Pacific Journal of Mathematics

Pacfc Journal of Mathematcs IRREDUCIBLE REPRESENTATIONS FOR THE ABELIAN EXTENSION OF THE LIE ALGEBRA OF DIFFEOMORPHISMS OF TORI IN DIMENSIONS GREATER THAN CUIPO JIANG AND QIFEN JIANG Volume 23 No. May 27

PACIFIC JOURNAL OF MATHEMATICS Vol. 23, No., 27 IRREDUCIBLE REPRESENTATIONS FOR THE ABELIAN EXTENSION OF THE LIE ALGEBRA OF DIFFEOMORPHISMS OF TORI IN DIMENSIONS GREATER THAN CUIPO JIANG AND QIFEN JIANG We classfy the rreducble weght modules of the abelan extenson of the Le algebra of dffeomorphsms of tor of dmenson greater than, wth fnte-dmensonal weght spaces.. Introducton Let W ν+ be the Le algebra of dffeomorphsms of the (ν+)-dmensonal torus. If ν =, the unversal central extenson of the complex Le algebra W s the Vrasoro algebra, whch, together wth ts representatons, plays a very mportant role n many areas of mathematcs and physcs [Belavn et al. 984; Dotsenko and Fateev 984; D Francesco et al. 997]. The representaton theory of the Vrasoro algebra has been studed extensvely; see, for example, [Kac 982; Kaplansky and Santharoubane 985; Char and Pressley 988; Matheu 992]. If ν, however, the Le algebra W ν+ has no nontrval central extenson [Ramos et al. 99]. But W ν+ has abelan extensons whose abelan deals are the central parts of the correspondng torodal Le algebras; see [Berman and Bllg 999], for example. There s a close connecton between rreducble ntegrable modules of the torodal Le algebra and rreducble modules of the abelan extenson ; see [Berman and Bllg 999; Eswara Rao and Moody 994; Jang and Meng 23], for nstance. In fact, the classfcaton of ntegrable modules of torodal Le algebras and ther subalgebras depends heavly on the classfcaton of rreducble representatons of and ts subalgebras. See [Bllg 23] for the constructons of the abelan extensons for the group of dffeomorphsms of a torus. In ths paper we study the rreducble weght modules of, for ν. If V s an rreducble weght module of some of whose central charges c,..., c ν are nonzero, one can assume that c,..., c N are -lnearly ndependent and c N+ = = c ν =, where N. We prove that f N, then V must have weght MSC2: prmary 7B67, 7B65; secondary 7B68. Keywords: rreducble representaton, abelan extenson, central charge. Work supported n part by NSF of Chna, grants No. 2776 and No. 579. 85

86 CUIPO JIANG AND QIFEN JIANG spaces whch are nfnte-dmensonal. So f all the weght spaces of V are fntedmensonal, N vanshes. We classfy the rreducble modules of wth fntedmensonal weght spaces and some nonzero central charges. We prove that such a module V s somorphc to a hghest weght module. The hghest weght space T s somorphc to an rreducble ( ν +W ν )-module all of whose weght spaces have the same dmenson, where ν s the rng of Laurent polynomals n ν commutng varables, regarded as a commutatve Le algebra. An mportant step s to characterze the ν -module structure of T. It turns out that the acton of ν on T s essentally multplcaton by polynomals n ν. Therefore T can be dentfed wth Larsson s constructon [992] by a result n [Eswara Rao 24]. That s, T s a tensor product of gl ν -module wth ν. When all the central charges of V are zero, we prove that the abelan part acts on V as zero f V s a unformly bounded -module. So the result n ths case s not complete. Throughout the paper,, + and denote the sets of complex numbers, postve ntegers and negatve ntegers. 2. Basc concepts and results Let ν+ = [t ±, t±,..., t± ν ] (ν ) be the rng of Laurent polynomals n commutng varables t, t,..., t ν. For n =(n, n 2,..., n ν ) ν, n, we denote t n tn tn ν ν by t n tn. Let be the free ν+ -module wth bass {k, k,..., k ν } and let d be the subspace spanned by all elements of the form ν r t r tr k, for (r, r) = (r, r,..., r ν ) ν+. = Set = /d and denote the mage of t r tr k stll by tself. Then s spanned by the elements {t r tr k p p =,,..., ν, r, r ν } wth relatons ν (2-) r p t r tr k p =. p= Let be the Le algebra of dervatons on ν+. Then { ν = f p (t, t,..., t ν )d p f p (t, t,..., t ν ) ν+ }, p= where d p = t p / t p, p =,,..., ν. From [Berman and Bllg 999] we know that the algebra admts two nontrval 2-cocycles wth values n : ν τ (t m tm d a, t n tn d b ) = n a m b m p t m +n t m+n k p, p=

ABELIAN EXTENSION OF LIE ALGEBRA OF DIFFEORMORPHISMS OF T n 87 τ 2 (t m tm d a, t n tn d b ) = m a n b ν p= m p t m +n t m+n k p. Let τ = µ τ + µ 2 τ 2 be an arbtrary lnear combnaton of τ and τ 2. Then the correspondng abelan extenson of s wth the Le bracket =, (2-2) [t m tm d a, t n tn k b ] = n a t m +n t m+n k b + δ ab The sum ν p= m p t m +n t m+n k p, [t m tm d a, t n tn d b ] = n a t m +n t m+n d b m b t m +n t m+n d a ( ν ) ( ν ) h = k d = = + τ(t m tm d a, t n tn d b ). s an abelan Le subalgebra of. An -module V s called a weght module f V = λ h V λ, where V λ = {v V h v = λ(h)v for all h h}. Denote by P(V ) the set of all weghts. Throughout the paper, we assume that V s an rreducble weght module of wth fnte-dmensonal weght spaces. Snce V s rreducble, we have k V = c, where the constants c, for =,,..., ν, are called the central charges of V. Lemma 2.. Let A = (a j ) (, j ν) be a (ν+) (ν+) matrx such that det A = and a j. There exsts an automorphsm σ of such that σ (t m k j ) = ν a pj t m AT k p, σ (t m d j ) = p= where t m = t m tm, B = (b j ) = A. ν b jp t m AT d p, j ν, p= 3. The structure of V wth nonzero central charges In ths secton, we dscuss the weght module V whch has nonzero central charges. It follows from Lemma 2. that we can assume that c, c,..., c N are -lnearly ndependent,.e., f N = a c =, a, then all a ( =,..., N) must be zero,

88 CUIPO JIANG AND QIFEN JIANG and c N+ = c N+2 = = c ν =, where N. For m = (m, m), denote t m tm by t m as n Lemma 2.. It s easy to see that V has the decomposton V = m ν+ V m, where V m = {v V d (v) = (γ (d ) + m )v, =,,..., ν}, wth γ P(V ) a fxed weght, and m = (m, m,..., m ν ) ν+. If V has fnte-dmensonal weght spaces, the V m are fnte-dmensonal, for m ν+. In Lemmas 3. 3.6 we assume that V has fnte-dmensonal weght spaces. Lemma 3.. For p {,,..., ν} and = t m k p, f there s a nonzero element v n V such that t m k p v =, then t m k p s locally nlpotent on V. Lemma 3.2. Let t m tm k p be such that m = (m, m) =, and there exsts a N such that m a = f N < p ν. If t m tm k p s locally nlpotent on V, then dm V n > dm V n+ m for all n ν+. Proof. Case : p {,,..., N}. We frst prove that dm V n dm V n+ m for all n ν+. Suppose dm V n = m, dm V n+ m = n. Let {w, w 2,..., w n } be a bass of V n+ m and {w, w 2,..., w m } a bass of V n. We can assume that m a = for some a ν dstnct from p, where m = (m, m) = (m, m,..., m ν ). Snce t m k p s locally nlpotent on V and V n+ m s fnte-dmensonal, there exsts k > such that (t m k p ) k V n+ m =. Therefore (t m d a ) k (t m k p ) k (w, w 2,..., w n ) =. On the other hand, by nducton on k, we can deduce that Therefore ( k t m k p = (t m d a ) k (t m k p ) k = k = k! k!! (k )! (k )! m a c p (t m k p ) k (t m d a ) k. k! k!! (k )! (k )! m a c p (t m k p ) k (t m d a ) k ) t m d a (w,w 2,...,w n ) = k! m k a ck p (w, w 2,..., w n ). Assume that ( k k! k! =! (k )! (k )! m a c p (t m k p ) k (t m d a ) k ) t m d a (w, w 2,..., w n ) = (w, w 2,..., w m )C, wth C m n, and that (3-) t m k p (w, w 2,..., w m ) = (w, w 2,..., w n )B,

ABELIAN EXTENSION OF LIE ALGEBRA OF DIFFEORMORPHISMS OF T n 89 wth B n m. Then BC = k! m k a ck p I. Ths mples that m n. So dm V n dm V n+ m for all n ν+. Also, by (3-) and the fact that r(b) = n, we know that m > n f and only f there exsts v V n such that t m k p v =. Snce t m k p s locally nlpotent on V, there exst an nteger s and w V n+s m such that (t m k p ) w =. Therefore (t m k p )t m k p w = t m k p (t m k p w) =. If t m k p w =, by the proof above, dm V n+s m m < dm V n+s m, contradctng the fact that dm V n+s m m dm V n+s m. Therefore (t m k p ) r w = for all r. Snce (t m k p ) s t m k p w = t m k p (t m k p ) s w = and (t m k p ) s w V n, t follows that there s a nonzero element v n V n such that t m k p v =. Thus n < m. Case 2: N < p ν. The proof s smlar to that of case, but we have to consder t m d p and t m k p nstead and use the -lnear ndependence of c,..., c N. Lemma 3.3. Let = t m k p and = t n k p be such that (m,..., m N ) =, (n,..., n N ) = f N < p ν, where m = (m, m,..., m ν ). () If t m k p s locally nlpotent on V, t m k q s locally nlpotent for q =,,..., ν. (2) If both = t m k p and = t n k p are locally nlpotent on V, then t m+ n k p s locally nlpotent. (3) If = t m+ n k p s locally nlpotent on V and (m + n,..., m N + n N ) = f N < p ν, then t m k p or t n k p s locally nlpotent. Lemma 3.4. For p ν, let = t m k p be such that (m,..., m N ) =, where m = (m, m,..., m ν ). Then t m k p or t m k p s locally nlpotent on V. Proof. The proof occupes the next few pages. We frst deal wth the case p N. Wthout losng generalty, we can take p =. Suppose the lemma s false. By Lemma 3.2, for any r ν+ we have dm V r+ m = dm V r = dm V r m, t m k V r = V r+ m, t m k V r = V r m. Fx r = (r, r) ν+ such that V r =. Let {v,..., v n } be a bass of V r and set v (k m) = c t k m k v, =, 2,..., n,

9 CUIPO JIANG AND QIFEN JIANG where k \ {}. Then {v (k m), v 2 (k m),..., v n (k m)} s a bass of V r+k m. Let B () m, m, B() m, m n n be such that t m k (v ( m), v 2 ( m),..., v n ( m)) = (v, v 2,..., v n )B () m, m c, t m k (v ( m), v 2 ( m),..., v n ( m)) = (v, v 2,..., v n )B () m, m c. Snce t m k and t m k are commutatve, t s easy to deduce that B () m, m = B() m, m. By Lemma 3., B () m, m s an n n nvertble matrx. Clam. B () m, m does not have dstnct egenvalues. Proof. Set c = /c. To prove the clam, we need to consder ct m k ct m k λ d, where λ. As n the proof of Lemma 3., we can deduce that f there s a nonzero element v n V such that (ct m k ct m k λ d)v =, then ct m k ct m k λ d s locally nlpotent on V. On the other hand, we have (ct m k ct m k λ d) l (v, v 2,..., v n ) = (v, v 2,..., v n )(B () m, m λ d)l. Therefore the clam holds. For p {, 2,..., ν}, let C p m,, C p m, m n n be such that Snce we have t m k p (v, v 2,..., v n ) = (v ( m),..., v n ( m))c (p) m,, t m k p (v ( m),..., v n ( m)) = (v, v 2,..., v n )C (p) m, m. c t m k t m k p (v, v 2,..., v n ) = t m k p c t m k (v, v 2,..., v n ), (3-2) C (p) m, m = B() m, m C(p) m,. Furthermore, by the fact that and t m k t m k t m k p (v, v 2,..., v n ) = t m k p t m k t m k (v, v 2,..., v n ) c c c c t m k q c t m k t m k p = t m k p c t m k t m k q,

ABELIAN EXTENSION OF LIE ALGEBRA OF DIFFEORMORPHISMS OF T n 9 we deduce that (3-3) B () m, m C(p) m, = C(p) m, B() m, m, C(p) m, C(q) = m, C(q) m, C(p) m,, p, q ν. Hence there exsts D n n such that {D B () m, m D, D C (p) D p ν} are m, all upper trangular matrces. If we set and then (w, w 2,..., w n ) = (v, v 2,..., v n )D w (k m) = c t k m k w, n, k \ {}, c t k m k (w ( m), w 2 ( m),..., w n ( m)) = (w,..., w n )D B () m, m D, t m k p (w, w 2,..., w n ) = (w ( m),..., w n ( m))d C (p) m, D. So we can assume that B () m, m, C(p), and C(p) m, m, m, for p ν are all nvertble upper trangular matrces. Furthermore, because ( ) t m l(v k p t m k c λ d, v 2,..., v n ) = (v, v 2,..., v n )(C (p) m, m λ d)l, the argument used n the proof of the clam shows that C (p) m, m also does not have dstnct egenvalues. For p N, set B (p) m, m = c p C (p) m, m and for p N denote by λ p the egenvalue of B (p) m, m. Let A (a) and A(a) k m, k m,k 2 m, for a ν and k, k, k 2 \ {}, be such that t k m d a (v, v 2,..., v n ) = (v (k m), v 2 (k m),..., v n (k m))a (a) t k m d a (v (k 2 m), v 2 (k 2 m),..., v n (k 2 m)) k m,, = (v (k m + k 2 m),..., v n (k m + k 2 m))a (a) k m,k 2 m. Case : ν >. Snce t m k = t m t m k =, t follows that there exsts a ν such that m a =, where m = (m, m 2,..., m ν ). Let b {,..., ν} be such that a = b. Consder (3-4) [t m d a, c t m k ] = m a c k, [t m d a, t m k b ] = m a k b. Case.: There exsts b {,,..., ν} such that b =, a and c b =. Then A (a) m, m = B() m, m A(a) m, + m a I, A (a) m, m C(b) m, = C(b) m, m A(a) m,.

92 CUIPO JIANG AND QIFEN JIANG By (3-2) and (3-3), A (a) + m m, a B () m, m = C (b) m, A(a) m, C(b) m,. B () m, m But the sum on the left-hand sde cannot be smlar to A (a), snce m m, a = and s an nvertble upper trangular matrx and does not have dfferent egenvalues. Thus ths case s excluded. Case.2: c b = for all b {,,..., ν}, b =, a. By (3-4) and (3-2), we have B () m, m A(a) m, B() m, m + ma B () m, m ma B (b) m, m = B () m, m C(b) m, A(a) m, C(b) m, B () m, m. (I) There exsts b = and a such that λ = λ b. Then m a B () m, m ma B (b) m, m s an nvertble upper trangular matrx and does not have dfferent egenvalues. As n case., we deduce a contradcton. (II) λ = λ b for all b {,..., ν} dstnct from a. (II.) Suppose frst that c a = (n ths case N = ν, a = ν) or c a = and λ a = λ (n ths case N = ν). Snce ν p= m pt m k p =, we have So ν p= ν p= m p t m k p c t m k =. m p C (p) m, m =, and therefore ν m p c p =, p= whch contradcts the assumpton that c,..., c N are -lnearly ndependent. (II.2) Now suppose c a =, λ a = λ and there exsts b = and a such that m b =. We deduce a contradcton as n case.2(i) by nterchangng a by b. (II.3) Suppose c a =, λ a = λ and m b = for all b {,..., ν} dstnct from a. Then m c λ + m a c a λ a =. The proof of ths case s the same as n case 2.2 below. Case 2.: ν =. In ths case a =. Case 2.: c a =. Snce [t m d, t m k ] = [t m k, t m d ] =, we have A () m, m = B() m, m A() m,, A() m, m = B() m, m A() m,. Therefore [t m d, t m d ](v, v 2,..., v n ) = (v, v 2,..., v n )B () [ m, m A (), ] m, A() m,.

ABELIAN EXTENSION OF LIE ALGEBRA OF DIFFEORMORPHISMS OF T n 93 At the same tme, we have [t m d, t m d ] = 2m d + m 2 ( µ + µ 2 )(m k + m k ), where τ = µ τ + µ 2 τ 2 as above. So (3-5) B () m, m [A() m,, A() m, ] = ( 2m (γ (d )+r )+m 2 ( µ +µ 2 )(m c +m c ) ) I, where γ s the weght fxed above. Snce γ s arbtrary, we can choose t such that 2m (γ (d ) + r ) + m 2 ( µ + µ 2 )(m c + m c ) =. But B () m, m s an nvertble trangular matrx and does not have dfferent egenvalues, n contradcton wth (3-5). Case 2.2: c a =. Snce we have [t m d, t m k ] = m k, [t m d, t m k ] = m k and [t m d, t m k ] = m k, [t m d, t m k ] = m k, [k t m d + k t m d, t m k ] = [k t m d + k t m d, t m k ] =. Therefore and k A () m, m + k A () m, m = B() m, m k A () m, m + k A () m, m = B() m, m ( k A () + k m, A () ( k A () + k m, A () m, m, ), ), [k t m d + k t m d, k t m d + k t m d ](v,..., v n ) = (v,..., v n )B () [ m, m k A () + k m, A (), k m, A () + k m, A () ] m,. At the same tme, we have [k t m d + k t m d, k t m d + k t m d ] = 2(m c + m c )(c d + c d ) (m c + m c ) 3 (µ µ 2 ) d. Snce c and c are -lnearly ndependent, we know that m c + m c =. As n case 2., we deduce a contradcton. Ths concludes the frst part of the proof. We next turn to the second major case, N < p ν. If N or N =, we have (m,..., m ν ) =, and the lemma follows from the frst part and Lemma 3.3. Otherwse, let t m k p = t m k p. Set = m tm d k and W = U( )v, where v V s s a homogeneous element. Snce c =, the sets {dm W (n,)+ s n } are not unformly bounded. But f nether t m k p

94 CUIPO JIANG AND QIFEN JIANG nor t m k p s locally nlpotent, then t k p and t k p are not locally nlpotent. So by Lemmas 3.2 and 3., dm V (n,)+ s = dm V s for all n, whch s mpossble snce dmv (n,)+ s dm W (n,)+ s. Ths proves Lemma 3.4 For p N, consder the drect sum t m p p d p k p, m p whch s a Vrasoro Le subalgebra of. Snce c p =, t follows from [Matheu 992] that there s a nonzero v p V r for some r ν+ such that (3-6) t m p p d p v p = for all m p + or (3-7) t m p p d p v p = for all m p. Lemma 3.5. If v p V r satsfes (3-6), the sets {t m p p k q m p +, q =,, 2,..., ν, q = p} are all locally nlpotent on V. Lkewse for (3-7), wth + replaced by. Proof. We only prove the frst statement. Suppose t s false; then by Lemma 3.3 t p k q s not locally nlpotent on V for some q {,,..., ν}, q = p. By Lemma 3.4, tp k q s locally nlpotent. Therefore there exsts k + such that So t 2 p d p(t p (t p k q) k v p =, (t p k q) k v p =. k q) k v p = kt p k q (t k q) k v p + (t k q) k tp 2 d pv p = kt p k q (t p k q) k v p =. Ths mples that t p k q s locally nlpotent, a contradcton. Lemma 3.6. If v p V r satsfes (3-6), the sets p {t m k p m = (m,..., m ν ) ν+, m p + } are all locally nlpotent on V. Lkewse for (3-7), wth + replaced by. Proof. Agan we only prove the frst statement. Wthout loss of generalty, we assume that p =. Let be the subspace of spanned by elements of whch are locally nlpotent on V. If t m k, for any m ν \{}, s not locally nlpotent on V, the lemma holds thanks to Lemmas 3.3 and 3.5. Suppose {t m k m ν } = {}. By Lemmas 3.2, 3.3 and 3.5, f t m k, then t m k /, and t m tm k for all m >. Case : Suppose t m t m k for any t m k. Then the lemma s proved. p

ABELIAN EXTENSION OF LIE ALGEBRA OF DIFFEORMORPHISMS OF T n 95 Case 2: Suppose there exsts = t m k such that t t m k /. Snce m = (m,..., m ν ) =, we can assume that m a = for some a {, 2,..., ν}. Let V r be such that dm V r = mn{dm V s V s =, s ν+ }. Case 2.: Assume t t m k / for any >. Let l + and consder (3-8) l = a t t m k t t m k v =, where v V r \{}. By Lemma 3.4, {t tm k, t tm k + }. So by Lemma 3.2, we have t tm k V r = t tm k V r = t tm d p V r = t tm d p V r =, +, p ν. Let j {,,..., l}. From (3-8) we have Therefore t j t m d a t j tm d a ( l = l = a t t m k t t m k )v =. a ( m a )t j k ( m a )t j k v = a j m 2 a c2 v =. So a j =, j =,,..., l. Ths means {t t m k t t m k )v l} are lnearly ndependent. Snce l can be any postve nteger, t follows that V r (,2m) s nfnte-dmensonal, a contradcton. Case 2.2: Assume there exsts l + such that t l t m k /, t l t m k. (I) Assume that t l t m k for any +. Let s > and consder s = a t l tm k t m k v =. Smlar to the proof above, we can deduce that V r (l,) s nfnte-dmensonal, n contradcton wth the assumpton that V has fnte-dmensonal weght spaces. (II) Assume there exsts s + such that t l t m k, t l t 2m k,..., t l t s m k, t l t (s +)m k /. Then there exst s 2, s 3,..., s k,... such that s s for = 2, 3,..., k,... and t l t( s s 2 s )m k, t l t( s s 2 s 2)m k,...,

96 CUIPO JIANG AND QIFEN JIANG t l t( s s 2 s s )m k, t l t( s s 2 s s )m k /. Assume that ( s a t l tm k t m k + Let = + + s 3 = s k = s 2 = a s +t 2l t (s +)m k t l t (s +)m k a s +s 2 +t 3l t (s +s 2 +)m k t 2l t (s +s 2 +)m k + a s + +s k +t kl t (s + +s k +)m k t (k )l t (s + +s k +)m k )v =. t jm d a t l t jm d a, j s, t l t(s + j)m d a t 2l t (s + j)m d a, j s 2,..., t (k )l t (s +s 2 + +s k + j)m d a t kl t (s +s 2 + +s k + j)m d a, j s k act on the two sdes of the above equaton respectvely. By Lemma 3.4, we deduce that a =, for =, 2,..., s, and that a s + +s j + = for =, 2,..., s j, 2 j k. Snce k can be any postve nteger, t follows that V r (l,) s nfnte-dmensonal, whch contradcts our assumpton. The lemma s proved. Lemmas 3. through 3.6 mmedately yeld the followng result. Theorem 3.7. Let V be an rreducble weght module of such that c,..., c N are -lnearly ndependent and N. Then V has weght spaces that are nfntedmensonal. Let + = ν t [t, t ±,..., t± ν ]k p ν p= p= p= = ν t [t, t±,..., t± ν ]k p ν = ν [t ±,..., t± ν ]k p ν p= p= t [t, t ±,..., t± ν ]d p, p= t [t, t±,..., t± ν ]d p, [t ±,..., t± ν ]d p. Then = +.

ABELIAN EXTENSION OF LIE ALGEBRA OF DIFFEORMORPHISMS OF T n 97 Defnton 3.8. Let W be a weght module of. If there s a nonzero vector v W such that + v =, W = U( )v, then W s called a hghest weght module of. If there s a nonzero vector v W such that v =, W = U( )v, then W s called a lowest weght module of. From Lemmas 3.2 and 3.6, we obtan: Theorem 3.9. Let V be an rreducble weght module of wth fnte-dmensonal weght spaces and wth central charges c =, c = c 2 = = c ν =. Then V s a hghest or lowest weght module of. In the remander of ths secton we assume that V s an rreducble weght module of wth fnte-dmensonal weght spaces and wth central charges c =, c = = c ν =. Set { {v V + v = } fv s a hghest weght module of, T = Then T s a -module and {v V v = } fv s a lowest weght module of. V = U( )T or V = U( + )T. Snce V s an rreducble -module, T s an rreducble -module. T has the decomposton T = m ν T m, where m = (m, m 2,..., m ν ), T m = {v T d v = (m + µ(d ))v, ν} and µ s a fxed weght of T. As n the proof n [Jang and Meng 23; Eswara Rao and Jang 25], we can deduce: Theorem 3.. () For all m, n ν, p =, 2,..., ν, we have dm T m = dm T n, t m k p T =, t m k (v (n),..., v m (n)) = c (v (m + n), v 2 (m + n),..., v n (m + n)), t m d (v (n), v 2 (n),..., v n (n)) = µ(d )(v (m + n), v 2 (m + n),..., v n (m + n)), where {v (),..., v m ()} s a bass of T and v (m) = c t m k v (), for =, 2,..., m.

98 CUIPO JIANG AND QIFEN JIANG (2) As an ( ν ν )-module, T s somorphc to Let F α (ψ, b) = V (ψ, b) [t ±,..., t± ν ] for some α = (α,..., α ν ), ψ, and b, where ν = [t ±,..., t± ν ], ν s the dervaton algebra of ν, and V (ψ, b) s an m-dmensonal, rreducble gl ν ( )-module satsfyng ψ(i ) = b d V (ψ,b) and ν t r d p (w t m ) = (m p + α p )w t r+m + r ψ(e p )w t r+m for w V (ψ, b). = M = Ind + + T or M = Ind + T. Theorem 3.. Among the submodules of M ntersectng T trvally, there s a maxmal one, whch we denote by M rad. Moreover V = M/M rad. 4. The structure of V wth c = = c ν = Assume that V s an rreducble weght module of wth fnte-dmensonal weght spaces and c = = c ν =. Lemma 4.. For any t r k p, t r k p or t r k p s locally nlpotent on V. Lemma 4.2. If V s unformly bounded, t r k p s locally nlpotent on V for any t r k p. Proof. For t r k p, by Lemma 4., t r k p or t r k p s nlpotent on V m for all m ν+. Snce V s unformly bounded,.e., max{dm V m m ν+ } <, there exsts N + such that (t r k p t r k p ) N V =, (t r k p t r k p ) N V = If the lemma s false, we can assume that t r k p s not locally nlpotent on V. Therefore for any = v V, we have t r k p v =. So (t r k p ) N V =. Let t 2 r d q be such that p = q and r q =. By the fact that [t 2 r d q, t r k p ] = r q t r k p, we deduce that t r k p (t r k p ) N V =, a contradcton. Lemma 4.3. If there exsts = v V such that t m k p v = for all m ν+ and p ν. Then (V ) =. Proof. Ths follows from (2-2), snce s commutatve and V s an rreducble -module. Theorem 4.4. If V s unformly bounded, t r k p V vanshes for any t r k p.

ABELIAN EXTENSION OF LIE ALGEBRA OF DIFFEORMORPHISMS OF T n 99 Proof. Let = t k p. If t k p V =, t s easy to prove that (V ) =. If t k p V =. Snce V s unformly bounded, by Lemma 4.2, there exsts l + such that (4-) (t k p t k p ) l V =, (t k p t k p) l V =. If there exsts s + such that (t k p ) s V =, (t k p ) s V =. By the fact that [t m d, t k p ] = t t m k p and [t m d p, t k p ] = t t m k, we have t r k p (t k p ) s V = t r k (t r k p ) s V = for all r ν+. If (t k p ) s V = for all s +. Then by (4-) there s r such that (t k p ) l (t k p ) l+ V = for all r, and (t k p ) l r (t k p ) l+r+ V =. So for any m ν+, we have t m d (t k p ) l r (t k p ) l+r+ V =, t m d p (t k p ) l r (t k p ) l+r+ V =. Therefore for all r ν+. t r k p (t k p ) l r (t k p ) l+r+ V =, t r k (t k p ) l r (t k p ) l+r+ V =, Case : ν 2 + +. By the precedng dscusson, there exst nonnegatve ntegers l and r, for =, 2, 4,..., ν, such that (t ν k ν ) l ν (tν k ν ) r ν (t ν 2 k ν 3 ) l ν 3 (t ν 2 k ν 3) r ν 3 (t k ) l (t k ) r V = and t m k p (t ν k ν ) l ν (tν k ν ) r ν (t ν 2 k ν 3 ) l ν 3 (t ν 2 k ν 3) r ν 3 (t k ) l (t k ) r V vanshes for all p ν and m ν+. By Lemma 4.3, the concluson of the theorem holds. Case 2: ν 2. Then there exst nonnegatve ntegers l and r, for =, 2, 4,..., ν 2, such that W = (t ν k ν 2 ) l ν 2 (t ν k ν 2) r ν 2 (t ν 3 k ν 4 ) l ν 4 (t ν 3 k ν 4) r ν 4 (t k ) l (t k ) r V s nonzero and (4-2) t m k p W = for all p ν and m ν+. By (2-), we know that (4-3) t m k ν W =,

CUIPO JIANG AND QIFEN JIANG for m ν+ such that m ν =. If there exsts t r k ν satsfyng t r k ν W =, let ν = span {t m d, t m d ν, t m k ν t m = t m tm t m ν ν, ν, m = (m,..., m ν ) ν, m ν+ }, W = U( ν )W. Then W = and t m k p W =, t n k ν W =, for all p ν, m ν+, and n ν+ such that n ν =. If there exsts = t m k ν such that t m k ν W =, we have (t m k ν ) l (t m k ν ) l W = and (t m k ν ) l (t m k ν ) l W = for some l +. As n the precedng proof, we can deduce that there exsts a nonzero v W such that for all n ν. Therefore t n k ν v = t m k p v = for all m ν+ and p ν. We have proved that (V ) =. References [Belavn et al. 984] A. A. Belavn, A. M. Polyakov, and A. B. Zamolodchkov, Infnte conformal symmetry n two-dmensonal quantum feld theory, Nuclear Phys. B 24:2 (984), 333 38. MR 86m:897 Zbl 66.73 [Berman and Bllg 999] S. Berman and Y. Bllg, Irreducble representatons for torodal Le algebras, J. Algebra 22: (999), 88 23. MR 2k:74 Zbl 942.76 [Bllg 23] Y. Bllg, Abelan extensons of the group of dffeomorphsms of a torus, Lett. Math. Phys. 64:2 (23), 55 69. MR 24h:222 Zbl 79.584 [Char and Pressley 988] V. Char and A. Pressley, Untary representatons of the Vrasoro algebra and a conjecture of Kac, Composto Math. 67:3 (988), 35 342. MR 89h:725 Zbl 66.722 [D Francesco et al. 997] P. D Francesco, P. Matheu, and D. Sénéchal, Conformal feld theory, Sprnger, New York, 997. MR 97g:862 Zbl 869.5352 [Dotsenko and Fateev 984] V. S. Dotsenko and V. A. Fateev, Conformal algebra and multpont correlaton functons n 2D statstcal models, Nuclear Phys. B 24:3 (984), 32 348. MR 85: 826 [Eswara Rao 24] S. Eswara Rao, Partal classfcaton of modules for Le algebra of dffeomorphsms of d-dmensonal torus, J. Math. Phys. 45:8 (24), 3322 3333. MR 25d:728 Zbl 7.72 [Eswara Rao and Jang 25] S. Eswara Rao and C. Jang, Classfcaton of rreducble ntegrable representatons for the full torodal Le algebras, J. Pure Appl. Algebra 2:-2 (25), 7 85. MR 26b:738 Zbl 7.79

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