ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Quadratic brackets from symplectic forms A. Yu. Alekseev, I. T. Todorov Vienna, Preprint ESI 33 (1993) July 5, 1993 Supported by Federal Ministry of Science and Research, Austria
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Quadratic brackets from sympletic forms Anton Yu.Alekseev Ivan T.Todorov October 2, 2001 Erwin Schrödinger International Institute for Mathematical Physics (ESI) Pasteurgasse 6/7, A 1090 Wien, Austria Abstract We give a physicist oriented survey of Poisson-Lie symmetries of classical systems. We consider finite dimensional geometric actions and the chiral WZNW model as examples for the general construction. An essential point is that quadratic Poisson brackets appear for group like variables. It is belived that after quantization they lead to quadratic exchange algebras. On leave from Steklov Mathematical Institute (LOMI), Fontanka 27, St.Petersburg 191011, Russia On leave from the Institute for Nuclear Research, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, BG 1784 Sofia, Bulgaria 1
1 Introduction. Good old canonical quantization still takes a respectable place among various quantization schemes. A geometric approach to the problem begins with a sympletic 2-form Ω on a phase space Γ. Then one should invert Ω and obtain a Poisson bracket (PB) algebra on Γ. Next, one selects a subset of PB to be represented by commutators upon quantization. We are only aware of a general principle for such a selection in the case when our physical system admits a symmetry algebra, that is, a Lie algebra G spanned by a set of generators X a satisfying linear PB relations {X a, X b } = f c abx c. (1) The skewsymmetry of the structure constants f in the pair of lower indices and the Jacobi identity f s abf d sc + f s bcf d sa + f s caf d sb = 0 (2) are nessary and sufficient for the existence of such a Lie algebra. If a set of variables Y µ transform linearly under the PB action of G we require that their covariance properties are preserved under quantization: {X a, Y µ } = f ν aµy ν [ ˆX a, Ŷµ] = i hf ν aµŷν. (3) In particular, the PB (1) goes into a commutation relation for ˆX a with structure constants i hfab. c The Heisenberg PB relations among coordinates and momenta {q i, p j } = δ i,j, {q i, q j } = 0 = {p i, p j }, can be viewed as a special case of ( 1) for a nilpotent Lie algebra. A special covariant family is given by the elements g of a matrix Lie group G whose Lie algebra is G. If G is spanned by matrices t a satisfying the commutation relations [t a, t b ] = fabt c c, (4) then the left covariance of g G under the PB action of G is expressed by {g, X a } = t a g. (5) Typically, the variable g satisfies quadratic PB relations [1]. Using the tensor notation g 1 = g 1, g 2 = 1 g (6) we can write the symplest example of quadratic brackets: { 1 g, 2 g} = 1 g 2 g r. (7) The Jacobi identity for the bracket (7) is equivalent to the classical Yang-Baxter equation [r 12, r 13 ] + [r 12, r 23 ] + [r 13, r 23 ] = 0. (8) Here r ij act in a triple product V 1 V 2 V 3 of finite dimensional spaces: r 12 = r 1 where 1 is the unit operator in V 3 etc. Quantization of the PB (7) leads to quadratic exchange relations 2 gg= 1 g 1 g 2 R. (9) 2
Here the matrix R is a solution of the quantum Yang-Baxter equation R 12 R 13 R 23 = R 23 R 13 R 12 (10) and R = R( h) 1 + i hr when h 0. There is a wide class of intresting examples where such R matrix is known. Nevetheless, it is still a challenge to find a regular scheme for quantization of quadratic brackets. The PB (7) have annoying property: they appear to violate a symmetry under right shifts, g gh 1. Remarkably, this symmetry can be restored, in some sence, if we admit non trivial PB for the parameters of the shift h. This is a generalization of supersymmetry where anticommutative parameters are required. If we postulate { 1 h, 2 h} = [r, 1 h 2 h], { 1 g, 2 h} = 0, (11) we discover that the fomula (7) survive after the right shift: {g 1 (h 1 ) 1, g 2 (h 2 ) 1 } = g 1 (h 1 ) 1 g 2 (h 2 ) 1 r. (12) We note that the product of two consecutive shifts h = h 1 h 2 has the same brackets (11) provided that the two factors have zero PB among themselves: h = h 1 h 2, { 1 h1, 2 h2} = 0, { 1 hi, 2 hi} = [r, 1 hi, 2 hi] { 1 h, 2 h} = [r, 1 h 2 h]. (13) The message of this symple example is that the symmetry group can carry a nontrivial PB. The backet respects the group multiplication. The Lie group supplied with such PB is called Poisson-Lie (PL) group. We should take into account the PB of transformation parameters when we check the symmetry with respect to PL group action. The PL symmetry does not preserve the PB if we treat transformation parameters just as c-numbers. It means that the notation of PL symmetry gives us a possibility to find a symmetry in examples where it seems to be broken. PL symmetries are viewed as classical counterparts of quantum group symmetries [6]. Suppose we have a physical system defined by the action principle x S = L(x, ẋ)dt. (14) If we suspect that the transformation x x + δ ε x is a symmetry, we can easely check it. The transformation δ ε should preserve the action S up to boundary terms. In the Section 2 we give a similar criterium [4] how to recognize a PL symmetry starting from the action principle. Usually the PL symmetry appears in the systems with quadratic PB. To get expirence of calculations with quadratic brackets we start the Section 3 with a system S = tr(i P u 1 u 1 2 p2 )dt, (15) where u is an element of compact semisimple Lie group G. P belongs to H +, positive Weyl chamber in the dual to corresponding Cartan subalgebra H. In the case of 3
sun p is just diagonal matrix with ordered eigenvalues: P = diag(p 1,..., p n ), p 1 p 2... p n. There is no PL symmetry in the system. But the bracket for variables u appears to be quadratic. This bracket was derived in [9] for the case of SU(2) and guessed for an arbitrary group G. We fill this gap and derive the PB for u directly from the action. The system ( 15) posess a one-parameter deformation (Eq.) so that in the deformed system there in neither left non right symmetry. But the left symmetry can be restored using the notion of PL symmetry. The PB were suggested in [9] and the action principle was proposed in [7],[8]. We put this two ideas together and derive qudratic PB from the action principle. We use this system as an example of PL symmetry. Let s mention that the action ( 15) is a classical model for spin and its deformation corresponds to quantum group analogue of particle spin. The infinite dimensional case of a chiral WZNW model, which has to a large extend inspired the present study, is considered in Section 4 (Relevant steps in working out the Lagrangean approach to WZNW models are contained in [8],[10] [13].) The basic variable is here the group valued field g(x) on the circle S 1. More precesely, g is a multivalued function on S 1 satisfying the quasiperiodicity condition g(x + 2π) = g(x)m. (16) The monodromy matrix M is a counterpart of the angular momentum matrix L. The symplectic form (84) on the chiral phase space Γ W Z involves an integral term and a discrete piece depending on the boundary values g 1 = g 0 and g 2 = g(2π) = g 1 M, reminiscent to the finite dimensional 2-form (69) studied in Section 3. After reviewing the basic properties of the form (84) (first introduced in [8]) including an elementary derivation of the condition that Ω is a closed form, dω = 0, we establish an intriguing new result: the presence of an infinite quantum symmetry of Ω under right shifts of g(x) satisfying certain boundary conditions. It appears, however, that the x-independent part of this Poisson-Lie symmetry (corresponding to the standard quantum group symmetry noted previously in various contexts cf.[8],[12] [16]) is sufficient to explain the cancellation between world sheet and quantum group monodromy, established earlier [15, 16]. 2 Quantum group symmetry of a symplectic form Our starting point will be a closed 2-form Ω on a manifold Γ. Suppose that it is non degenerate. The condition dω = 0 (17) implies the existence, at least locally, of a 1-form Θ such that Ω = dθ, Θ = P i dx i, P i = P i (x). (18) Hence, if we choos a Hamiltonian H = H(x) on Γ, we can define a Lagrangean L and the corresponding action S as: L = P i (x)ẋ i H(x), S = L dt = (d 1 Ω H dt), (19) 4
where a dot (on x) denotes, as usual, a time derivative. We recall that an infimitesimal transformation x x + δ ε x of Γ with parametrs ε a δ ε x = ε a δ a x (20) defines a classical symmetry. If (i) it leaves the Hamiltonian invariant: δ ε H = H(x + δ ε x) H(x) = O(ε 2 ), (21) (ii) it preserves Ω. It follows from the definition that the action is invariant (up to boundary terms) provided that ε is time independent. If ε a = ε a (t) than δ ε S = X a ε a dt (+ boundary terms). (22) Functions X a generate the symmetry via PB: δ a f = f x δ ax = {f, X a }. (23) A Poisson-Lie (PL) symmetry of the classical phase space Γ is a generalization of the above concept but the condition (ii) is replaced by the weaker requirement δ ε S = ε a A a (+ boundary terms) (24) where the connection 1-forms A a satisfies the zero curvature condition [4] da a = F bc a A b A c. (25) The constants Fa bc form a skew symmetric tenzor with respect to the upper indices and must obey the Jacobi identity (2). Hence, they can be viewed as the structure constants of the Lie algebra G. We note that (22) appears as a special case of (24),(25) where A a = dx a and hence da a = 0. Let τ a be (say, matrix) generators of G such that [τ a, τ b ] = F ab c τ c. (26) Let us define the group element g = g (x), x Γ as a solution of the equation (d A a τ a )g = 0. (27) The group like variable g is a substitute for all generators X a. It is easy to check that δ a f = tr(τ a {f, g }g 1 ) (28) The PB is not invariant under such generalized quantum symmetry transformation. Instead we have a complicated derivation in the r.h.s. {f 1, f 2 }(x + δ ε x) = {f 1 (x + δ ε x), f 2 (x + δ ε x)} x + f 1 (δ ε x i ) f 2 (δ ε x j ) F ab x i ε a x j ε b c ε c. (29) 5
If we ascribe to the PL symmetry parameters ε a the PB {ε a, ε b } = F ab c ε c (30) than we can restore the PB symmetry by considering f(x + δ ε x) as a function of both x and ε and including the derivation in the r.h.s. of (29) in the definition of PB: {f 1, f 2 }(x + δ ε ) = {f 1 (x + δ ε x), f 2 (x + δ ε x)} x + {f 1 (x + δ ε x), f 2 (x + δ ε x)} ε. (31) Now we shall discuss a basic example of the algebra G useful for next Sections. Let t a be the matrix generators of a Lie algebra G satisfying (4). Assuming that G is semisimple we choose a Cartan subalgebra with basis t j = H j (1 j rank G and split the remaining generators into t α and t α where α enumerates the positive roots. Setting ε = ε a t a (32) we define a PB between the commuting matrices by the classical r-matrix relation [1] where r is skewsymmetric: 1 ε= ε 1 and 2 ε= 1 ε (33) { 1 ε, 2 ε} = [r, 1 ε + 2 ε] (34) r = α>0(t α t α t α t α ). (35) We thus arrive at a new PB Lie algebra G with structure constants F c ab related to those of G by Fc αβ = fαβ c if α > 0, β > 0, Fc αβ = fαβ c if α < 0, β < 0, Fα αi = fαi α if α > 0, (36) Fα αj = fαj α if α < 0, Fc αβ = Fc αβ = 0 if α > 0, β < 0. (Here, as before, i, j label the Cartan generators of G.) Lie algebra G can be embadded into the direct sum of two Borel subalgebras of G spanned by (t α, H i ) and by (t α, H i ) (α > 0), respectively. More precisely, a pair (b +, b ) belongs to G if the matrices b ± have opposite diagonal elements: (b +, b ) G if b diag + + b diag = 0. (37) 6
We shall also use a group G with Lie algebra G. It is contained in the product of Borel subgroups of G (the subgroups of upper and lower triangular matrices in the case of SL (n, C)): (L +, L ) G if L diag + L diag = 1 (38) The product in G is defined component wise: (L +, L )(M +, M ) = (L + M +, L M ), (39) while the connection form A = dg g 1 is given by A = (dl + L 1 +, dl L 1 ). (40) It is important to note that the PB (34) for infinitesimal matrices can be exponentiated resulting in Sklyanin s quadratic algebra [1] for (matrix) group elements (cf.(11)), { 1 v, 2 v} = [r, 1 v 2 v], (41) where we have used again the tensor notation (6). The global version of the formula (??) looks like follows: {f 1, f 2 }(x v ) = {f 1 (x v ), f 2 (x v )} x + {f 1 (x) v, f 2 (x v )} v, (42) where x v is a point x shifted by the group element v. 3 Chiral T G phase space and its quantum group deformation. Let G be a simple compact Lie group with an ordered Cartan-Weyl basis of its Lie algebra. We define the (left) chiral phase space Γ = Γ(G, H) as the product of G with the dual H of a real Cartan subalgebra H of Lie algebra G factored by the Weyl in group which amounts taking a Weyl chambe H + of H. For a unimodular matrix Lie group G the space H is spanned by real traceless diagonal n n matrices P. (For G = SL(n) the positive Weyl chamber consists of P s with ordered eigenvalues: p 1 p n.) The (undeformed) classical Lagrangian, invariant under left shifts of u G, is L = tr(ip u 1 u 1 2 P 2 ). (43) The resulting system has been proposed [12] as a finite dimensional analogue of a chiral WZNW model. The corresponding quantum mechanical state space contains every finite dimensional irreducible representation of G with multiplicity equal to one. The infinitesimal left shifts δ ε u = εu, δ ε P = 0. (44) 7
satisfy conditions (i) and (ii) of Section 2. If t a is a basis in G so that then the generator X a of the symmetry ε = ε a t a, (45) X a = tr t a u P u 1 (46) appears in the variation of the action (22). The symplectic form Ω 0 associated to the Lagrangean (43) can be written either in terms of P and u 1 du or as a bilinear form on G G, where Ω 0 = i tr (dp u 1 du P (u 1 du) 2 ), (47) Ω 0 = i 2 tr(dl du u 1 + dp u 1 du), (48) l = u P u 1. (49) The equivalence between (47) and (48) is established using the identity dl = [du u 1, l] + u dp u 1. Here and in what follows we omit the wedge sign in the products of exterior differentials. In order to bring Ω 0 to a canonical form we first introduce a basis of left invariant 1-forms Θ a on G setting i u 1 du = Θ a t a = Θ j H j + α>0(θ α t α + Θ α t α ) (50) where the sum in α runs over the positive roots of G, while the first term involves a sum (in j) over the Cartan basis. In the special case G = SL n we have t α = e jk j < k, t α = e kj = e jk, H j = e jj e j+1 j+1, 1 j n 1, (51) where e jk are the (n n) Weyl matrices characterized by the product formula e ij e kl = δ jk e il. (52) More generally, the basis in G is characterized by the commutation relations [H j, t ±α ] = ±2 α jα t αj 2 ±α, [t αj, t αj ] = H j (53) where α j (j = 1,, rank G) are the simple roots of G. Next we expand P in a basis dual to {H j }: P = P j h j ( H +) (54) where (h i, H j ) = tr h i H j = δ i j. (55) 8
In the case of SL(n) we have Setting further we transform (47) into h j = e jj 1 n n 1 (1 = e ii ). (56) i=1 [t α, t α ] = H α, (57) (P, H α ) = P α (58) Ω 0 = dp j Θ j + i α P α Θ α Θ α. (59) The canonical expression (59) for Ω 0 has the advantage of being readily invertible. To write the corresponding PB we introduce the vector fields V i and V ±α dual to Θ i and Θ ±α (in the sense that < Θ a, V b >= δ a b ): V i = tr u H i u, The resulting PB is given by the skew product If f 1,2 are two functions of P i and u then V ±α = tr u t ±α u. (60) P = V j + i 1 V α V α. (61) P j α>0 P α {f 1, f 2 } = P(f 1, f 2 ). (62) In particular, the non-vanishing PB for the basic phase space variables P and u are { 1 u, 2 u} = 1 u 2 u r 0 (P ), (63) { 1 u, 2 P } = 1 u ρ, (64) where r 0 is an r matrix depending on the dnamical variables P α : r 0 (P ) = α i P α (t α t α t α t α ), (65) while ρ = j H j h j. (66) The matrix r 0 (p) can be regarded as a classical analogue of 6j-symbol (for detailes see [?]). Remark. The linear relation (64) is readily quantized, [ 1 u, 2 P ] = i h 1 u ρ. (67) 9
The quantum R matrix corresponding to (63) was proposed in [9]. It would be intresting to work out a general algorithm permitting to derive it starting from the classical action principle. We now turn to our main example, the quantum group deformation of Γ. Let us introduce the exponentiated angular momentum matrix L and its Gauss decomposition: L = ue iγp u 1 = L + L 1. (68) Here L ± belong to the Borel subgroups B ± of G generated by H j and t ±α and product of their diagonal parts is equal to one cf.(38). The deformed counterpart of the symplectic form (47) looks as follows [8, 7]: Ω γ = tr{dp i u 1 du 1 2γ (eiγp u 1 du e iγp u 1 du + L 1 + dl + L 1 dl )}. (69) In verifying that Ω γ is closed we use the relations 1 6γ tr(l 1 dl) 3 = 1 6γ tr(l 1 dl L 1 + dl + ) 3 = 1 2γ d(trl 1 + dl + L 1 dl ); (70) 1 6γ tr(l 1 dl) 3 = 1 2γ tr{(u 1 du) 2 (e iγp u 1 du e iγp e iγp u 1 du e iγp )} + + i 2 tr{u 1 du(e iγp u 1 du e iγp + e iγp u 1 du e iγp 2u 1 du)dp } = = i(1 cos γp α )dp α Θ α Θ α (71) and tr{(idp 1 2γ eiγp u 1 du e iγp )u 1 du} = dp j Θ j + i α>0 sin γp α Θ α Θ α. (72) γ Let us mention that the first formula (71 is exactly the same as we use to calculate d 1 tr(dg g 1 ) 3 in the WZNW model. The product of Gauss components leads to bosonisation of WZNW action. It is also from (72) that the first two terms in (69) reproduce the undeformed expression (59) in the limit γ 0. As we shall see shortly, the third term vanishes when γ 0 (in accord with (71)). The form (69) possesses a quantum symmetry under left shifts δ ε u = εu, δ ε P = 0: δ ε Ω γ = d tr(ε(t)(dl + L 1 + dl L 1 )). (73) The derivation of the formula (73) is straightforward but quite long. We leave it as an exercise for intrested reader (see also [7] where formula (73) proved in more general setting). Now we proceed to computing the PB of the dynamical variables on Γ. The PB of L ± are easier to find see [8, 7]. The calculation of the PB for a pair of u s requires some work (although their expression has been conjectured correctly in [9]). 10
We shall first compute { 1 u, 2 u} at the group unit and will then use the Lu-Weinstein theorem of Section 2 to derive the PB for arbitrary u s. For u = 1 the right invariant form i du u 1 coincides with the left invariant one (50) while L ± = e ±i γ 2 P. We find il 1 + dl + = γ 2 dp + α>0 2t α Θ α sin γ 2 P α il 1 dl = γ 2 dp + α>0 2t α Θ α sin γ 2 P α (74) and hence we can calculate the symplectic structure at the group unit Ω γ u=1 = dp j Θ j + 2i γ This gives a counterpart of the PB form (61) P γ u=1 = V j e i γ 2 Pα sin( γ α>0 2 P α)θ α Θ α. (75) γ + P j 1 e V iγpα α V α. (76) Now the PB of u and P at the group unit can be written dawn: { 1 u, 2 P } u=1 = ρ, { 1 u, 2 u} u=1 = r γ (P ), (77) where r γ (P ) = α>0 γ 1 e iγpα (t α t α t α t α ). (78) To evaluate the PB for arbitrary u we use the quantum symmetry under left shifts and Eq.(29): { 1 u, 2 u} = {v 1 u 1, v 2 u 2 } u v=u,u=1 +{v 1 u 1, v 2 u 2 } v v=u,u=1 = = v 1 v 2 r γ (P ) v=u +γ[r, v 1 v 2 ] v=u = 1 u 2 u r γ (P ) + γ[r, 1 u 2 u] = γr 1 u 2 u 1 u 2 u r γ (P ). (79) where r is the classical skew symmetric r matrix (35) and r γ(p ) is equal to r γ (P ) = γr r γ = α>0 The PB of u and P just reproduces (64). γe iγpα e iγpα 1 (t α t α t α t α ). (80) 4 Chiral WZNW model. The phase space Γ W Z of chiral WZNW model is spanned by quasiperiodic G valued functions g on the circle S 1 satisfying (16) with x independent monodromy matrix 11
M which is part of the dynamical variables. One expresses the symplectic form on Γ W Z in terms of the left invariant form g 1 (x) dg(x), the boundary values and the Gauss decomposition of the monodromy g 1 = g(0), g 2 = g(2π) (81) M = g 1 1 g 2 = M + M 1, (82) where we require that the diagonal elements of M ± cancel each other (as in the case of L ± ), M+ diag M diag = 1. (83) The symplectic form on Γ W Z is written as [8] Ω = 1 2π 2γ tr{ g 1 dg (g 1 dg)dx + dg 1 g1 1 dg 2 g2 1 M+ 1 dm + M 1 dm }, (84) 0 where γ = 4π (k is the level in the classical case and the height, i.e., the level k plus the dual Coxeter number in the quantum case). The relative coefficients of the three terms in (84) are in fact determined by the requirement that the 2 form is closed, dω = 0. (85) In proving (85) we use the following easily verifiable identities tr 2π 0 g 1 dg (g 1 dg)dx = 1 3 tr{(g 1 2 dg 2 ) 3 (g 1 1 dg 1 ) 3 }; (86) tr d(m 1 + dm + M 1 dm ) = 1 3 tr(m 1 dm) 3. (87) It is straightforward to check that Ω is invariant under periodic left shifts δ ε g = ε(x)g(x), ε(0) = ε(2π). (88) We shall demonstrate that the form (84) also possesses an infinite quantum symmetry under right shifts: δ ε g(x) = g(x)ε(x), (89) where ε is either x independent (the known case of a finite quantum group symmetry) or satisfies the boundary conditions ε(0) = ε + ε i H i + ε α t α, ε(2π) = ε ε i H i + ε α t α. (90) α>0 α>0 In verifying the invariance of the form (84) we use the following relations: δ ε (g 1 dg) = [g 1 dg, ε] + dε, (91) 12
tr{2d δ ε tr g 1 dg (g 1 dg)dx = ε (g 1 dg)dx + g 1 2 dg 2 (dε + [g 1 2 dg 2, ε]) g 1 1 dg 1 (dε + + [g 1 1 dg 1, ε + ])}; (92) δ ε tr(dg 1 g1 1 dg 2 g2 1 ) = (93) = tr(dε + g1 1 g1 1 dg 2 g2 1 g 1 dε g2 1 dg 1 g1 1 g 2 ); δm ± = ε ± M ±, tr(dm M 1 dε dm + M+ 1 dε + ) = 0 (94) δ ε tr(m 1 + dm + M 1 dm ) = tr(m 1 dm dε + dm M 1 dε + ). (95) The result for the sum of three terms then reads δ ε Ω = 2tr d{ ε (g 1 dg)dx + ε + g1 1 dg 1 ε g2 1 dg 2 }. (96) The formula (96) shows that the WZNW symplectic form Ω satisfies conditions of the Section 2 and thus the model enjoys PL symmetry. Acknowledgments The authors thank the Erwing Shrödinger International Institute for Mathematical Physics for hospitality and support during the course of this work. I.T. acknowledges partial support by the Bulgarian Science Foundations for Scientific Reseach under contract F11. References [1] E.K.Sklyanin, On an algebra generated by quadratic relations, Uspekhi Mat.Nauk 40:2(1985)214. [2] L.D.Faddev, N.Yu.Reshetikhin, L.A.Takhtajan, Quantization of Lie groups and Lie algebras, Algebra and Analysis 1:1 (1989)178 (English transl.:leningrad Math.J.1(1990)193-225) (Acad.Press, Boston 1989) pp.129-139 [3] M.A.Semenov-Tian-Shansky, Dressing transformations and Poisson group actions, Publ.RIMS, Kyoto Univ.21:6(1985) 1237-1260 [4] J.N.Lu, A.Weinstein, Poisson-Lie groups, dressing transformations and Bruhat decompositions, J.Diff.Geom.31 (1990)501 13
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