A natural differential calculus on Lie bialgebras with dual of triangular type

Transcription

1 Centrum voor Wiskunde en Informatica REPORTRAPPORT A natura differentia cacuus on Lie biagebras with dua of trianguar type N. van den Hijigenberg and R. Martini Department of Anaysis, Agebra and Geometry AM-R

2 Report AM-R9519 ISSN CWI P.O. Box GB Amsterdam The Netherands CWI is the Nationa Research Institute for Mathematics and Computer Science. CWI is part of the Stichting Mathematisch Centrum (SMC), the Dutch foundation for promotion of mathematics and computer science and their appications. SMC is sponsored by the Netherands Organization for Scientific Research (NWO). CWI is a member of ERCIM, the European Research Consortium for Informatics and Mathematics. Copyright Stichting Mathematisch Centrum P.O. Box 94079, 1090 GB Amsterdam (NL) Kruisaan 413, 1098 SJ Amsterdam (NL) Teephone Teefax

3 A Natura Dierentia Cacuus on Lie Biagebras with Dua of Trianguar Type N. van den Hijigenberg CWI P.O. Box 94079, 1090 GB Amsterdam, The Netherands R. Martini University of Twente P.O. Box 217, 7500 AE Enschede, The Netherands Abstract We prove that for a specic cass of Lie biagebras, there exists a natura dierentia cacuus. This cass consists of the Lie biagebras for which the dua Lie biagebra is of trianguar type. The dierentia cacuus is expicity constructed with the hep of the R- matrix from the dua. The method is iustrated by severa exampes. AMS Subject Cassication (1991): 16S30, 16W30, 17B35, 17B37, 81R50. Keywords & Phrases: Universa enveoping agebras, Lie agebras, Hopf agebras, Quantum theory, Quantum groups, Coour Lie superagebras, non-commutative geometry. Note: The rst author is supported by NWO, Grant N Introduction In [1] the authors discuss dierentia cacui of Poincare-Birkho-Witt type on the universa enveoping agebra of a Lie agebra. It is shown in [2] that these dierentia agebras can be equipped with a dierentia Hopf agebra structure (for the denition see e.g. [3]) extending the standard Hopf agebra structure of the enveoping agebra. This Hopf agebra structure turns out to pay a prominent roe in the quantization of these dierentia cacui. A Quantized Universa Enveoping agebra (QUE agebra) on a Lie agebra g is a Hopf agebra deformation U h(g) of the enveoping agebra U(g) (see e.g. [4]). Its cassica imit determines a co-poisson bracket on U(g) whose restriction to g is caed a cocommutator and denes a Lie biagebra structure on g. In [5] the authors investigate deformations of De Rham compexes on U(g) in order to obtain a De Rham compex on the QUE agebra U h(g). They show that a necessary condition for the existence of such deformations can be interpreted as a compatibiity between the dierentia operator and the co-poisson bracket. One coud say there are two main strategies to utiize this compatibiity condition in order to construct De Rham compexes on QUE agebras. The rst one is to start with a certain dierentia cacuus on U(g) and compute a compatibe cocommutator. This is the point of view presented in [5]. The second approach is to consider a certain quantization of the Lie agebra g, take its cassica imit and try to construct a compatibe dientia operator d. This is the approach we wi use in this paper. In fact we wi prove that, for a specic cass of Lie biagebras, there exists a natura compatibe dierentia cacuus. This specic cass consists of the Lie biagebras for which the dua Lie biagebra is of trianguar type. 1

4 2 2 The compatibiity condition Let g be a nite dimensiona Lie biagebra over the ed of compex numbers C with commutator [ ; ] and cocommutator : g! g g. A cocommutator is a 1-cocyce with the additiona property that its transpose : g g! g denes a commutator on g. We note that g denotes the agebraic dua of g. With respect to a basis fx p g p of g one can write (2.1) [X p ; X q ] = C pq X (X p ) = p k Xk X where C pq and p k are the so-caed structure constants with respect to the given basis of g. Throughout this paper we wi make use of the Einstein summation convention. A dierentia cacuus on U(g) of PBW type can be described by the universa enveoping agebra of an N-graded coour Lie superagebra L (for the denition of coour Lie superagebra see [6]) that extends g in the foowing way (see [2]). A basis of L is given by fx p ; ^Xp g p where the eements X p have degree zero and the eements ^X p have degree one. The commutator of L is an extension of the commutator of g of the form (2.2) [X p ; ^X q ] = A pq the corresponding 2-cocyce is dened by ^X [ ^X p ; ^X q ] = 0; (2.3) (m; n) = (?1) mn m; n 2 N: The commutator of L is constructed such that the inear operator d : L! L dened by d(x p ) = ^Xp and d( ^Xp ) = 0 is a graded derivation of degree 1. For the structure constants this yieds the foowing conditions: A pq (2.4) (2.5) A jk A i m? A ik p A jp m = C ij A k m A pq? A qp = C pq Condition (2.4) expresses the Jacobi identity and condition (2.5) the derivation property of d. The dierentia operator is the graded derivation on U(L) that uniquey extends d. In order to obtain a more intrinsic description of L one can dene the inear map : g! g(g) by (see [1]) (2.6) (X p )(X q ) = A pq X : Conditions (2.4) and (2.5) are equivaent to (2.7) ([x; y]) = (x) (y)? (y) (x) (2.8) [x; y] = (x)(y)? (y)(x) for a x; y 2 g. Such a map is therefore caed a mutipicative representation of g (see [7]), it competey determines the dierentia cacuus on U(g). In order to construct a De Rham compex on the QUE agebra U h(g), which reduces to the previousy described dierentia cacuus on U(g) when h is put equa to zero, the cocommutator, which is the cassica imit of U h(g), and the dierentia operator d must be compatibe in a certain sense. This compatibiity condition, which is introduced in [5], can be described as foows. Dene the extension of from g to L by (2.9) d = (d id + d) = d ; where : L! L is dened as the inear map satisfying (x) = (?1) p x for a x 2 L p. With respect to the given basis of L the extension ooks ike (2.10) ( ^Xp ) = p k ( ^Xk X + X k ^X ): The dierentia d and the cocommutator are compatibe if and ony if this extension denes a cocommutator on L. In that case we ca L a coour Lie bisuperagebra. Hence, the probem of constructing a compatibe dierentia cacuus on the Lie biagebra g is to nd a mutipicative representation of g that denes a coour Lie superagebra L with the additiona property that the induced extension of dened by (2.9) is a cocommutator. In order to study this probem we wi ook at it from the dua point of view.

5 3 3 The dua point of view From the theory of Manin-tripes (see e.g. [4]), we earn that a Lie biagebra structure on g induces a Lie biagebra structure on g. The commutator of g is the transpose of the cocommutator of g and the cocommutator of g is the transpose of the commutator of g. For instance, if the Lie biagebra g is described by (2.1) then the structure of g is given by (3.1) [ p; q] = pq ( p) = C k p k where the eements p are dened to be dua to the basis eements X q, i.e. (3.2) p 2 g < p; X q >= q p and p q denotes the Kronecker deta symbo. Evidenty this is aso vaid for coour Lie bisuperagebras. We consider L to be the dua of the coour Lie bisuperagebra L with basis fx p ; ^Xp g p and structure maps as described by (2.1), (2.2) and (2.10), that represents the compatibe dierentia cacuus on the Lie biagebra g. As basis of L we choose f p ; ^ p g p which is dened dua to the basis of L, i.e. (3.3) < p; X q >= q p < p; ^X q >= 0 < ^ p; X q >= 0 < ^ p; ^X q >= q p: From the duaity between L and L we nd the foowing expressions for the commutator and cocommutator of L : (3.4) [ p; q] = pq [ p; ^ q] = pq ^ [ ^ p; ^ q] = 0; (3.5) ( p) = Cp k k ( ^ p) = A p k k ^? A p ^ k k : The dierentia operator d on L satises (3.6) d [ ; ] = [ ; ] d d = d : We : g! g to be the transpose of the dierentia d. It p) = 0 ^ p) =? p and from (3.6) it foows that = [ ; ] = [ ; = [ ; ] (@ id If we ook at the construction of a compatibe dierentia cacuus on U(g) from the point of g then the probem is to nd a mutipicative representation of g. This representation describes the commutator between eements of L 0 and L 1 and is such that the rst condition of (3.6) is satised. The second condition of (3.6) is then used to extend the cocommutator and one needs to verify whether the extension meets the conditions of a cocommutator on L. However, duay we see that the coour Lie superagebra structure of L given by (3.4) is uniquey determined by the commutator of g and the second property from (3.7). Hence, in the dua formuation the probem is to nd an extension of the cocommutator of g which is a cocommutator on L and satises the rst condition of (3.7). Evidenty, these probems are equivaent. However, in case the dua Lie biagebra g has a trianguar structure there is a simpe and natura soution which is evident from the dua point of view. 4 A canonica compatibe dierentia cacuus Let us suppose that the dua Lie biagebra g has a trianguar structure. This means that its cocommutator is a coboundary, i.e. : g! g g can be written by means of an R-matrix R 2 g g as () = R, with a corresponding R-matrix which is antisymmetric and a soution of the Cassica Yang Baxter Equation (CYBE). The dot denotes the action of the Lie agebra g on the tensor product g g induced by the adjoint representation. The antisymmetry of R is reected by the condition (R) =?R where : g g! g g represents the ip operator on the tensor product. In terms of the dua basis f pg p we can write (4.1) R = k k yieding (4.2) ( p) = k ( q pk q + q pk q):

6 4 The CYBE is usuay written as (4.3) [jr; Rj] = [R 12; R 13] + [R 12; R 23] + [R 13; R 23] = 0: In this notation R 12 denotes the eement k k 1 in U(g ) U(g ) U(g ) and [jr; Rj] is considered as eement of g g g U(g ) U(g ) U(g ). In order to extend from g to L it seems quite natura to preserve the coboundary property. Hence, we propose the foowing extension: (4.4) ( ^) = ^ R ^ 2 (L )?1 = (L 1 ) : Evidenty this extension denes a cocommutator: is a 1-cocyce since it is a 1-coboundary by construction and the commutator properties of the bracket on L dened by its transpose are a direct consequence of the trianguarity properties of the matrix R. Due to the fact is a derivation on L and (R) = 0 it foows that the rst condition of (3.7) is satised. Hence this extension denes a compatibe dierentia cacuus on U(g). With respect to the basis f p ; ^ p g p one nds that (4.5) which in comparison with(3.5) yieds (4.6) ( ^ p) = ^ p R = k ( q pk ^ q + q p k ^ q) A p k = kq pq: In order to obtain a description of the corresponding mutipicative representation we study in more detai the coboundary property of. By denition of we have (4.7) < R; x y >=< (); x y >=< ; [x; y] > 2 g ; x; y 2 g and by (4.1) (4.8) R = st ([; s] t + s [; t]): We focus on the rst term and take = k. Due to (4.9) < [ k; s] t; X p X q >=< ks t; Xp X q >= p ks q t we can write the transpose of the action of the rst component of R, which we wi denote by 1 : g! g g, as (4.10) 1(X p X q ) =? sq p sk Xk or equivaenty (4.11) 1 =?R 13 ( id) where R 13 denotes the map from g g g to g which is dened by etting R act on the rst and third component of the tensor product. Simiary one derives that (4.12) 2 =?R 12 (id ) which is dened as the transpose of the action of the second component of R. Hence, combining (4.7), (4.11) and (4.12) we obtain (4.13) [x; y] =?(R 13 ( id) + R 12 (id ))(x y) x; y 2 g: In order to rewrite this expression we note that (4.14) From this we derive that id = ( id) 12: R 12 (id ) = R ( id) 12 =?R ( id) 12 =?R 13 ( id) 12 and by substiting this in (4.13) we obtain (4.15) [x; y] = R 13 ( id) ( 12? id)(x y) x; y 2 g From the reasoning above we concude that the mapping : g! g(g) dened by (4.16) (x)(y) = R 13 ( id) 12(x y) satises (2.7) and (2.8), i.e. it denes a mutipicative representation of g. From this reasoning we can extract the foowing resut. Theorem 1 Let g be a Lie biagebra with corresponding dua Lie biagebra g of trianguar type with R-matrix R 2 g g. Then, the map : g! g(g) dened by (4.16) is a mutipicative representation on g and this mutipicative representation denes a dierentia cacuus on U(g) which is compatibe with the cocommutator of g.

7 5 5 Exampes In this section we present some exampes. 5.1 The 2-dimensiona sovabe Lie agebra Consider the 2-dimensiona Lie biagebra S over C with basis fa; bg and structure maps (5.1) [a; b] = b and (a) = (a b? b a) = a ^ b (b) = 0 We dene the dua basis f; g of S, the structure maps of S are given by (5.2) [; ] = and () = 0 () = ^ : Evidenty S is trianguar with R-matrix R = ^. The extension of according to formua (4.4) is (5.3) (^) = ^ ( ^ ) = ^ ^ ( ^) = ^ ( ^ ) =? ^ ^ The corresponding mutipicative representation of S is described by (5.4) (a) =? (b) = 0 0? 0 and this denes the dierentia cacuus on U(S) given by (5.5) [a; ^a] =?^a [a; ^b] = 0 [b; ^a] =?^b [b; ^b] = 0 We remark that the Lie biagebras S and (S ) op are isomorphic, an isomorphism ' : S! S is given by '(a) =?; '(b) =. 5.2 The Heisenberg Lie agebra Consider the 2n + 1-dimensiona Heisenberg Lie biagebra H n with basis fc; p i; q ig 1in and structure maps (5.6) [c; p i] = [c; q i] = 0 [p i; q j] = i jc (c) = 0 (p i) = p i ^ c (q i) = q i ^ c: We dene the dua basis f; i; ig 1in of Hn and obtain as structure maps [; i] =? i [; i] =? i [ i; j] = 0 (5.7) P () = n i ^ i ( i) = 0 ( i) = 0 i=1 One can easiy verify that Hn is trianguar with R-matrix R = 1 2Pi i ^ i. The extension of is given by (5.8) (^) = 1 2 nx i=1 (^ i ^ i + i ^ ^ i) (^ i) = 0 ( ^ i) = 0 which gives the foowing dierentia cacuus on U(H n) (5.9) [p i; ^q j] = 1 2 i j^c [q i; ^p j] =? 1 2 i j^c and a other commutators are equa to zero. For instance the mutipicative representation in case n = 1 ooks ike (5.10) (c) = 0 (p) = ! (q) = 0? 1 2 0!

8 6 5.3 The dua of the non-standard s 2 We consider the 3-dimensiona Lie biagebra K with basis f; ; g and structure given by (5.11) [; ] = [; ] = 0 [; ] =? () = ^ () = 2 ^ () =?2 ^ The corresponding dua basis of K wi be denoted by fh; e; fg and its structure is given by (5.12) [h; e] = 2e [h; f] =?2f [e; f] = h (h) = h ^ e (e) = 0 (f) = f ^ e: From this we see that K is isomorphic to the Lie agebra s 2 equipped with the non-standard cocommutator determined by the R-matrix R = 1 2 h ^ e. The extension of is given by (5.13) (^h) = h ^ ^e (^e) = e ^ ^e ( ^f) = ^f ^ e? 1 2 h ^ ^h and the corresponding mutipicative representation of K is described by (5.14) () = ?1 0! () = 0 () = 0 0? ! The dierentia cacuus on U(K) is given by (5.15) [; ^] = ^ [; ^] = 0 [; ^] = ^ [; ^] =? ^ [; ^] = 0 [; ^] = 0 [; ^] = 0 [; ^] = 0 [; ^] =? 1 ^ The dua of the Virasoro agebra In this exampe we consider an innite dimensiona Lie agebra. By V we denote the Virasoro agebra, this Lie agebra has a basis fe p; cg p with commutator given by (5.16) [e p; e q] = (q? p)e p+q + 0 p+q q 3? q c [c; ep] = 0 p; q 2 Z: 12 The Virasoro agebra can be equipped with a trianguar Lie biagebra structure by means of the R-matrix R = c ^ e 0, this yieds the foowing cocommutator (5.17) (e p) = pe p ^ c (c) = 0: The dua Lie biagebra V consists of forma series in ff p ; g p. These eements are dened to be dua with respect to the given basis of the Virasoro agebra. The commutator and cocommutator of V are given by [f p ; f q ] = 0 [; f p ] =?pf p (f p ) = X q (2q? p)f p?q f q () = X q q 3? q 12 f?q f q : The extension of the cocommutator of V to the coour Lie superagebra with basis fe p; ^e p; c; ^cg p according to (4.4) is given by (5.18) (^e p) = p(^e p c? c ^e p) (^c) = 0: The corresponding mutipicative representation on V is described by (5.19) (f p )(f q ) = 0 (f p )() = 0 ()(f p ) =?pf p ()() = 0: We remark that in this exampe V pays the roe of g and V the roe of g. So, in fact we used the trianguar structure of V (V ) to obtain a mutipicative representation of V.

9 7 6 The quasi trianguar case In this section we discuss the possibiity of extending the resut of Theorem I to the case where g is of quasi trianguar type. A Lie biagebra is said to be quasi trianguar if its cocommutator is a coboundary determined by an R-matrix that satises the CYBE. In contrast to the trianguar case, the R-matrix does not need to be antisymmetric. In genera one can write R = R a + R s where R a and R s denote the antisymmetric and the symmetric part of R respectivey, i.e. (R a) =?R a and (R s) = R s. Since the cocommutator satises =?, the symmetric part of R is g-invariant. Suppose we extend to L as described by (4.4), i.e. ( ^) = ^ R = ^ R a + ^ R s. The antisymmetry condition for the extension impies that R s needs to be invariant X under the adjoint action of the odd part of L. We can write (6.1) R s = i i (R s) = R s i where both f ig i and f ig i are ineary independent sets in g. Then (6.2) ^ R s = X i ([ ^; i] i + i [ ^; i]) = X i ([; ^ i] i + i [; ^ i]) = 0 impies that both sets are invariant under the adjoint action of 2 g. Hence, the extension wi ony be we dened if a terms in R s consist of centra eements of g. However, in that case (6.3) 0 = [jr; Rj] = [jr a + R s; R a + R sj] = [jr a; R aj] which impies that g is of trianguar type with R-matrix R a. The concusion is therefore that the construction of section 3 can not be appied to 'proper' quasi trianguar Lie biagebras such as for instance doubes of Lie biagebras (see e.g. [4]). References [1] R. Martini, G.F. Post and P.H.M. Kersten (1995). Dierentia Cacuus on Universa Enveoping Agebras of Lie Agebras. Memorandum 1261, University of Twente, Enschede. [2] N. van den Hijigenberg and R. Martini (1995). Dierentia Hopf Agebra Structures on the Universa Enveoping Agebra of a Lie agebra. CWI-report R9515, Center for Mathematics and Computer Science, Amsterdam. [3] G. Matsiniotis (1990). Groupes quantiques et structures dierentiees (Quantum groups and dierentia structures). C.R. Acad. Sci. Paris Ser I. 311, [4] V. Chari and A. Pressey (1994). Quantum Groups. Cambridge University Press, Cambridge. p [5] N. van den Hijigenberg, R. Martini and G.F. Post (1995). Quantization of Dierentia Cacui on Universa Enveoping Agebras, Preprint. [6] Yu.A. Bahturin, A.A. Mikhaev, V.M. Petrogradsky and M.V. Zaicev (1992). Innite Dimensiona Lie Superagebras. Water de Gruyter, Berin-New York. p. 13. [7] G.F. Post and R. Martini (1994). Mutipicative representations of Lie agebras. Preprint, University of Twente, Enschede.