is enough to

check for the generators for F(u) . Conversely, suppose that $I E Horn- (V, ( a ~ ) ~ ~ ) . We may as well F ( u )

assume that ,II, E Homa (V, ( @ L ) " ~ ) . We define

cp(v @ u ) = + ( v ) t ~ for any v @ ,u E av.

Obviously, cp preserves the right action of ?(u). To show that it preserves

the left action, it is enough to prove that

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

2152 LIU

- THEOREM 3.3.1. Let 0 E Z ( U ) be a central character, 1 E A ( 8 ) a weight; let

F mCI G be pojective 8 - fu1~ to r s of type o for ?(u), Define the homomor-

phism of vector spaces

Proof ( a ) Since F is a projective 0-funct,or of type a. hy defii~itioll,

there exists a decomposition Fv(0) = F@F'. where V is a finite dinm~sional

U-module of type a. This gives the followiiig deco~llpositioil

This decompositioll ilnplies that it suffices to prove the theorell1 ollly for

F = Fv(8). We may as well assume V is irreducible. Similarly, we may

assume G = FL(B) for L a finite dinlellsio~~al irreducible U-module of type

0. Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS

and the latter space is isolnorphic to

However,

dim HoniFi,)(V g L*. ? ( u ) $ ~ ) > dim H o i ~ l + ~ ~ ) ( V @ L*,I(U):)

= diin(V @ L*)o (Theorem 2.1.4 ).

This proves (*) and also colrlpletes t l ~ proof of the theorem.

4. Decompositions of Projective Functors

4.1. Let F be a projective functor of type o. Tlie following proposi-

tion shows that its restriction F m ( 6 ) to the subcategory G y e ) (Note that

G m ( 0 ) Z z f ) is completely determined by its restrictiou F(0) to the

subcategory M ( 0 ) .

PROPOSITION 4.1.1. Let F, G he projective fullctors of type a. Then each

morphisni cp : F ( 0 ) ---+ G(6) call bc extended to a morpllim + : Fw(6) -+

Ga5(8). If cp is an iso~aorphism, thcll so is G. If lf = G and cp is an idempo-

tent, then we can choose @ to be an idempotent.

Proof. Use a similar argurnent as in ([2] 3.7).

As a consequelice of Theoreni 3.3.1 a d t l ~ c above propositioii, we have

COROLLARY 4.1.2. Let F , G be projective functors of type 0, a maxi-

n1a1 weight, 6 a central cl~aracter sudl that X E A($) . Then, each iso-

morphism: F M ( ~ ) --t GM(X) can be extended to an isoiuorpllism of the

functors: FDO(B) - Gm(6); a ~ ~ d each decon~posit io~~ FM(X) = @ M, can

be extellded to a decomposjtiol~ of the functor F""(6) = @ F Y ( 0 ) with

F,O"(B)M(X) = M,. Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

LIU

4.2. Lct F he a project,ive ftmctor of type a . Tim

Each of t,hc funct,ors F o Pr(H) is rlet,err~li~led by its restriction to the sub- -..

category M"(0). (R.ecal1 that G x ( 0 ) GZf.) Hewe: Corollary 4.1.2

implics that F o Pr(0) decomposes illto a direct sun1 of a finite i ~ u n ~ h e r of

illtlecomposable projcctivc fui~ct~ors of type n, i.e., wc llitve proved

THEOREM 4.2.1. Every projt.ctivc frrnctor of type n decomposes into a dircct

su~i t of i i ldecon~~osable ptojectivc, firrictors of type n.

If F is it11 il~dec:oi~~posnl:)le yrojectivc functor of type n. t,Ilen F = F o

Pr(0) for some % E Z ( U ) . So, F M ( J ) = 0 if X $ A ( % ) : if E A ( % ) a d

J\ is doininant,, tllci~. by Cvrollary 4.1.2, FM(X) is iitdecorrlyosable: by 1.5

( f ) and Lell~llln 3.1.2 (iv), ~ h d ( S ) is project,ivc in 0. So, F M ( ~ ) is all

indeconiposable projective o\ljt:ct in 0. Hei~cc, F M ( ~ ) = P(,$) for somc

4 d .

5. Classificatiotl of Projective Functors

5.1. All the resldts wc have obtained t l ~ s far linld for q r i t l~cr a n iude-

terminate or a non-zero coinplex utiin1)er whir11 is not :t mot of 1. However,

for conveaience. in 5.1 and 5.2 we asvunle that q E * which is not a root of

Fix a dominant weight 2~ E A(0) for each ccutral cl iarxter 0 E Z ( U ) .

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS 2157

We say two elements [ = ( 2 0 , @) and [' = (5$, @') are equivalent. denoted by

[ [ I , if and oilly if B = 0', @ = 0-0 + (7 + 5, $' = 00 + 0- + @' for (P, (P' E b*

such that cp' = w p for some w E W X B = { W E W I 1 ~ x 0 = ~ 0 ) Let E , be the

fanlily of equivalence classes: So/ N'. An element c = ( > i e , $) is said to be

written properly if @ = 06 + a + 5 for cp E b* such that cp 5 WX,(cp). Every

element in E, can be written properly (cf. [2] 3.3).

5.2. We can now prove the following partial classification theorem:

THEOREM 5.2.1. For each ele~nent 5 = ( 2 ~ , 4 ) E E,, there is an indecompos-

able projective functor Fg of type a , a unique one up to isomorpl~ism, such

that

(i) F < ( M ( X ) ) = 0, if X @ A(0).

(ii) If E = ( i e , 4 ) E Z, is written properly, then F ( ( M ( 2 0 ) ) = ~ ( , 4 ) . The map: ( - Fg is a bijectiou of E , with the set of isomorpl~isin

classes of indecomposable projective functors of type o for P ( U ) . Proof. ( a ) We have already seen (see 4.2) that if F is an indecompos-

h

able projective functor of type o , then F = F o Pr(B) for some 0 E Z ( U ) ,

and F ( M ( 2 e ) ) = ~ ( 4 ) for some 6 E U^O. There is a finite dimensional irre-

ducible highest weight F(u)-module V of type o such that F ( M ( > i e ) ) is a

direct suinnland of F v ( M ( 2 e ) ) . So, - is a weight of V BF M ( Z O ) , hence

(20 ,4 ) E EO,.

It is easy to see that Fv is an exact functor. Hence it induces an

endomorphism on the Grotlmidieck group K ( O ) , we denote it by F;. By 1.5 ..--.

(h), if is4, ~ y ~ 6 . y . ) > 0 for 4 E U0 2nd V as in the above, then (20, *) E I;.

We define for a projective functor F of type u

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

2158 LIU

Then, if F v = $Fa, we have S(FV) = U S(F,) and S m ( F V ) C U Sm(F,).

(b) If F is an indecoinposable plojective functor of type a, the11 S m ( F )

consists of only one class in Z,.

In fact, since F = F o Pr(0) and F ( M ( g e ) ) = P ( J ) for some 0 E zF) A

and 6 E U o , it follows from 1.5 (11) combined witli 1.5 (e) that 4 5 @ for

any $ satisfying {6+, ~ ~ 6 % ~ ) > 0. Now, suppose (ke ,$) 6 Sm(F) witli

(F7 = 00 + a + @', it is easy to see tliat cp' 2 $1' witli respect to the order

relation defined in ([2] 1.5). Then, it follows from ([2] lelimia 1.5 (iii)) tliat

cp' E WXb(l:") . therefore, (20, ii.) -' (20.6) .

(c) We have also seen in the above that ( g e , 4 ) in (b) is written properly.

Thus, we have assigned each irideco~llposable projective functor F an element

E = (lie, J ) in So, and if ( 2 0 , g ) is written properly, we have F ( M ( 2 e ) ) =

~(4). -

Conversely, let [ = (Re. $) be an element 111 S,, where 4 = 00 + (T + $'. - -

Since 1 j -2~ = n+$'-% E o+P(II), there is a finite dimensiollal irreducible

highest weight U-module V of type a having 4 - go as an extreme weight.

It is clear that ( g o , 4) E Sm(Fv) . The fiuictor Fv can be decolnposed into

a direct suln of indecoinposable projective functors: Fv = @ F, (see 4.2.1).

It follows from (a) that 0;ie 4) E Sm(F,) for some a, hence, F, 1s assigned

to (go, 4). and the proof is complete.

5.3. In this and uext subsertion we return to the usual situation. q is

either an indeter~ninate or a non-zero complex liumber which is not a root of Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS 2159

unity. If F and G are projective functors of different type, say, F of type 01

and G of 02. it is easy to see that the first part ( (a) and (b)) of the proof of

Theorem 3.3.1 is still valid aud the i m p ij; is still injective. When F = Fv

and G = FL, one has for inaximal

if 01 $ a2 (mod rZ0); dim H O ~ ~ ( , , ( V @ M ( X ) , L @ M ( ~ ) ) =

dim(V* @ L ) ,,-,, , otherwise.

Hence, if 01 and 0 2 are not congruent. we have HO~II~, , ,~(F, G) = 0. If al

and u2 are congruent, the situation is more complicated. Our first result is:

THEOREM 5.3.1. Let F and G be projective functors of type al and uz ,

respectively. Suppose that 01 and 02 are conguent but a1 l r # o2 l r . T11ezl

the space H o i n ~ ~ , , ~ ~ ( F , G) = 0.

It is enough to prove that the space HornFunct (FL(,,+-;;), %Zt;)) O

for any A , p E p + P+(II). By 3.2 (see also [2] 1.3), this space is isomorphic to

Horn i ~ ( u ) , i i u ) , ( L ( ~ ~ +I) @F ?(u). ~ ( 0 2 + 2) @F F ( u ) ) . In order to prove

the theorem we show the following

LEMMA 5.3.2. Let V and L be two U-modules. One has

where the dual action on V* is defined via the antipode S.

Proof of the lenma. Define a map P

- as follows: For any homomorphism p E H O ~ ( ~ ( ~ ) , ? ( ~ ) ) ( L @F F(U), V BF - F ( U ) ) , let P(cp)(v' @ 1 @ u) = X I ; v*(,u&)uk if y ( l @ u) = X I ; ,uk @ U I ; . One

checks that ,B is the required isomorphism (cf. [I] 7.3).

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

LIU

Proof of the theorem. By the above lemma, it is enough t,o prove

for 0 E I';' but olr # 0 arid X E p + P+(II)

);). The11 210 iilust be non-zero and of Uo-weight X - p under the adjoint -

action. Since cp preserves tlie right actiou of F(U), we have

In particular

- - @ E,) = u0Ez for all i. ' ~ ( ~ u , + ~ - ~

Tlie following lemma conlpletes the proof of the theorenl.

LEMMA 5.3.3. T h e r e is no t ro f 0 in ?(u) such that (5.3.1) holds.

Proof. Suppose that LLO # 0 in ?(u) satisfies (5 .3 .1) . Sirlcc ulr # 0.

there exists at least one i such that cr(K,,) = -1. On t,he other hand, there

exists LL E -2P+(IT) s l~ch that uh = I(p~uO E F(U). Then (5 .3 .1) illlplies

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS

Using (5.3.2) one can easily compute

An induction shows

This is impossible because the left lialid side is zero for sufficiently large m

while the right hand side will never be zero. (Recall that q is not a root of

unity and U is an integral domain.)

5.4. The intuitive reason for us to believe that (5.3.1) could not be

true is some "positivity" of the structural constants . Write u g as a lin-

ear conlbination of basis elenlents introduced in 2.3. Let F ~ K E ~ , where

K = K z . . . K2","", be the highest degree term that occur with non-zero

coefficients. Then

E ~ F ~ K E ' = q S F k ~ E i E r + lower degree terms

for some s E Z. So we need only focus on EaEr.

We would like to have (Recall that q is not a root of unity)

E,.Er = q"ErEi + lower degree terms in U+ for some s E Z, (5.4.1)

This is the case in the following example, though the proof is difficult

in general.

Example 5.4.1. Let g = A2 and let a = (€2, €1) (see Example 2.4.1) with

€2 = -1. Then a E Fa0 and a(K,,) = -1. Take the reduced expression

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

2162 LIU

S ~ S Z S ~ for 7110. Then PI = a l . /32 = s la2 = a1 + a 2 and P3 = slszal = a2.

So Ep, = El. Ep, = TlE2 = -El E2 + q-lE2 El and Ep, = TlT2E1 =

E z . Now we have E r = ET' (-El E2 + q - 1 E 2 ~ I ) ' 2 ~ F . There are following

possibilities.

( i ) r l = 0: We have

(iii) rl = 2: We have

E2Er = ((q + q - 1 ) ~ l ~ 2 ~ l - E ~ E ~ ) E ~ ~ E ; ~ (by(1.1.4))

= ((9 + q - ' ) ~ i ( y ~ p , + ~ E I Ea) - E : E ~ ) E ~ E ~ ~

= (q2 + ~ ) E ~ E ~ ~ + ~ E ; ~ + q 2 ~ : ~ 2 ~ g ~ ; 3 = (q2 + + ) E ~ E ~ + ' E ~ ~ + y - ' . 2 f 2 ~ r ~ 2 (by (i)).

(iv) r - 1 2 3: Repeatedly use (i)-(iii).

So we always liave

E2 Er = q" ErE2 + lower degree terms

for some s E Z. This implies

- E2uo = ~ " F ~ K E ' E ~ + lower degree terins.

wllicll verifies Leiiliiiit 5.3.3.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS 2163

Theorem 5.3.1 and the above exaniple show that there may be no natural

transforlnations between two projective functors of different type even if their

restrictions to Verlna modules are isomorphic. This is drastically different

from the classical case.

5.5. The discussion in 5.3 and 5.4 leads us to the following

DEFINITION 5.5.1. Let 01 and a2 be two elenlents in I?;. We say a1 and o2

are equivalent, denoted by a1 02, if they are congruent and 01 l r = a21r

Obviously, if a1 and a2 are equivalent, one has F ~ ( o , + % ) F ~ ( n 2 + ~ ) for

any X E p + P+(II); and the ison~orpliisin can be obtained by composing

F ~ ( 6 1 + ~ ) F ~ ( n 2 - c l + 3 . In this way we can identify projective functors

wit11 equiva1c:lt type.

An i~nlnediate consequence of Theorem 5.3.1 is the following

COROLLARY 5.5.2. Two indecomposable projective functors which are not of

equivalent type are not isoinorphic.

5.6. In this subsection we assume again that q is non-zero co~nplex

lzunlber that is not a root of unity. We can now introduce the following

definition of projective functors for ?(u).

- DEFINITION 5.6.1. (i) A functor F : %zf - M Z f is called a projective

functor for ?(u) if it is isomorphic to a direct summand of the functor

for a finite dilnensional U-module V. FvLG,, - (ii) Let Q E Z ( U ) be a central character. A functor F : M ( 6 ) - M is

called a projective 6-functor for F ( u ) if it is isomorphic to a direct sunlrnand

of the functor F v ( Q ) for a finite dimensional U-module V. where F v ( Q )

denotes the restrictiou of Fv to Z ( 0 ) .

Fix an arbitrary set of representatives T = {a l , 02,. . . , a,) for the equiv-

alence classes of I?;/ N.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

2164 LIU

- It follows from 5.5 that every projective f ~ i i ~ c t o r for F ( U ) is a direct suni

of projective functors of type a with o running over T. Therefore, we can -

clmsify all indecomposable projective functors for F ( U ) as follows.

Recall that we have fixed a dominant weight E A(H) for each ce~itral - character 0 E Z ( U ) and we can write ke = ~ e + % ~ with as E I?; and Xe E IJ*

Let

We say two elei~leiits [ = (ie, 8) and [' = (2;. @ I ) are equivalellt, denoted

by ( wr E l . if and only if 0 = 0'. 3 = a0 + o + $. @' = as + a + $7 for the same

a E T and for p. pr E f ~ * such that pr = 20p for some ro E Wx, = {,u, E W / [ U X e = xH}. Let E be tlie family of equivalence classes: So/ N'. An elemelit,

< = ( i s . $) is sxid t,o be writ,t,en properly if cp = 0s + a + @ for p E fi* suc l~

tallat p 5 W,, (p) . Every element in E can be writtell properly (cf. [2j 3.3).

It is easy to see that Z is the disjoint uilioii of all Z,: IT E T.

THEOREM 5.6.2. For each eleme~lt [ = (j&j, 4) E E, ~ ~ J E I . C is ill1 il~deco~npos-

able projective fullctor FE , a ~mique one up to isorl~orphism, s7dl that

(i) F [ ( M ( X ) ) = O. ifX 6 A ( @ ) .

(ii) If < = (ge.,dl) E E is writtc~l properly. tltllen F E ( M ( g e ) ) = ~ ( 4 ) . The map: [ H FE is a bijectiou of E with the set of isoritllolpllis~n classes

- of indecoinposable projective functors for F(U) .

The det,ails of the proof are left to the reader.

5.7. Tl i~ls far. we have established t,lie projcckive functor theory for

t l ~ c siinply coilnected quant,ized clivelopillg algebras witlloiit ally exceptions

at all! I t Ilas as ~niuiy ~pplications as in t,lie classical case (sce [2]). For

example, in ((131 5.11). it is applied to prove that for certaiii pairs of central Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPlNG ALGEBRAS 2165

characters (0,0'), the categories M w ( 0 ) and Mm(O') are equivalent and the

equivalence is given by certain projective functors ([I31 tlieorelii 5.11.1). It is

also applied to prove that the lattice of two sided ideals in ?(u) containing

Je is isomorphic to tlie submodule lattice of M ( f ) , where 1;. is a certain

weight in A(0) (1131 theorem 5.11.2). It may also be applied to tlie study of

the multiplicities in Verma modules and to that of Harish-Cliandra modules.

However, these are beyond the scope of this paper.

Appendix

The liiaiii idea of tlie proof of Theorem 2.2.1 is due to Joseph and Letzter

(see [7] and [B]), there is a complete proof for tlie adjoint quantized enveloping

algebras (with some exceptions) in [13], we will present a proof for the simply

coimected quailtized enveloping algebras in this appendix.

Let us first fix sollie notations about filtered and graded rings and mod-

ules.

Let A be a 2-filtered ring (or module), and the filtration is exliaustive,

. . 5 A-1 C_ A. 2 Al c . . . 5 A, c A,+, . . . (a 00 +ns

with A = A i Let Gr rA = @ A,/A,-1 be tlie associated graded 2=-00 z = - a 2

ring (or module). Let I3 be a subspace of A, then

+m

is an exhaustive filtration of B and GrrB = $ (B n A,)/(B n is n=-cc

canonically identified as a subspace of Gr7A. For any n E A! if a E A, and

a 4 An-ll then Gra denotes the image of a in A,/A,-1: a l~olnogelleous

elenleiit in GrFA. Similarly. if D is a subspace of QrFA (not necessarily

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

LIU

is ail exhaustive filtration of D and GrFD = $,D,,/D,-1 is canonically

identified as a graded suhspace of GrFA. If b E GTFA (not necessarily

homogeneous) let Grb denote the leading llolrlogelleous component of b. i.e..

tlie lligllest degree term of b.

We define an ad-invariant filtration for U as follows.

Notice that in the expressioll of the fundamental weight wi as a rational

coinbinatioii of the siillple soots, the sum of the coefficients may be a fraction

with the denoniinator 2 or 3 for some types, namely, with 2 for type B,, C,,

D,. E7 and with 3 for type E6. So. ill order to have I{&,, haviug integral

degree for all i. we set deg E; = 0, cleg Fi = 6, deg K,, = -6 a i d deg Kg,' =

G f o r i = l . 2 , . . . , n.

Siricc the Serre relat,ions (1.1.4) and (1.1.5) are homogeneous, deg is

well defined on U+. U - and UO. respectively.

By the triangular decomposition, each ,u E U can be written uniquely

as u = x i ,uiu:.uT wit,h u.;, and u: rnononiials in U - . U0 and U + .

respectively. We define deg (,u,u:ut) = deg u; + deg Z L ~ + deg$ and

deg u = nlaxi{deg (U,U;U~)}. Let

Tlieil. a siinilar arglili~erlt xs ill ([7] 2.2 and 2.3) shows that {FnV},Ez

forrm a filtration of U. and this filtsat,iol~ is preserved by t,lie itd,joint i~rtioil,

i.e.. FT"U is a submodule of U under tlle adjoint act,ioli for each 97r E 2.

Using the same iiletliod as in ( [7] 2.2- 2.3) , one proves that t,llis filtration

gives rise to a graded algel~ra

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS

and it is an integral domain (see [G] 4.10).

As in U, the multiplication map also gives an isomorphisn~ of U- @F

U0 8~ U + onto GrrU as vector spaces.

Another advantage of this filtration is that F ( U ) is positively and finitely

filtered, i.e., F m ( U ) = 3 ( U ) n F m U are 0 for all m < 0, and are finite

dimensional for all m 2 0. (See [7] and [13].)

Let Y = Y(U) = GrrZ(U). Let MaxlY(U) denote the set of ideals of

codimension 1 in Y(U). It is the same as HornF-,l,(Y(U),F). Let Y+ E

MaxlY(U) be the ideal generated by all homogeneous elements of deg > 0.

LEMMA Al. For all Yx f MaxlY(U), one has

Remark. We want the equality here: not just an iso~norphism of graded

vector spaces as in ([7] 7.1).

Proof. As in ([7] 7.1), the multiplication map

is an isomorphism of vector spaces. Since Y, E MaxlY(U) and only the

scalars have degree 0, we actually have

The rest of the proof follows similarly as in (171 7.1). Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

For any Yx E MaxlY(U). Let J, = (GTFF(U))Y,.

LIU

COROLLARY A2. For all Yx E MaxlY(U), one has

Proof. Using the same argumellt as in ([7] 7.2) but with the ecluality in

the above lemma, which is necessary even to get an isomorphism of graded

vector spaces, we actually get the equality in the corollary, instead of an

iso~iiorpliislll of graded vector spaces. And the equality is needed in the

proof of tlie next result.

Since Grr3(CT) is a seinisilllple U module. the subinodule (Grr3(U))Y+

admits a graded U subinodule ?-t as a complement.

PROPOSITION A3. The n~ultiplication map

% @ F Y ( U ) ---t GTFF(U)

h @ y i---, h, y

is an isomolphism of adU moduIes.

For tlie proof. we just modify that of ([7] 7.3). Surjectivity follows

similarly as in ( [ 5 ] 8.2.2). For injectivity. suppmc: h, @ yz E 7-i & Y(U) Z

so that C h,y, = 0 in Gr3F(U). Without loss of generality, assnme that

the h,'s are linearly iiidepeldent and all y, # 0. For any Y, E MaxlY(U)

corresponding to x E HoKI~-,~,(Y(U). F), we have

So. C h,y, = O iinplies C ~(y . ) t l , t J , since Y(U) is a polynoo,ial algebra I Z

(See [7] or Theorem 4.15 in [13]). lience. there exists a x such that ~ ( y , ) # 0

for some 9,. So, (*) inlplies

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTlZED ENVELOPING ALGEBRAS 2169

Since 7-l is a complement of (GrrF(U))Y+ and G ~ F J , = GTFF(U)Y+, we

must have 2 n Jx = 0, (The equality in Corollary A1 is needed to get this.),

so, (**) is impossible.

Recall that F m ( U ) = 3 ( U ) n FmU, for all m E N. Since 3 ( U )

is a semisimple adU module , for each m 2 0, there exists an adU-module

complement of Fm-'(U) in Fm(U) , denoted by 3,(U). Then F ( U ) =

e m E ~ F m ( U ) as adU modules and the grading

induces an isomorpl~ism of adU modules

And the direct sum ,B of all the Pm's is an isomorpllism of adU modules

Let H = ,F1(H). Then, as in ( [ 5 ] 8.2.4). one shows the following

theorem, whicll is called by Joseph and Letzter the Kostant's separation of

variables for quantized enveloping algebras. (see [7]).

THEOREM A4. T h e multiplication map

H @ F Z ( U ) ---+ F ( U )

h @ z hz

is an isomorphisnl of U-modules.

Using the formula for the determinant det(H,) in Proposition 1.3.2 and

the same argument as in [8], one shows (For the details, see [8] 9.1)

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

2170

A

THEOREM A5. For. all X E UO, one has

H n A n n q U ) M ( X ) = 0.

LIU

Consequently,

Proof of Theorem 2.2.1: Obviously, we have

To show the reverse inclusion, it is enough to show that for any finite dimen-

sional simple U-module L ( z + 3 one has

Indeed, by Theorel11 A5, [ F ( U ) / A n n F ( u l M ( X ) : L(?+ a] 2 [H : L(& + a] =

[ T ( U ) / . F ( U ) kerOi : L(G+a] . ( The last equality follows from Theorem A4.)

In general case, it is obvious that

b Suppose u = x,=l T ( X ~ ) - ' U , Y ~ E AllnKpC,~p(U)7, M ( X ) , where XI,. . . , Xk E

-R+(n) . We rewrite u in the for111

2=1 sfi

It is easy to see t,liat ,u annihilat,es hd(X) if and only if ,u' does. Then the

above argument shows that u' E F ( U ) ker Ox. Hence u E K ~ ' T ( u ) ~ , ker Ox.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

QUANTIZED ENVELOPING ALGEBRAS 2171

ACKNOWLEDGMENTS

I am truly grateful to Jens Jantzen for his instruction and inspiration

I have received through numerous discussions with him. And I would also

like to tliank the NSF for their support through a group grant wliile I was

writing this paper.

REFERENCES

[I] H. H. Andersen, J . C. Jantzen and W. Soergel, Representations of quan-

turn groups at a p-th root of uni ty and of semisimplc groups i n charac-

teristic p: independence of p, preprint.

[2] J. N. Bernstein and S. I. Gelfand, Tensor products o f finite and infi-

nite dimensional representations of semisimple Lie algebras, Compositio

Mathernatica, Vol. 41. Fasc.2, 1980, 245-285.

[3] J. N. Bernstein, I. M. Gelfand and S. I. Gelfand, O n a c a t e g o v of g

modules, Funct. Anal. Appl. 10 (1976), 87-92.

[4] C. DeConcini and V. G. Kac, Representations of quantum groups at roots

of 1, In Operator algebras, unitary representations, enveloping algebras

and i m a r i a n t theory, Progress in Math.. Birkliauser, Bostoil 92 (1990).

471-506.

[5] J . Dixmier, Algkbres envelopantes, Caliiers Scie~itifiques no. 37,

Gauthier-Villars, Paris 1974.

[GI A. Joseph and G. Letzter, Localfiniteness of the adjoint action for quan-

tized enveloping algebras, J . Algebra, 153 (1992), 289-318.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014

LIU

[7] A. Joseph axid G. Letzt,er, Sepciratzon of vanalrles for qunntzzed envelop-

zng algehms, prepriut.

[8] G. Letzter, Representation theory for quai~tized enveloping algebras,

preprint.

[9] G. Lusztig, Quantum deformcltioi~s of certain simple modules over En-

d o p i n g Algebms, Adv. in Matli. 70 (1988). 237--249.

[lo] G. Lusztig. Quu7~tuin g r o u p ( (~t roots of 1. Geom. Dtd. 35 (1990).

89-114.

[I l l G. Lusztig, Canonicud basis arising from qua~~tizerl envelopin,g algebras.

J . Amer. Math. Soc. 3 (1990), 447--498.

[I31 C . Liu. Tensor pl-orlucts offinit,e and infinite din~ensionnl representc~tzons

of qu(mtizet1 enveloping cl,lgdms, P11.D. thesis, Uiliversit,y of Oregon,

1993.

Received: February 1993

Revised: August 1993

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

05:

29 1

7 O

ctob

er 2

014