A Poisson-Lie Group Structure on the Diffeomorphism Group of a Circle

The Erwin Schrodinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria

A Poisson-Lie group Structure on the Dieomorphism 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 Schrodinger 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 described 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 ∈ ∧ 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 dieomorphism groups”.

1. Introduction. The Lie algebra X(S1) of smooth vector elds on the circle S1 may be identied 1 1 with the space CR∞(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 complexication V = CC∞(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 innite dimensional Lie group Di(S1) of smooth dieomorphisms of the circle, while the complexication does not correspond to any reasonably dened 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 innite dimensional examples). The rst observation is that for nite dimensional Lie algebra g any 2–cocycle ω on g is non–degenerate on a subalgebra,

1991 Mathematics Subject Classication. 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 ω to be invertible, when the 2–cocycle ω g g is viewed as a ∈ map ω : g g; ω(, ). 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 = ω . 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) , = ( ) ( ), { 1 2} R(1) 2 R(2) 1 where R : g g is dened 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 innite dimension, one encounters many problems related to the facts that we must deal with innite 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–dened, 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 suciently 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 dieomorphisms” seems to be much more dicult 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 cocycle in V :

(2.1) ω(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 ω(em, en) = m m, n.

This cocycle is actually degenerate, since ω(e0, ) = 0 (there is no nondegenerate cocycle on W), but it leads to a symplectic structure on the innite dimensional homogeneous space Di(S1)/ Rot(S1) (cf. [KY]). As a Lie subalgebra on which ω has the trivial kernel we take V0 = f V : f(0) = 0 , where we identify functions on the circle with the 2–periodic functions{ ∈ on R. On}the 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 ∞

bm : m = 0 b { 6 } ∪ { ∞} in W with

(2.3) [b , bn] = nbn nb . ∞ ∞ Clearly,

3 (2.4) ω(bm, bn) = m m, n

for m, n = 0 and ω(b , ) = 0. Passing to the formal inverse of ω in W0 we get the ”r–matrix”6 ∞ 1 1 (2.5) r = bm b m W0 W0. 2 m3 ∧ ∈ ∧ m=0 X6 The sum is formally innite, but it satises formally the classical Yang–Baxter equation and denes 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. ∞

Theorem 1. Considering the basis bn : n = 0 b of the Witt algebra as above, we nd that { 6 } ∪ { ∞}

1 1 (2.7) r = bm b m W W 2 m3 ∧ ∈ ∧ m=0 X6 satises 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 , ∞ m2 ∧ ∞ m=0 X6

and denes a Lie bialgebra structure on W. The corresponding bracket , = on the formal dual is given by { }

1 1 3 m n b , b = (n m) + b + b b ; { m n} m n m+n n3 m m3 n 1 (2.9) bn , b = b . { ∞} n2 ∞ 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 ∞ ∞

c , c = (n m)c + mc nc ; { m n} m+n m n (2.10) cm , c = mc , { ∞} ∞

so the algebra (W , , ) is isomorphic to (W , [ , ]) and (W , , ) is just the 0 { } 0 { } 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 {W}is given by

bm (bn ) = (2m n)bn m + nbn mm,n( bk + b ); ∞ k=0 X6 (2.11) bm (b ) = mb ; ∞ ∞ b (bn ) = nbn ; ∞

b (b ) = kbk. ∞ ∞ k=0 X6

Observe that this coadjoint action is in fact dened 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 ∞ formula (1.4) gives however a well dened bracket on W . This bracket cannot be extended to W 0, since we should sum up not convergent series of coecients. This is also the reason that we cannot dene the corresponding Lie algebra structure on W W as one usually does in nite dimension to get a Manin triple. 3. Poisson bracket obtained by the hermitian duality We shall nd now the bracket dened by our r–matrix using the hermitian duality

1 (3.1) f, g = fgdt. h i 2 1 ZS A POISSON–LIE STRUCTURE ON THE DIFFEOMORPHISM GROUP OF A CIRCLE 5

Since the r–matrix has the form 1 1 1 (3.2) r = em e m + e0 em 2 m3 ∧ m3 ∧ m=0 m=0 X6 X6 in terms of the orthogonal basis em : m Z , we get a well–dened mapping R : L2(S1) L2(S1) : { ∈ } → 1 R(em) = (e0 e m), m = 0, m3 6 1 (3.3) R(e ) = e . 0 m3 m m=0 X6 2 1 Observe that W0 is embeded into L (S ) by identifying bm with em, m = 0, while 2 1 6 the element b is not from L (S ), since it can be formally written in the form ∞ ∞ b = em. ∞ m= X∞ It satises however (also formally) R(b ) = 0. Note that R does not map V into itself, since R(e ) = F, where ∞ 0 1 1 t(t )(t 2) (3.4) F (t) = e = sin(mt) = , 0 t 2, m3 m m3 6 m=0 m=0 X6 X6 is only a class C1 function on the circle. We can now write R as an integral operator

(3.5) R(f)(x) = G(x, y)f(y)dy. 1 ZS The kernel is a class C1 function given clearly by 1 G(x, y) = e (x)F (y) + F (x)e (y) e (x)e (y), 0 0 m3 m m m=0 X6 so (3.6) G(x, y) = i(F (x + y) F (x) F (y)). Since the coadjoint acjon of V relative to the hermitian pairing is given by h(g) = (2h g + hg0), we get the following. 0 1 Theorem 2. The r–matrix (3.2) denes a bracket on V = CC∞(S ) relative to the hermitian pairing by

f, g (x) = H(x, y)(g(x)f(y) f(x)g(y))dy+ { } 1 ZS (3.7) + G(x, y)(g0(x)f(y) f 0(x)g(y))dy, 1 ZS where G(x, y) = i(F (x + y) F (x) F (y)), H(x, y) = 2G0 (x, y) = 2i(F 0(x + y) F 0(x)), x and 1 F (x) = sin(mx). m3 m=0 X6 This bracket is antisymmetric and satises the Jacobi identity if only the appearing terms make sense. 6 JANUSZ GRABOWSKI

4. Poisson–Lie bracket on ]Di(S1)

Since our r–matrix (3.2) is purely imaginary, let us consider the real form r0 = ir : 1 (4.1) r = sin(mt) cos(mt) + 1 F (t). 0 m3 ∧ ∧ m=0 X6 In the nite dimensional case any r–matrix r0 gives a Lie–Poisson structure P on the corresponding Lie group which can be written in the form P = rl rr, 0 0 l r where r0 and r0 and the left and right invariant tensors on the group corresponding 1 to r0. In our case r0 is not even a bivector tangent to Di(S ), since the vector eld F is of class C1 only. Passing however to suciently ”smooth” functions on Di(S1), we can derive a well–dened bracket. We shall use the evaluation functions, so it will be much more convenient to describe the bracket on the universal covering group ]Di(S1) of the identity component of Di(S1) rather than on Di(S1) itself. The group ]Di(S1) can be viewed as the group of those smooth dieomorphism g of R for which g(t+2) = g(t)+2. We consider clearly the Lie algebra of ]Di(S1) as consisting of 2–periodic functions on R. For any R we have the evaluation function on ]Di(S1) : (g) = g(). For left and right ∈invariant vector elds generated by X X(S1) we obviously have: ∈ l d d (4.2) X ()(g) = (g exp(tX)) = g(exp(tX)()) = g0()X(); dt t=0 dt t=0 | | d d (4.3) Xr()(g) = (exp(tX) g) = exp(tX)(g()) = X(g()). dt t=0 dt t=0 | | This easily implies the following. Theorem 3. The r-matrix (4.1) denes a Lie–Poisson bracket on ]Di(S1) which 1 on the evaluation functions , C∞(]Di(S )) has the form ∈ (4.4) , (g) = g0()g0()K(, ) K(g(), g()), { } 1 for K(x, y) = F (x y) F (x) + F (y) and F (t) = m=0 m3 sin(mt). 6 References P [BM] Beggs E. and Majid S., Matched pairs of topological Lie algebras corresponding to Lie bialgebra structures on diff(S1) and diff(R), Ann. Inst. Henri Poincare 53 (1990), 15– 34. [Dr1] Drinfel’d V. G., Hamiltonian structures on Lie groups, Lie bialgebras, and the geometrical meaning of classical Yang-Baxter equations, [Russian] English translation: Sov. Math. Dokl. 27, 68–71 (1983), Dokl. Akad. Nauk SSSR 268 (1983), 285-287. [Dr2] Drinfel’d V. G., Quantum groups, Proc. Int. Congress Math., Berkeley, California (1986). [Dr3] Drinfel’d V. G., On constant, quasiclassical solutions of the Yang–Baxter quantum equation, Sov. Mat. Dokl. 28 (1983), 667–671. [KY] Kirillov A.A. and Yuriev D. V., Kahler geometry of the innite–dimensional homogeneous 1 1 space M = Diff+(S )/Rot(S ), Functional Anal. Appl. 21 (1987), 284–294.

Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland. E-mail address: [email protected]