The Erwin SchrÄodinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria A Poisson-Lie group Structure on the Di®eomorphism Group of a Circle J. Grabowski Vienna, Preprint ESI 26 (1993) June 4, 1993 Supported by Federal Ministry of Science and Research, Austria A POISSON{LIE STRUCTURE ON THE DIFFEOMORPHISM GROUP OF A CIRCLE Janusz Grabowski The Erwin SchrÄodinger Institute for Mathematical Physics, Vienna Abstract. Starting from the Gelfand{Fuks{Virasoro cocycle on the Lie algebra X(S1) of the vector ¯elds on the circle and applying the standard procedure de- scribed by Drinfel'd in ¯nite dimension, we obtain a classical r{matrix (i.e. an element r X(S1) X(S1) satisfying the classical Yang{Baxter equation), a Lie bialgebra 2 ^ structure on X(S1), and a sort of Poisson{Lie structure on the group ]Di®(S1) of dif- feomorphisms. Quantizations of such Lie bialgebra structures may lead to "quantum di®eomorphism groups". 1. Introduction. The Lie algebra X(S1) of smooth vector ¯elds on the circle S1 may be identi¯ed 1 1 with the space CR1(S ) of (real) smooth functions on S equipped with the Lie bracket (1:1) [f; g] = fg0 f 0g: ¡ 1 We can pass to the complexi¯cation V = CC1(S ) of this space considering complex imt valued function with the same bracket (1.1). The elements em(t) = ie = sin(mt) i cos(mt) satisfy the commutation relations ¡ ¡ (1.2) [e ; e ] = (m n)e ; m n ¡ m+n so they span the Witt algebra W: The Lie algebra of vector ¯elds X(S1) can be regarded as the Lie algebra of the in¯nite dimensional Lie group Di®(S1) of smooth di®eomorphisms of the circle, while the complexi¯cation does not correspond to any reasonably de¯ned complex group. There is a strategy to obtain Lie bialgebras (and hence Poisson{Lie groups) starting from 2{cocycles described by Drinfel'd [Dr1] (cf. also [Dr2] and [BM] for some in¯nite dimensional examples). The ¯rst observation is that for ¯nite dimensional Lie algebra g any 2{cocycle ! on g is non{degenerate on a subalgebra, 1991 Mathematics Subject Classi¯cation. 17B66, 22E65, 58D05, 81R50. Key words and phrases. r{matrix, Poisson{Lie structure, Lie bialgebra, vector ¯eld, Witt algebra, Virasoro cocycle. Typeset by -TEX AMS 1 2 JANUSZ GRABOWSKI so we can consider ! to be invertible, when the 2{cocycle ! g¤ g¤ is viewed as a 2 ­ map ! : g g¤; » !(»; ): An invertible 2{cocycle can be also viewed as a left{ invariant symplectic¡! 7!form on¢ the corresponding simply connected Lie group. Put 1 now r = !¡ : Then r viewed as an element in g g obeys the classical Yang{Baxter equation ­ (1:3) [r12; r13] + [r12; r23] + [r13; r23] = 0 (cf. [Dr1]). Next, Drinfel'd showed how to obtain from such an "r{matrix" a Lie bialgebra structure ± on g: One puts namely ± = @r; where @ is the coboundary operator in the Lie algebra complex with values in g g under the adjoint action. ­ Now the dual @¤ : g¤ g¤ g¤ is a Lie bracket on g¤: This bracket may be also described by ­ ¡! (1.4) ´ ; ´ = ® (´ ) ® (´ ); f 1 2g R(´1) 2 ¡ R(´2) 1 where R : g¤ g is de¯ned by ¡! (1.5) R(´) = r; ´ = r( ; ´): h i ¢ Both Lie brackets give rise to a Lie bracket on g g¤ with the obvious pairing on the last space being ad{invariant (this is what is ©called Manin triple). If one wants to repeat the above procedure in in¯nite dimension, one encounters many problems related to the facts that we must deal with in¯nite sums, we must decide what is understood as the dual, what is the meaning of tensor products, etc. For example, it is clear that the Witt algebra tells us the structure of the Lie algebra of vector ¯elds, while the Witt basis can be understood only as a topological basis and it is not clear if we should take for the dual distributions or, say, the same algebra with the pairing given by the hermitian product. Since we do not want to deal with such questions here, we shall ¯rst consider our structure on a formal level and then we shall study the bracket obtained by using the hermitian pairing on the topological level. This bracket will be well{de¯ned, but it will not give a Lie algebra structure on the whole dual, since the bracket of two smooth functions may be only continuous. We ¯nd however this structure su±ciently interesting to present it. Finally, we obtain the corresponding Lie{Poisson structure on ]Di®(S1) {the 1 universal covering of the identity component Di®+(S ) which, again, is not strictly "smooth", since it uses C1{ vector ¯elds which formally do not belong to the Lie algebra of this group. We are however able to calculate the corresponding bracket, which for the evaluation functions gives us continuous functions on ]Di®(S1): The Lie bialgebras obtained from classical r-matrices can be always canonically quantized [Dr3], but the question of constructing "quantum group of di®eomor- phisms" seems to be much more di±cult and remains open. Note that Beggs and Majid [BM] considered Lie bialgebra structures on the Lie algebra X(S1); but related to cohomologically trivial cocycles. 2. Lie bialgebra structure on the Witt algebra The 2{cocycle we shall start with is the well{known Gelfand{Fuks{Virasoro co- cycle in V : (2.1) !(f; g) = i f 00g0dt 1 ZS A POISSON{LIE STRUCTURE ON THE DIFFEOMORPHISM GROUP OF A CIRCLE 3 which on the Witt basis gives 3 !(em; en) = m ±m; n: ¡ This cocycle is actually degenerate, since !(e0; ) = 0 (there is no nondegenerate cocycle on W), but it leads to a symplectic structure¢ on the in¯nite dimensional homogeneous space Di®(S1)= Rot(S1) (cf. [KY]). As a Lie subalgebra on which ! has the trivial kernel we take V0 = f V : f(0) = 0 ; where we identify functions on the circle with the 2¼{periodic functionsf 2 on R: Ongthe level of the Witt algebra, W = W V is spanned by b = e e ; m = 0: We have the commutation rules 0 \ 0 m m ¡ 0 6 (2.2) [b ; b ] = (m n)b mb + nb m n ¡ m+n ¡ m n with the convention b0 = 0: Putting now b = e0; we get a new basis 1 ¡ bm : m = 0 b f 6 g [ f 1g in W with (2.3) [b ; bn] = nbn nb : 1 ¡ 1 Clearly, 3 (2.4) !(bm; bn) = m ±m; n ¡ for m; n = 0 and !(b ; ) = 0: Passing to the formal inverse of ! in W0 we get the "r{matrix"6 1 ¢ 1 1 (2:5) r = bm b m W0 W0: 2 m3 ^ ¡ 2 ^ m=0 X6 The sum is formally in¯nite, but it satis¯es formally the classical Yang{Baxter equation and de¯nes the mapping R : W ¤ W as in (1.5), where we regard 0 ¡! 0 the formal dual W ¤ of W to be generated by elements b¤ ; n = 0; such that 0 0 n 6 b ; b¤ = ± : Of course, we can regard r as an element in W W and R as a h m ni m;n ^ mapping R : W ¤ W given by ¡! 1 R(bm¤ ) = b m; ¡m3 ¡ (2.6) R(b¤ ) = 0: 1 Theorem 1. Considering the basis bn : n = 0 b of the Witt algebra as above, we ¯nd that f 6 g [ f 1g 1 1 (2.7) r = bm b m W W 2 m3 ^ ¡ 2 ^ m=0 X6 satis¯es the classical Yang{Baxter equation. The coboundary ± = @r : W W W ¡! ^ 4 JANUSZ GRABOWSKI is given by k m k ±(bk) = ¡ bm+k b m + bk bm; m3 ^ ¡ m3 ^ m=0 m=0 X6 X6 1 (2.8) ±(b ) = bm b ; 1 m2 ^ 1 m=0 X6 and de¯nes a Lie bialgebra structure on W: The corresponding bracket ; = ±¤ on the formal dual is given by f¢ ¢g 1 1 3 m n b¤ ; b¤ = (n m) + b¤ + b¤ b¤ ; f m ng ¡ m n m+n n3 m ¡ m3 n ¶ 1 (2.9) bn¤ ; b¤ = b¤ : f 1g n2 1 The proof is just a matter of calculations. 3 Remark. Changing the basis in W ¤ by putting cn¤ = n bn¤ ; c¤ = b¤ ; we see imme- diately that 1 1 c¤ ; c¤ = (n m)c¤ + mc¤ nc¤ ; f m ng ¡ m+n m ¡ n (2.10) cm¤ ; c¤ = mc¤ ; f 1g 1 so the algebra (W ¤; ; ) is isomorphic to (W ; [ ; ]) and (W ¤; ; ) is just the 0 f¢ ¢g 0 ¢ ¢ f¢ ¢g semidirect product of W0¤ and a 1{dimensional algebra relative to the 1{dimensional representation cm¤ m of W0¤: Note that the brac7! ket ; may be obtained also by the use of formula (1.4). The coadjoint action ® of f¢W¢gis given by ®bm (bn¤ ) = (2m n)bn¤ m + nbn¤ m±m;n( bk¤ + b¤ ); ¡ ¡ ¡ 1 k=0 X6 (2.11) ®bm (b¤ ) = mb¤ ; 1 1 ®b (bn¤ ) = nbn¤ ; 1 ®b (b¤ ) = kbk¤: 1 1 ¡ k=0 X6 Observe that this coadjoint action is in fact de¯ned in the complete dual W 0 of W consisting of all formal series in bn¤ and b¤ : Because of the particular form of R; the 1 formula (1.4) gives however a well de¯ned bracket on W ¤: This bracket cannot be extended to W 0; since we should sum up not convergent series of coe±cients. This is also the reason that we cannot de¯ne the corresponding Lie algebra structure on W W ¤ as one usually does in ¯nite dimension to get a Manin triple.