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Generalized double bracket vector fields

Petre Birtea P. Birtea: Department of Mathematics, West University of Timişoara
petre.birtea@e-uvt.ro
Zohreh Ravanpak Z. Ravanpak: Department of Mathematics, West University of Timişoara
zohreh.ravanpak@e-uvt.ro
 and  Cornelia Vizman C. Vizman: Department of Mathematics, West University of Timişoara
cornelia.vizman@e-uvt.ro
Abstract.

By introducing a metriplectic tensor, we generalize the double bracket vector fields defined on semi-simple Lie algebras to the case of Poisson manifolds endowed with a pseudo-Riemannian metric. We also construct a generalization of the normal metric on an adjoint orbit of a compact semi-simple Lie algebra such that the above vector fields, when restricted to a symplectic leaf, become gradient vector fields. We illustrate the newly introduced objects to a variety of examples and carefully discuss complications that arise when the pseudo-Riemannian metric does not induce a non-degenerate metric on parts of the symplectic leaves.

Key words and phrases:
Poisson manifold, Riemannian manifold, double bracket vector field, Lie Poisson structures, Poisson-Lie groups
2020 Mathematics Subject Classification:
53D17, 58D17, 17B20

1. Introduction

On a Riemannian manifold, one of the questions is how can one compute the gradient vector field of a cost function. This problem appears, for instance, in designing steepest descent algorithms on Riemannian manifolds, like matrix Lie groups and homogeneous spaces. To answer this, one needs to have sufficient information on the metric in order to solve the equation that defines the gradient vector field. This is oftentimes not possible or leads to very tedious computations.

A more amenable case is when our manifold of interest is realized as a submanifold in an ambient Riemannian manifold with a simpler Riemannian geometry, so that one can explicitly construct the orthogonal projection operator that relates the gradient vector fields in the two manifolds. This approach has been extensively applied in constrained optimization problems, e.g. in [1][13].

In the often encountered case of a submanifold described by a set of constraint functions, the gradient vector field on the submanifold, with respect to the induced metric, can be realized as a restriction of a vector field (called embedded gradient vector field) defined on the ambient space. This construction has been worked out in [5][6].

Sometimes it happens that not enough components of the constraint map that defines the submanifold are available. This case appears, for instance, when one does not explicitly know enough Casimirs to describe a symplectic leaf of a Poisson manifold. Nevertheless, in the case of the Lie-Poisson structure for a compact Lie algebra, the gradient vector field on the regular symplectic leaves has been obtained as the restriction of a vector field on the ambient space, namely the double bracket vector field. Unlike in the previous setting, the metric on the symplectic leaf will not be the induced one, but the so called normal metric.

Our goal is to obtain an analogue of the above scenario in the setting of a general Poisson manifold equipped with a (pseudo-)Riemannian metric.

More precisely, following an idea presented in [5], we construct a symmetric contravariant tensor that couples the Poisson structure and the Riemannian structure (no compatibility conditions required). This type of tensors are called in the literature metriplectic tensors [16][17][20]. Our newly constructed tensor allows to associate a vector field to every smooth function in a naturally way. We call this vector field the generalized double bracket vector field as, for the case of a compact semi-simple Lie algebra, we recover the classical double bracket vector field. The generalized double bracket vector field has some useful properties. It is tangent to all the symplectic leaves. Moreover, when restricted to such a leaf, it proves to be of gradient type with respect to a natural metric that generalizes the normal metric.

All the constructions above work more generally for a pseudo-Riemannian structure on the Poisson manifold. Thus, non-compact semi-simple Lie algebras can be included. The trade off is that in this case one needs to restrict to what we will call good symplectic leaves, namely those for which the induced metric is non-degenerate.

However, new complications arise in the nonlinear Poisson setting. An important question is if the restriction of the indefinite signature metric to the symplectic leaves is non-degenerate. While for the special case of a non-compact semi-simple Lie algebra this happens on some exceptional leaves only, in the general case, the situation is more intricate and this question will occupy us for a good part of this paper. We will in particular look at the case of the Lie algebra 𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{R}) and a huge class of Poisson structures on 3\mathbb{R}^{3} generalizing it non-linearly, while keeping a rescaling of the Killing form of 𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{R}) in the ambient space 3\mathbb{R}^{3}. This will lead to an interesting and sophisticated interplay between the Poisson and the pseudo-Riemannian structures.

Our constructions that generalize the double bracket vector fields are suitable for the study of various dynamical properties of dissipative systems, analogous to other methods, e.g. those used in [7][18].

Structure of the paper

We start Section 2 by recalling the classical double bracket vector field in the linear case of a semi-simple Lie algebra 𝔤\mathfrak{g}. In order to generalize this setting, following an idea presented in [5], we construct a metriplectic tensor that couples a Poisson structure with a pseudo-Riemannian structure. We introduce the generalized double bracket vector field and prove that in the particular case of a semi-simple Lie algebra it becomes the classical double bracket vector field.

In Section 3, we construct a pseudo-Riemannian metric on symplectic leaves, that we call double bracket metric. This generalizes the normal metric defined on an adjoint orbit of a compact semi-simple Lie algebra. Its construction is limited to good symplectic leaves, or more generally to green zones of symplectic leaves, notions that we define in the body of the paper. We then show that the restriction of the generalized double bracket vector field to a good symplectic leaf or a green zone in a symplectic leaf is the gradient of a smooth function with respect to this double bracket metric.

In Section 4, we provide a large class of Poisson structures on 3\mathbb{R}^{3} and study their symplectic leaves. These include the Lie-Poisson manifold 𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{R})^{*} as well as a related Poisson-Lie group. But they are much more general and, e.g., permit one to construct examples of symplectic leaves of arbitrary genus as explained in Section 5.

In Section 6, we introduce the pseudo-metric gg which we put on the Poisson manifolds of the preceding section. This then permits us to discuss where problems with the non-degeneracy of gg pulled back to symplectic leaves arise. We call the set RR of such points the red zone, as it prohibits applying the constructions of Section 3.

In Section 7, the information about symplectic leaves gathered in Section 4 is combined with the one on the red zone. Intersecting a leaf SS with RR yields what we call red lines; the remainder of the leaf, S\(RS)S\backslash(R\cap S), then provides the green zones. Leaves not having any such an intersection are simply good leaves. It is precisely the good leaves or, more generally, the green zones, where the Theorem 3.4 will be applicable. In the final Section 8 we compute the structures induced on the leaves for the class of our examples and illustrate Theorem 3.4 by means of them.

2. Metriplectic tensor. Generalized double bracket vector field

In this section we introduce the metriplectic tensor and the generalized double bracket vector field. These objects are constructed in a very general setting of a Poisson manifold (M,Π)(M,\Pi) endowed with a pseudo-Riemannian metric gg, with no compatibility conditions required. We recover, as a particular case, the classical double bracket vector field.

We start by recalling the classical setting of the double bracket vector field. Let (𝔤,[,])(\mathfrak{g},[\ ,\ ]) be a semi-simple Lie algebra with κ:𝔤×𝔤\kappa\colon{\mathfrak{g}}\times{\mathfrak{g}}\to\mathbb{R} the Killing form, hence a non-degenerate, symmetric, Ad-invariant bilinear form. The following vector field, called the double bracket vector field, has been introduced by Brockett [11], [12], see also [8], in the context of dynamical numerical algorithms and linear programming:

L˙=[L,[L,N]],\dot{L}=[L,[L,N]], (2.1)

where L𝔤L\in\mathfrak{g} and NN is a fixed regular element in 𝔤{\mathfrak{g}}. It turns out that the double bracket vector field is tangent to the adjoint orbits of the Lie algebra 𝔤{\mathfrak{g}}, orbits that are the symplectic leaves for the linear Poisson bracket on 𝔤{\mathfrak{g}}. More precisely, by identifying the Lie algebra 𝔤{\mathfrak{g}} with its dual 𝔤{\mathfrak{g}}^{*} using the Killing form, the Kirillov-Kostant-Souriau (KKS) Poisson bracket on 𝔤\mathfrak{g}^{*} transforms into the linear Poisson bracket on 𝔤{\mathfrak{g}}:

{F,G}𝔤(L)=κ(L,[F(L),G(L)]).\{F,G\}_{{\mathfrak{g}}}(L)={\kappa}(L,[\nabla F(L),\nabla G(L)]). (2.2)

In the case of a compact semi-simple Lie algebra 𝔤{\mathfrak{g}}, it has been proved that the double bracket vector field (2.1), when restricted to a regular adjoint orbit S𝔤S\subset{\mathfrak{g}}, is a gradient vector field with respect to the normal metric. We recall briefly this construction, for details see [9], [10]. For every LSL\in S consider the orthogonal decomposition with respect to the minus Killing form κ\kappa, that is 𝔤=𝔤L𝔤L{\mathfrak{g}}={\mathfrak{g}}_{L}\oplus{\mathfrak{g}}^{L}, where 𝔤L=Im(adL){\mathfrak{g}}_{L}=\operatorname{Im}(\operatorname{ad}_{L}) and 𝔤L=Ker(adL){\mathfrak{g}}^{L}=\operatorname{Ker}(\operatorname{ad}_{L}). The linear space 𝔤L{\mathfrak{g}}_{L} can be identified with the tangent space TLST_{L}S and 𝔤L{\mathfrak{g}}^{L} with the normal space. One can endow the adjoint orbit SS with the normal metric [4], also called standard metric [2],

νS([L,X],[L,Y])=κ(XL,YL),{\nu}^{S}([L,X],[L,Y])=-{\kappa}(X^{L},Y^{L}), (2.3)

where XL,YLX^{L},Y^{L} are the normal components according to the above orthogonal decomposition of XX, respectively YY.

Theorem 2.1 ([9], [10]).

Let NN be a fixed regular element in the compact semi-simple Lie algebra 𝔤\mathfrak{g}. The gradient of the linear function H:𝔤H\colon\mathfrak{g}\to\mathbb{R} defined by H(L)=κ(L,N)H(L)={\kappa}(L,N) restricted to a regular adjoint orbit SS, taken with respect to the normal metric, is

νS(H|S)(L)=[L,[L,N]].\nabla_{\nu^{S}}(H|_{S})(L)=[L,[L,N]]\,.

We move to a more general case: (M,Π,g)(M,\Pi,{g}) a Poisson manifold endowed with a pseudo-Riemannian metric.

Definition 2.1.

We call metriplectic tensor the following symmetric contravariant 2-tensor :Ω1(M)×Ω1(M)𝒞(M){\mathcal{M}}\colon\Omega^{1}(M)\times\Omega^{1}(M)\to{\cal C}^{\infty}(M),

(α,β):=g(Πα,Πβ).\mathcal{M}(\alpha,\beta):={g}(\sharp_{{}_{\Pi}}\alpha,\sharp_{{}_{\Pi}}\beta)\,.

When α=dF\alpha=\mathrm{d}F and β=dG\beta=\mathrm{d}G, where F,G𝒞(M)F,G\in{\cal C}^{\infty}(M), we have

(dF,dG)=g(XF,XG).{{\mathcal{M}}(\mathrm{d}F,\mathrm{d}G)={g}(X_{F},X_{G}).}
Lemma 2.2.

The following identity =ΠgΠ\sharp_{{}_{\mathcal{M}}}=-\sharp_{{}_{\Pi}}\circ\flat_{g}\circ\sharp_{{}_{\Pi}} holds, where g\flat_{g} is taken relative to the pseudo-Riemannian metric gg on MM.

Proof.

The proof is the following straightforward computation

(α)(β)\displaystyle(\sharp_{{}_{\mathcal{M}}}\alpha)(\beta) =(α,β)=g(Πα,Πβ)=Πβ(g(Πα))=Π(β,gΠ(α))\displaystyle=\mathcal{M}(\alpha,\beta)=g(\sharp_{{}_{\Pi}}\alpha,\sharp_{{}_{\Pi}}\beta)=\sharp_{{}_{\Pi}}\beta(\flat_{g}(\sharp_{{}_{\Pi}}\alpha))=\Pi(\beta,\flat_{g}\circ\sharp_{{}_{\Pi}}(\alpha))
=Π(gΠ(α),β)=(ΠgΠ)(α)(β),\displaystyle=-\Pi(\flat_{g}\circ\sharp_{{}_{\Pi}}(\alpha),\beta)=-(\sharp_{{}_{\Pi}}\circ\flat_{g}\circ\sharp_{{}_{\Pi}})(\alpha)(\beta),

for all α,βΩ1(M)\alpha,\beta\in\Omega^{1}(M). ∎

In the finite dimensional case, the symmetric matrix associated to the contravariant tensor {\mathcal{M}} is given by

[]=[Π]T[g][Π]=[Π][g][Π].[{\mathcal{M}}]=[{\Pi}]^{T}[{g}][{\Pi}]=-[\Pi][g][\Pi]. (2.4)

Metriplectic tensors similar with \mathcal{M}, that combine Poisson structures with Riemannian structures, have been previously introduced, for instance, in the works of J.P. Morrison [16, 17] and I. Vaisman [20].

We introduce the generalization of the double bracket vector field on a nonlinear Poisson manifolds.

Definition 2.2.

Let (M,g,Π)(M,g,\Pi) be a pseudo-Riemannian manifold equipped with a Poisson structure. For a smooth function GG on MM, we call the vector field

G:=idG,\partial_{\mathcal{M}}G:=-{i}_{\mathrm{d}G}{\mathcal{M}}\,, (2.5)

the generalized double bracket vector field.

Remark 2.1.

By Lemma 2.2, one sees that the generalized double bracket vector field G\partial_{\mathcal{M}}G is closely related to the Hamiltonian vector field XGX_{G}. For G𝒞(M)G\in{\cal C}^{\infty}(M), we have

G=(Πg)(XG)=ig(XG)Π,\partial_{\mathcal{M}}G={(\sharp_{\Pi}\circ\flat_{g})(X_{G})}=i_{\flat_{g}(X_{G})}\Pi, (2.6)

hence G\partial_{\mathcal{M}}G is tangent to all the symplectic leaves of Π\Pi. Note also that G=(dG)\partial_{\mathcal{M}}G=-\sharp_{\mathcal{M}}(\mathrm{d}G) by (2.5).

The vector field G\partial_{\mathcal{M}}G is a natural generalization of the double bracket vector field defined on a semi-simple Lie algebra 𝔤\mathfrak{g}. The Killing form provides an isomorphism κ:𝔤𝔤\flat_{\kappa}\colon\mathfrak{g}\to\mathfrak{g}^{*} which transports the Poisson structure from 𝔤\mathfrak{g}^{*} to the linear Poisson structure on 𝔤\mathfrak{g} in (2.2).

Lemma 2.3.

On the semi-simple Lie algebra 𝔤\mathfrak{g}, with the linear Poisson structure (2.2), the Hamiltonian vector field with Hamiltonian function GC(𝔤)G\in C^{\infty}(\mathfrak{g}) is

XG(L)=[L,G(L)], for all L𝔤,X_{G}(L)=[L,\nabla G(L)],\text{ for all }L\in\mathfrak{g}, (2.7)

where the gradient is taken with respect to the Killing metric κ\kappa on 𝔤\mathfrak{g}.

Proof.

The Hamiltonian vector fields with Hamiltonian functions GG and G¯=Gκ{\bar{G}}=G\circ\sharp_{\kappa} on 𝔤\mathfrak{g}^{*} are κ\kappa-related, hence it is enough to show that the Hamiltonian vector field XG¯X_{\bar{G}} is

XG¯(ξ)=κ([L,G(L)]), for all ξ=κ(L)𝔤.X_{\bar{G}}(\xi)=\kappa([L,\nabla G(L)]),\text{ for all }\xi=\flat_{\kappa}(L)\in\mathfrak{g}^{*}.

By the definition of the Lie-Poisson bracket on 𝔤\mathfrak{g}^{*}:

Πξ(L,L)=(ξ,[L,L])\Pi_{\xi}(L,L^{\prime})=(\xi,[L,L^{\prime}])

for all L,L𝔤𝔤L,L^{\prime}\in\mathfrak{g}\cong\mathfrak{g}^{**}. Thus, for ξ=κ(L)\xi=\flat_{\kappa}(L),

Πξ(G(L),L)=(ξ,[G(L),L])=κ(L,[G(L),L])=κ([L,G(L)],L).\Pi_{\xi}(\nabla G(L),L^{\prime})=(\xi,[\nabla G(L),L^{\prime}])=\kappa(L,[\nabla G(L),L^{\prime}])=\kappa([L,\nabla G(L)],L^{\prime}). (2.8)

On the other hand, the Hamiltonian vector field with Hamiltonian function G¯\bar{G} satisfies

Πξ(dG¯(ξ),L)=(XG¯(ξ),L).\Pi_{\xi}(\mathrm{d}\bar{G}(\xi),L^{\prime})=(X_{\bar{G}}(\xi),L^{\prime}). (2.9)

Since G(L)=dG¯(ξ)\nabla G(L)=\mathrm{d}\bar{G}(\xi), where dG¯(ξ)𝔤=𝔤\mathrm{d}\bar{G}(\xi)\in\mathfrak{g}^{**}=\mathfrak{g}, the identity (2.7) follows from (2.8) and (2.9). ∎

Theorem 2.4.

Let (𝔤,[,])({\mathfrak{g}},[\cdot,\cdot]) be a semi-simple Lie algebra endowed with the Poisson bracket defined by (2.2). Then the nonlinear double bracket vector field coincides with the classical double bracket vector field:

G(L)=[L,[L,G(L)]].\partial_{\mathcal{M}}G(L)=[L,[L,\nabla G(L)]].
Proof.

As before, we work on 𝔤\mathfrak{g}^{*}, so let ξ=κ(L)\xi=\flat_{\kappa}(L). The formula (2.6) implies

(G¯(ξ),L)\displaystyle(\partial_{\mathcal{M}}\bar{G}(\xi),L^{\prime}) =Πξ(κ(XG¯)(ξ),L)=Πξ(XG(L),L)=(ξ,[XG(L),L])\displaystyle=\Pi_{\xi}(\flat_{\kappa}(X_{\bar{G}})(\xi),L^{\prime})=\Pi_{\xi}(X_{G}(L),L^{\prime})=(\xi,[X_{G}(L),L^{\prime}])
=κ(L,[XG(L),L])=κ([L,XG(L)],L),\displaystyle=\kappa(L,[X_{G}(L),L^{\prime}])=\kappa([L,X_{G}(L)],L^{\prime}),

hence G¯(κ(L))=κ([L,XG(L)])\partial_{\mathcal{M}}\bar{G}(\flat_{\kappa}(L))=\kappa([L,X_{G}(L)]). By using the fact that the nonlinear double bracket vector fields on 𝔤\mathfrak{g} and 𝔤\mathfrak{g}^{*} are κ\kappa-related, we get G(L)=[L,XG(L)]\partial_{\mathcal{M}}G(L)=[L,X_{G}(L)]. Now the Lemma 2.3 yields the conclusion.

3. Gradient nature of the generalized double bracket vector field

In this section, we prove that the generalized double bracket vector field defined in Definition 2.2, when restricted to a leaf, is a gradient vector field with respect to a metric, that we call double bracket metric. This metric generalizes the normal metric defined on the regular adjoint orbits of a semi-simple compact Lie algebra. In the case when the ambient metric is of indefinite signature, one has to be careful when dealing with submanifolds as they might not be pseudo-Riemannian with respect to the induced metric [19].

As a consequence of Lemma 2.2, we have the inclusion ImImΠ\operatorname{Im}\sharp_{\mathcal{M}}\subseteq\operatorname{Im}\sharp_{\Pi}. Points where the two images do not coincide merit special attention:

Definition 3.1.

A point mMm\in M is called \mathcal{M}-regular if ImΠ|m=Im|m\operatorname{Im}\sharp_{\Pi}|_{m}=\operatorname{Im}\sharp_{\mathcal{M}}|_{m} and mm is called \mathcal{M}-singular otherwise.

Proposition 3.1.

Let SS denote a symplectic leaf of (M,g,Π)(M,g,\Pi) and ι:SM\iota\colon S\hookrightarrow M. Then the two-tensor gindS:=ιgg_{\textnormal{ind}}^{S}:=\iota^{*}g induced by gg on SS is degenerate at sSs\in S if and only if ss is \mathcal{M}-singular.

Proof.

Since Π:TMTS\sharp_{{}_{\Pi}}\colon T^{*}M\rightarrow TS is surjective, gindSg^{S}_{\textnormal{ind}} is non-degenerate iff gindS(Πα,Πβ)=0g^{S}_{\textnormal{ind}}(\sharp_{{}_{\Pi}}\alpha,\sharp_{{}_{\Pi}}\beta)=0 for all βTM\beta\in T^{*}M implies αkerΠ\alpha\in\ker\sharp_{{}_{\Pi}}. The induced metric gindSg^{S}_{\textnormal{ind}} coincides with gg upon evaluation on vectors tangent to SS. By Definition 2.1, one has

gindS(Πα,Πβ)=β(α),g^{S}_{\textnormal{ind}}(\sharp_{{}_{\Pi}}\alpha,\sharp_{{}_{\Pi}}\beta)=\beta(\sharp_{{}_{\mathcal{M}}}\alpha)\,,

for all α,βTM\alpha,\beta\in T^{*}M. The right-hand-side vanishes for all βTM\beta\in T^{*}M iff αker\alpha\in\ker\sharp_{{}_{\mathcal{M}}}. Consequently, gindSg^{S}_{\textnormal{ind}} is degenerate iff KerΠKer\operatorname{Ker}\sharp_{{}_{\Pi}}\subsetneq\operatorname{Ker}\sharp_{{}_{\mathcal{M}}}. This is equivalent with the strict inclusion ImImΠ\operatorname{Im}\sharp_{{}_{\mathcal{M}}}\subsetneq\operatorname{Im}\sharp_{{}_{\Pi}}, hence the conclusion. ∎

Remark 3.1.

For a Riemannian metric gg, all points in MM are \mathcal{M}-regular. In particular, the \mathcal{M}-distribution Im\operatorname{Im}\sharp_{\mathcal{M}} coincides with the characteristic distribution of the Poisson manifold, hence it is integrable. On the other hand, in the case of a pseudo-Riemannian metric gg with signature, the integrability of this \mathcal{M}-distribution is not guaranteed.

Definition 3.2.

We call a symplectic leaf SMS\subset M a good symplectic leaf if the induced metric gindSg^{S}_{\textnormal{ind}} is non-degenerate. Equivalently, SS is a good symplectic leaf if all its points are \mathcal{M}-regular.

Example 3.1.

Consider M=4(w,x,y,z)M=\mathbb{R}^{4}\ni(w,x,y,z) equipped with the pseudo-Riemannian metric111We omit writing the symmetrized tensor product, so, e.g., 2dwdx2\,\mathrm{d}w\mathrm{d}x stands for dwdx+dxdw\mathrm{d}w\otimes\mathrm{d}x+\mathrm{d}x\otimes\mathrm{d}w. g=2dwdx+2dydzg=2\,\mathrm{d}w\mathrm{d}x+2\,\mathrm{d}y\mathrm{d}z and the Poisson bivector field Π:=xy\Pi:=\partial_{x}\wedge\partial_{y}. Every symplectic leaf SS is a plane characterized by w=w0w=w_{0}, z=z0z=z_{0} for some (w0,z0)2(w_{0},z_{0})\in\mathbb{R}^{2}. Here, on every SS, the induced metric vanishes identically,

gindS=0,g^{S}_{\textnormal{ind}}=0\,,

since dw|S=0\mathrm{d}w|_{S}=0 and dz|S=0\mathrm{d}z|_{S}=0. Correspondingly, as a direct matrix calculation shows easily, see (2.4), the metriplectic tensor vanishes identically as well, =0\mathcal{M}=0, so that Im={0}Vect(x,y)=ImΠ\operatorname{Im}\sharp_{\mathcal{M}}=\{0\}\neq\mathrm{Vect}(\partial_{x},\partial_{y})=\operatorname{Im}\sharp_{\Pi}. Every point m4m\in\mathbb{R}^{4} is \mathcal{M}-singular. All leaves are bad in this example.

In what follows we will define the double bracket metric on a good symplectic leaf and we will prove that this metric generalizes the normal/standard metric on adjoint orbits of a semi-simple compact Lie algebra. However, all the statements below will be applicable also to \mathcal{M}-regular regions of a symplectic leaf SS, called green zones in Section 6 below.

Definition 3.3.

The double bracket metric on a good symplectic leaf SS (or in a green zone of a leaf SS) is the pseudo-Riemannian metric defined as

τDBS(X,Y):=(gindS)1(iXωS,iYωS),X,Y𝔛(S){\tau}_{\operatorname{DB}}^{S}\left(X,Y\right):=({g}_{\textnormal{ind}}^{S})^{-1}\left({i}_{X}\omega^{S},{i}_{Y}\omega^{S}\right),\quad X,Y\in\mathfrak{X}(S) (3.1)

where ωS\omega^{S} is the induced symplectic form on the leaf SS and (gindS)1({g}_{\textnormal{ind}}^{S})^{-1} is the co-metric tensor associated to the pseudo-Riemannian metric gindS{g}_{\textnormal{ind}}^{S} induced on SS by the ambient metric.

Lemma 3.2.

The following identity τDBS=ωSgindSωS\flat_{\tau^{S}_{\operatorname{DB}}}=-\flat_{\omega^{S}}\circ\sharp_{g^{S}_{\textnormal{ind}}}\circ\flat_{\omega^{S}} holds on every good symplectic leaf.

Proof.

The proof is the following straightforward computation

(τDBSX)(Y)\displaystyle(\flat_{\tau^{S}_{\operatorname{DB}}}X)(Y) =τ(X,Y)=(gindS)1(iXωS,iYωS)=(ωSY)(gindSωSX)\displaystyle=\tau(X,Y)=({g}_{\textnormal{ind}}^{S})^{-1}\left({i}_{X}\omega^{S},{i}_{Y}\omega^{S}\right)=(\flat_{\omega^{S}}Y)(\sharp_{{g}_{\textnormal{ind}}^{S}}\flat_{\omega^{S}}X)
=ωS(Y,gindSωSX)=ωS(gindSωSX,Y)=ωSgindSωS(X)(Y)\displaystyle=\omega^{S}(Y,\sharp_{{g}_{\textnormal{ind}}^{S}}\flat_{\omega^{S}}X)=-\omega^{S}(\sharp_{{g}_{\textnormal{ind}}^{S}}\flat_{\omega^{S}}X,Y)=-\flat_{\omega^{S}}\sharp_{{g}_{\textnormal{ind}}^{S}}\flat_{\omega^{S}}(X)(Y)

for all X,Y𝔛(S)X,Y\in\mathfrak{X}(S). ∎

In the finite-dimensional case, for xSx\in S, the matrix associated with the co-metric tensor (gindS)1({g}_{\textnormal{ind}}^{S})^{-1} is given by

[gindS(x)]1=[g(x)|TxS×TxS]1,[{g}_{\textnormal{ind}}^{S}(x)]^{-1}=[{g}(x)|_{{T_{x}S\times T_{x}S}}]^{-1},

and consequently, the matrix associated with the double bracket metric τDBS{\tau}_{\operatorname{DB}}^{S} has the expression

[τDBS(x)]=[ωS(x)]T[gindS(x)]1[ωS(x)]=[ωS(x)][gindS(x)]1[ωS(x)].[{\tau}_{\operatorname{DB}}^{S}(x)]=[\omega^{S}(x)]^{T}[{g}_{\textnormal{ind}}^{S}(x)]^{-1}[\omega^{S}(x)]=-[\omega^{S}(x)][{g}_{\textnormal{ind}}^{S}(x)]^{-1}[\omega^{S}(x)].
Lemma 3.3.

The following identity =ιτDBSι\sharp_{{}_{\mathcal{M}}}=\iota_{*}\circ\sharp_{\tau_{\operatorname{DB}}^{S}}\circ\iota^{*} holds over every good symplectic leaf SS, where ι:SM\iota\colon S\to M denotes the immersion of the leaf into the ambient manifold.

Proof.

The proof is the following straightforward computation

\displaystyle\sharp_{{}_{\mathcal{M}}} =ΠgΠ=ιωSιgιωSι\displaystyle=-\sharp_{{}_{\Pi}}\circ\flat_{g}\circ\sharp_{{}_{\Pi}}=-\iota_{*}\circ\sharp_{\omega^{S}}\circ\iota^{*}\circ\flat_{g}\circ\iota_{*}\circ\sharp_{\omega^{S}}\circ\iota^{*}
=ιωSgindSωSι=ιτDBSι,\displaystyle=-\iota_{*}\circ\sharp_{\omega^{S}}\circ\flat_{g_{\textnormal{ind}}^{S}}\circ\sharp_{\omega^{S}}\circ\iota^{*}=\iota_{*}\circ\sharp_{\tau_{\operatorname{DB}}^{S}}\circ\iota^{*},

using Lemma 2.2 at step one and Lemma 3.2 at step four. ∎

Remark 3.2.

In the case when the pseudo-Riemannian metric gg on MM is Riemannian, all symplectic leaves are good leaves. Moreover, the characteristic distribution of the metriplectic tensor \mathcal{M} (i.e. the image of \sharp_{{}_{\mathcal{M}}}) coincides with the characteristic distribution of the Poisson tensor Π\Pi (i.e. the image of Π\sharp_{{}_{\Pi}}). This is a direct consequence of the identity from Lemma 3.3.

The commutative diagram below summarizes Lemma 2.2, Lemma 3.2, and Lemma 3.3.

TM|S\textstyle{T^{*}M|_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Π\scriptstyle{\sharp_{\Pi}}ι\scriptstyle{\iota^{*}}\scriptstyle{\sharp_{\mathcal{M}}}TM|S\textstyle{TM|_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}𝔤\scriptstyle{\flat_{\mathfrak{g}}}TS\textstyle{T^{*}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ωS\scriptstyle{\sharp_{\omega^{S}}}τDBS\scriptstyle{\sharp_{\tau_{DB}^{S}}}TS\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ι\scriptstyle{\iota_{*}}𝔤indS\scriptstyle{\flat_{{\mathfrak{g}}_{ind}^{S}}}TM|S\textstyle{TM|_{S}}TM|S\textstyle{T^{*}M|_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ι\scriptstyle{\iota^{*}}Π\scriptstyle{-\sharp_{\Pi}}TS\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ι\scriptstyle{\iota_{*}}TS\textstyle{T^{*}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ωS\scriptstyle{-\sharp_{{\omega}^{S}}}

In analogy with the compact semi-simple Lie algebra case, we have the following result.

Theorem 3.4.

Let MM be a smooth manifold equipped with a pseudo-Riemannian structure and with a Poisson structure. On a good symplectic leaf SS, the generalized double bracket vector field G\partial_{\mathcal{M}}G, for GC(M)G\in C^{\infty}(M), is minus the gradient vector field of G|SG|_{S} with respect to the double bracket metric:

(G)(x)=τDBS(G|S)(x),xS.(\partial_{\mathcal{M}}G)(x)=-\nabla_{\tau^{S}_{\operatorname{DB}}}{(G|_{S})}(x),\quad x\in S. (3.2)
Proof.

Using the Lemma 3.3, we have,

(G)|S\displaystyle(\partial_{\mathcal{M}}G)|_{S} =((dG))|S=ιτDBS(ι(dG)|S)\displaystyle=(\sharp_{{}_{\mathcal{M}}}(\mathrm{d}G))|_{S}=\iota_{*}\sharp_{\tau_{\operatorname{DB}}^{S}}(\iota^{*}(\mathrm{d}G)|_{S})
=ιτDBS(d(G|S))=ιτDBS(G|S),\displaystyle=\iota_{*}\sharp_{\tau_{\operatorname{DB}}^{S}}(\mathrm{d}(G|_{S}))=-\iota_{*}\nabla_{\tau^{S}_{\operatorname{DB}}}{(G|_{S})},

hence the equality in (3.2). ∎

Next, we will show that for the case of a compact semi-simple Lie algebra the double bracket metric introduced above and the normal metric (2.3) coincide up to a sign.

Theorem 3.5.

Let 𝔤{\mathfrak{g}} be a semi-simple compact Lie algebra and let SS be a regular adjoint orbit in 𝔤\mathfrak{g}. Then

τDBS=νS.{\tau}_{\operatorname{DB}}^{S}=-{\nu}^{S}.
Proof.

Compactness implies that the Killing form has index zero, thus every symplectic leaf SS of 𝔤\mathfrak{g} is a good symplectic leaf.

It is enough to show that for all the functions on S𝔤S\subset\mathfrak{g} of the form H(L)=κ(L,N)H(L)=\kappa(L,N), with NN regular element in SS, the gradient taken with respect to the normal metric νS{\nu}^{S} is the opposite of the gradient taken with respect to the double bracket metric 𝝉DBS\boldsymbol{\tau}_{\operatorname{DB}}^{S}. By Theorem 2.1 we have νSH(L)=[L,[L,N]]{\nabla}_{\nu^{S}}H(L)=[L,[L,N]] for all LSL\in S. On the other hand, by Theorem 2.4, we also have H(L)=[L,[L,H(L)]]=[L,[L,N]]\partial_{\mathcal{M}}H(L)=[L,[L,\nabla H(L)]]=[L,[L,N]]. Combined with Theorem 3.4, it yields that νSH(L)=τDBSH(L){\nabla}_{\nu^{S}}H(L)=-{\nabla}_{\tau^{S}_{\operatorname{DB}}}H(L), thus νS=τDBS{\nu}^{S}=-{\tau}_{\operatorname{DB}}^{S}. ∎

4. A class of Poisson structures on 3\mathbb{R}^{3} and their symplectic leaves

In this section we work with a large class of Poisson structures on 3\mathbb{R}^{3} that illustrate all the new notions introduced in the previous sections. Among them, we specify three examples of a linear Poisson structure, a quadratic Poisson structure and a Poisson Lie group. Symplectic leaves and green zones of each case, where our main theorem is applicable, are carefully studied in the next sections.

The canonical basis of the Lie algebra 𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{R}),

𝐞1=(0100),𝐞2=(0010),𝐞3=(1001),{\bf e}_{1}=\left(\begin{array}[]{cc}0&1\\ 0&0\end{array}\right),~{}~{}~{}{\bf e}_{2}=\left(\begin{array}[]{cc}0&0\\ 1&0\end{array}\right),~{}~{}~{}{\bf e}_{3}=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right),

gives rise to the standard form of its Lie brackets:

[𝐞1,𝐞2]=𝐞3;[𝐞1,𝐞3]=2𝐞1;[𝐞2,𝐞3]=2𝐞2.[{\bf e}_{1},{\bf e}_{2}]={\bf e}_{3};\qquad[{\bf e}_{1},{\bf e}_{3}]=-2{\bf e}_{1};\qquad[{\bf e}_{2},{\bf e}_{3}]=2{\bf e}_{2}. (4.1)

They induce a linear Poisson structure on the dual. More precisely, rescaling the basis elements, x:=𝐞1/2x:={\bf e}_{1}/\sqrt{2}, y:=𝐞2/2y:={\bf e}_{2}/\sqrt{2}, and z:=𝐞3/2z:={\bf e}_{3}/2, and considering them as coordinates on the dual 𝔰𝔩(2,)3\mathfrak{sl}(2,\mathbb{R})^{*}\cong\mathbb{R}^{3}, we get

{x,y}lin=z,{z,x}lin=x,{z,y}lin=y.\{x,y\}_{\mathrm{lin}}=z\>,\qquad\{z,x\}_{\mathrm{lin}}=x\>,\qquad\{z,y\}_{\mathrm{lin}}=-y\>. (4.2)

These can be considered as the fundamental Poisson brackets on 3\mathbb{R}^{3}, extended to all smooth functions by means of the Leibniz rule.

We now consider the following non-linear generalization of these brackets:

{x,y}=U(z)+V(z)xy,{z,x}=x,{z,y}=y,\{x,y\}=U(z)+V(z)\,xy\>,\qquad\{z,x\}=x\>,\qquad\{z,y\}=-y\>, (4.3)

where UU and VV are arbitrarily chosen smooth functions. For the choice U=idU=\mathrm{id} and V=0V=0 we regain the formulas (4.2). It is easy to verify that (4.3) satisfies the Jacobi identity and thus defines a Poisson structure. Such Poisson structures appeared in the study of two-dimensional gravity models [15].

A nice feature of these brackets is that a Casimir function, a non-constant function in the center of the Poisson bracket, can be found explicitly:

Lemma 4.1 ([15]).

Let PP be a primitive of the function VV, P(z)=V(z)P^{\prime}(z)=V(z), and QQ such that Q(z)=U(z)exp(P(z))Q^{\prime}(z)=U(z)\exp(P(z)). Then the function CC(3)C\in C^{\infty}(\mathbb{R}^{3}) defined by

C(x,y,z):=xyexp(P(z))+Q(z)C(x,y,z):=xy\exp{\left(P(z)\right)}+Q(z) (4.4)

is a Casimir function of the brackets (4.3).

The generic symplectic leaves are obtained simply from putting CC equal to some constant cc\in\mathbb{R}, C(x,y,z):=cC(x,y,z):=c. This will permit us to visualize the leaves in different cases.

While the generic leaves are two-dimensional, there are also point-like singular leaves. They occur when the right-hand side of (4.3) vanishes, i.e. when (x,y)=(0,0)(x,y)=(0,0) and zz is a zero of the function UU. The singular symplectic leaves are thus restricted to the zz-axis and lie at such values of zz. Let us denote the union of all singular symplectic leaves of (M,Π)(M,\Pi) by

sing:={(0,0,z)3|U(z)=0}.{\cal L}_{sing}:=\{(0,0,z)\in\mathbb{R}^{3}|U(z)=0\}\,. (4.5)

All other leaves are regular.

Let us now depict some of the symplectic leaves in particular cases:

Example 4.1 (Linear brackets).

For the brackets (4.2), the Casimir function (4.4) becomes

Clin(x,y,z)=xy+12z2.C_{\mathrm{lin}}(x,y,z)=xy+\tfrac{1}{2}z^{2}. (4.6)

To plot some of its well-known level surfaces, it is convenient to use the coordinates

X:=z,Y:=x+y2,T:=xy2X:=z\>,\quad Y:=\frac{x+y}{\sqrt{2}}\>,\quad T:=\frac{x-y}{\sqrt{2}} (4.7)

which gives

2Clin=X2+Y2T2.2C_{\mathrm{lin}}=X^{2}+Y^{2}-T^{2}\,. (4.8)

In this example, there is precisely one singular symplectic leaf, which lies at the origin. The level surface Clin=cC_{\mathrm{lin}}=c for c=0c=0 splits into three leaves, the singular leaf at the origin and the two cones222We still call them “cones” despite of that the tips are not included. with T>0T>0 and T<0T<0, which are regular leaves. Choosing c>0c>0, we obtain a one-sheeted hyperboloid, which topologically is a cylinder while for c<0c<0 we obtain the two-sheeted hyperboloids, two regular leaves of trivial topology each. See Fig. ​1 for an illustration.

(a) a
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(b) b
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(c) c
Figure 1. Symplectic leaves on 𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{R})^{*}: (A) for c=1c=-1, (B) for c=0c=0, and (C) for c=1c=1

As special non-linear cases of (4.3), we first consider the following one:

Example 4.2 (Quadratic brackets).

With the choice

Uqua(z):=3z21,Vqua(z):=0,U_{\mathrm{qua}}(z):={3z^{2}-1}\quad,\qquad V_{\mathrm{qua}}(z):=0\,, (4.9)

one obtains quadratic brackets from (4.3). The Casimir (4.4) takes the form

Cqua=xy+z3z,C_{\mathrm{qua}}=xy+z^{3}-z\,, (4.10)

when choosing P=0P=0 and Q=z3zQ=z^{3}-z. (Changing the integration constants for PP and QQ, leads to the (irrelevant) redefinition CquaeaCqua+bC_{\mathrm{qua}}\mapsto\,e^{a}\,C_{\mathrm{qua}}+b for some a,ba,b\in\mathbb{R}).

There are now precisely two singular leaves, at (0,0,13)(0,0,\tfrac{1}{\sqrt{3}}) and (0,0,13)(0,0,-\tfrac{1}{\sqrt{3}}), as ±13\pm\tfrac{1}{\sqrt{3}} are the two zeros of the function UU, cf. (4.9). These happen for the critical values c=±23c=\pm 2\sqrt{3} of the Casimir function, which implies that the level set splits into regular and singular leaves for those values. For all other values we have two-dimensional and thus regular symplectic leaves. If we choose c=1c=1, for example, we obtain a topologically trivial symplectic leaf, depicted in Fig. ​2.

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Figure 2. Quadratic bracket: symplectic leaf for c=1c=1

On the other hand, for c=0c=0, we find a symplectic leaf with two holes, i.e. essentially two cylinders orthogonal to one another glued together to form a single leaf—see Fig. ​3. Now, topologically, SS is a genus one surface with one puncture.

(a) a
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(b) c
Figure 3. Quadratic bracket: symplectic leaf SS for c=0c=0: (A) seen from the side (B) seen from above. Topologically SS is a punctured torus.

We finally also provide an example where the function VV is non-zero:

Example 4.3 (Poisson-Lie group [3]333For an easier comparison with [3], use the coordinates x1:=zx_{1}:=z, x2:=2yx_{2}:=\sqrt{2}\,y, and x3:=2xx_{3}:=\sqrt{2}\,x in the formulas below and rescale CgrpC_{\mathrm{grp}} by a factor of 12\tfrac{1}{2}.).

Consider the three-dimensional book Lie algebra 𝔤~\widetilde{\mathfrak{g}}, described by the Lie brackets [z~,x~]=ηx~[\widetilde{z},\widetilde{x}]=-\eta\widetilde{x}, [z~,y~]=ηy~[\widetilde{z},\widetilde{y}]=-\eta\widetilde{y}, and [x~,y~]=0[\widetilde{x},\widetilde{y}]=0 for some appropriate choice of generators x~\widetilde{x}, y~\widetilde{y}, and z~\widetilde{z} and a non-vanishing real parameter η\eta, whose significance will become visible shortly. The couple (𝔤~,𝔰𝔩(2,))\left(\widetilde{\mathfrak{g}},\mathfrak{sl}(2,\mathbb{R})\right) forms a well-known Lie bialgebra. Integrating 𝔤~\widetilde{\mathfrak{g}} to its unique connected and simply connected Lie group 𝒢\cal G thus leads to a Poisson-Lie group. It turns out that, as a manifold, 𝒢3{\cal G}\cong\mathbb{R}^{3}, and its Poisson structure is given by (4.3) for the choice

Ugrp(z):=1e2ηz2η,Vgrp(z):=η.U_{\mathrm{grp}}(z):=\frac{1-e^{-2\eta z}}{2\eta}\quad,\qquad V_{\mathrm{grp}}(z):=\eta. (4.11)

We now see that η\eta can be considered as a deformation parameter here: in the limit of sending η\eta to zero, we get back the linear brackets (4.2). For an appropriate choice of integration constants, (4.4) yields

Cgrp=xyeηz+cosh(ηz)1η2C_{\mathrm{grp}}=xye^{\eta z}+\frac{\cosh(\eta z)-1}{\eta^{2}} (4.12)

which also reduces to the 𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{R})-Casimir (4.6) in the limit. This example provides a particularly interesting 1-parameter family of deformations of the linear Poisson structure on 𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{R})^{*} within the infinite-dimensional deformation space governed by the two functions UU and VV as it corresponds to a Poisson-Lie group.

(a) a
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(b) b
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(c) c
Figure 4. Symplectic leaves on Poisson-Lie group for the deformation parameter η=1\eta=1: (A) for c=1c=-1, (B) for c=0c=0, and (C) for c=1c=1. For η0\eta\to 0, they more and more approach the leaves shown in Fig. ​1.

5. Topological nature of certain symplectic leaves

To better understand the topological nature of the symplectic leaves for different choices of UU and cc, still keeping VV zero for simplicity, it is convenient to consider the following function:

hc(z):=Q(z)c.h_{c}(z):=Q(z)-c\,. (5.1)

The number and kind of zeros of this function determines the topology.

Let us first consider Example 4.1, where we depict this function for the three values c=1c=-1, c=0c=0, and c=+1c=+1 in Fig. ​5A.

(a) a
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(b) c
Figure 5. (A) Function hch_{c} for the linear Poisson structure on 𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{R})^{*}, orange color for h1h_{-1}, black color for h0h_{0} and blue color for h1h_{1}. (B) Function hch_{c} for the quadratic Poisson structure, orange color for h0h_{0} and blue color for h1h_{1}.

We see that h1h_{-1} has no zeros and that the corresponding leaf is topologically trivial, cf. Fig. ​1A, while h1h_{1} has two simple roots and the leaf SS for c=1c=1 is a cylinder topologically. This qualitative behaviour of hch_{c} and the corresponding leaves remains if one does not change the sign of cc. For the special value c=0c=0, on the other hand, we have one multiple root of this function and precisely there we find a singular leaf together with two regular ones.

In fact, the latter observation is not a coincidence: Recall that all singular leaves lie on the zz- or XX-axis for values where the function UU vanishes, see (4.5). By the definition (5.1), we see that for every value of zz where UU vanishes also hch_{c}^{\prime} vanishes. So singular leaves appear for values of cc were hch_{c} has a multiple zero.

Let us now look at the second example, Example 4.2, and depict the graph of h1h_{1} and h0h_{0}—see Fig. ​5B. For c=1c=1 the function has one simple zero and the leaf is a plane topologically. For c=0c=0, the function has three simple zeros and the corresponding leaf SS the fundamental group π1(S)=𝔽2\pi_{1}(S)=\mathbb{F}_{2}, the free group with two generators.

More generally, the situation is as follows. If the function hch_{c} has n1n\geq 1 simple zeros, then SS is a Riemann surface of genus [n+12]1\left[\tfrac{n+1}{2}\right]-1, where the square brackets denote the integer part of the enclosed number, with one puncture if nn is odd and two punctures if nn is even. So, for n=3n=3 we obtain a punctured torus, in agreement with what we had found above for its fundamental group. For n=4n=4 we get a torus with two punctures (boundary components)—see Fig. ​6, while for n=5n=5 already a genus two surface with one puncture—see Fig. ​7. Such leaves arise, for example, if we take Q(z)=2(z2)(z1)(z+1)(z+2)Q(z)=2(z-2)(z-1)(z+1)(z+2) and Q(z)=2(z2)(z1)z(z+1)(z+2)Q(z)=2(z-2)(z-1)z(z+1)(z+2), respectively, and choose c=0c=0 (so that h0=Qh_{0}=Q). By reverse engineering, the Poisson brackets yielding such leaves can be obtained from the brackets (4.3) when choosing U=QU=Q^{\prime} and V=0V=0. For completeness, we mention that if hch_{c} has no zeros, one obtains two planar symplectic leaves, and if it has zeros of some multiplicity, the leaf contains singular ones.

(a) a
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(b) c
Figure 6. The symplectic leaf when h(z)=2(z2)(z1)(z+1)(z+2)h(z)=2(z-2)(z-1)(z+1)(z+2), in (A) from the side and in (B) from above. Topologically it is a genus one surface with two punctures or boundary components.
(a) a
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(b) c
Figure 7. The symplectic leaf when h(z)=2(z2)(z1)z(z+1)(z+2)h(z)=2(z-2)(z-1)z(z+1)(z+2), in (A) from the side and in (B) from above. Topologically it is a punctured genus two surface.

Evidently, the freedom in choosing the function UU in the brackets (4.3) permits one to obtain much more intricate symplectic leaves than in the linear case shown in Figure 1. Fixing any integer kk\in\mathbb{N}, by an appropriate choice UU, we can create a sample of symplectic leaves which topologically are Riemann surfaces of genus kk—with optionally one or two boundary components.

6. Degeneracy structure of the metriplectic tensor: red zones

The second main ingredient is the choice of a metric on the space where the leaves are embedded in, i.e. here on the Poisson manifold 3\mathbb{R}^{3}. In the case of the linear 𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{R})^{*} brackets, there is a natural candidate for such a metric, the Killing metric κ\kappa of the Lie algebra 𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{R}). It is of indefinite signature, corresponding to the fact that SL(2,)\operatorname{SL}(2,\mathbb{R}) is a non-compact Lie group. After a trivial rescaling, g:=12κg:=\tfrac{1}{2}\kappa, this metric takes the form

g=2dxdy+dz2=dT2+dX2+dY2,g=2\,\mathrm{d}x\mathrm{d}y+\mathrm{d}z^{2}=-\mathrm{d}T^{2}+\mathrm{d}X^{2}+\mathrm{d}Y^{2}\,, (6.1)

written in both of the coordinate systems used before. Recall that we omit writing the symmetrized tensor product; correspondingly, dz2\mathrm{d}z^{2} stands for dzdz\mathrm{d}z\otimes\mathrm{d}z.

As mentioned in the introduction, we do not want to make a sophisticated choice for the metric on the given Poisson manifold, potentially one that would adapt to the symplectic leaves of the given Poisson structure, for example. The metric should be simple and thus we do not also deform (6.1) along with the brackets.

The metric (6.1) allows us to identify the given 3\mathbb{R}^{3} with a 2++1 dimensional Minkowski space, TT being a time coordinate and XX and YY spatial coordinates. To decide in a case by case study where the potential metric gSg^{S} induced by gg on a given symplectic leaf becomes degenerate, may be cumbersome. In general, one needs an atlas to cover such a leaf—even in the case of the two-dimensional leaves under consideration here. Therefore, one needs to see if a degeneracy of gSg^{S} in some coordinate system on the leaf is due to an intrinsic degeneracy or happens because of a bad choice of coordinates (like when writing the standard metric on 2\mathbb{R}^{2} in polar coordinates). Fortunately, this is not necessary.

Using the Definition 2.1 or, equivalently, Equation (2.4), the matrix corresponding to the metriplectic tensor \mathcal{M} in the basis xx\partial_{x}\otimes\partial_{x}, xy\partial_{x}\otimes\partial_{y}, \ldots, zz\partial_{z}\otimes\partial_{z} takes the form

[]=(x2xyW2xWxyW2y2yWxWyW2xy)\left[\mathcal{M}\right]=\begin{pmatrix}x^{2}&-xy-W^{2}&xW\\ -xy-W^{2}&y^{2}&yW\\ xW&yW&-2xy\end{pmatrix} (6.2)

where we introduced the abbreviation

W=U(z)+xyV(z).W=U(z)+xy\,V(z)\,. (6.3)
Proposition 6.1.

Let (3,g,Π)(\mathbb{R}^{3},g,\Pi) be one of the pseudo-Riemmanian Poisson manifolds considered in this section. The points m=(x,y,z)3m=(x,y,z)\in\mathbb{R}^{3} for which the function

f(x,y,z):=2xy+W2(x,y,z)f(x,y,z):=2xy+W^{2}(x,y,z)\, (6.4)

vanishes are precisely the points of one of the two classes below:

  • mm is an \mathcal{M}-singular point

  • {m}\{m\} is a singular symplectic leaf.

Proof.

We always have ImImΠ\operatorname{Im}\sharp_{\mathcal{M}}\subseteq\operatorname{Im}\sharp_{\Pi}. Thus whenever Πm\Pi_{m} vanishes—i.e. at the points msingm\in{\cal L}_{sing}—also m\mathcal{M}_{m} does. But there are no further points where \mathcal{M} vanishes, as inspection of (6.2) shows. Since (6.4) vanishes at these points, too, we proved the second item in the proposition.

It remains to show that all the remaining points m3m\in\mathbb{R}^{3} for which ff vanishes are those where rkm=1\mathrm{rk}\,\mathcal{M}_{m}=1. First let us show that the condition is sufficient: Replacing (xy+W2)-(xy+W^{2}) by xyxy in the first two lines and 2xy-2xy by W2W^{2} in the third line, we see that each of the three lines is proportional to (xyW)(x\quad y\quad W). So, at points where (6.4) vanishes, the rank of \mathcal{M} is one.

To see that f=0f=0 is also necessary for rkm=1\mathrm{rk}\,\mathcal{M}_{m}=1, note first that if both xx and yy vanish, then the rank of \mathcal{M} cannot be odd. Suppose first then that x0x\neq 0: Subtract the first line multiplied by 1xW\tfrac{1}{x}W from the third. Then the new third line becomes (0f)(0\quad*\quad f). So, f=0f=0 is necessary for rank one of \mathcal{M} in this case. A similar argument for y0y\neq 0—or noting that \mathcal{M} remains unchanged under the diffeomorphism (x,y,z)(y,x,z)(x,y,z)\mapsto(y,x,z)—provides the same result. ∎

The pseudo-Riemannian Poisson manifolds (M,g,Π)(M,g,\Pi) considered in this section are defined by M=3M=\mathbb{R}^{3}, the pseudo-Riemannian metric (6.1), and the bivector (cf. (4.3))

Π=W(x,y,z)xyxxz+yyz,\Pi=W(x,y,z)\,\partial_{x}\wedge\partial_{y}-x\,\partial_{x}\wedge\partial_{z}+y\,\partial_{y}\wedge\partial_{z}\,, (6.5)

where the function WW is given in (6.3) and UU and VV can be chosen arbitrarily. Our main theorem, Theorem 3.4, is applicable whenever we exclude \mathcal{M}-singular points, while singular symplectic leaves—all pointlike in our case, see (4.5)—are admissible. Thus, in view of Prop. 6.1, let us define the set

R:={(x,y,z)3|f(x,y,z)=0}\sing,R:=\{(x,y,z)\in\mathbb{R}^{3}|f(x,y,z)=0\}\backslash{\cal L}_{sing}\,, (6.6)

with sing{\cal L}_{sing} defined in (4.5). More explicitly, the function ff is given by

f(x,y,z)=U2+2(1+UV)xy+V2(xy)2,f(x,y,z)=U^{2}+2(1+UV)\,xy+V^{2}\,(xy)^{2}\,, (6.7)

where UU and VV depend on zz only. RR is a smooth manifold. We will henceforth call RR the red zone of the given triple (M,g,Π)(M,g,\Pi).

Let us illustrate this for the three examples highlighted in Section LABEL:sec:4.1:

Example 6.1.

For the case (𝔰𝔩(2,),κ,Π𝔰𝔩(2,))(\mathfrak{sl}(2,\mathbb{R})^{*},\kappa,\Pi_{\mathfrak{sl}(2,\mathbb{R})^{*}}) of Example 4.1, resulting from U(z)=zU(z)=z and V=0V=0 in (6.5), we see that f(x,y,z)=2C(x,y,z)f(x,y,z)=2C(x,y,z), cf. (4.6). Thus, the red zone RR, depicted in Fig. ​8A, agrees precisely with the two regular symplectic leaves obtained from Clin=0C_{lin}=0 when the origin is excluded, see Fig. ​1B. Both these leaves are thus “bad” symplectic leaves.

(a) a
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(b) b
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(c) c
Figure 8. The red zones RR for the three main examples, all with respect to the metric gg in (6.1):
(A) The Lie Poisson manifold 𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{R})^{*}, Example 4.1,
(B) Example 4.2 with the bivector Πqua\Pi_{qua}, cf. (6.8), and
(C) the Poisson-Lie group of Example 4.3 for the choice η:=1\eta:=1.
In all three cases, RR is invariant under YYY\to-Y as well as, separately, under TTT\to-T. In (A) and (B)(B), RR has two connected components—recall the the singular leaves, all at the conic tips of the red surfaces, do not belong to RR. In (C), RR has four connected components

That, for a generic choice of UU and VV, symplectic leaves are not entirely included in the forbidden red zones becomes well obvious already when looking at the function ff for Example 4.2. The corresponding surface RR is drawn in Fig. ​8B and has more or less nothing to do with symplectic leaves of the Poisson bivector

Πqua=(3z21)xyxxz+yyz,\Pi_{qua}=(3z^{2}-1)\,\partial_{x}\wedge\partial_{y}-x\,\partial_{x}\wedge\partial_{z}+y\,\partial_{y}\wedge\partial_{z}\,, (6.8)

depicted in Figs. 2 and 3 for two values of the Casimir function. As mentioned in the general discussion above, RR does not contain the singular leaves. For the bivector (6.8), these lie at (x,y,z)=(0,0,±13)(x,y,z)=(0,0,\pm\tfrac{1}{\sqrt{3}}), corresponding to X=±13X=\pm\tfrac{1}{\sqrt{3}}, Y=0Y=0 and T=0T=0. These two points are well visible in Fig. ​8B; they coincide with the conic tips of the red surface, to which they do not belong.

The red zones for Example 4.3, the Poisson-Lie group, are drawn in Fig. ​8C. They are already quite intricate to imagine graphically and it is not obvious at first sight, if not some of the symplectic leaves, such as those depicted in Fig. ​4, would lie inside the four connected components of RR. That this is not the case becomes evident when looking at the intersection with the symplectic leaves of (M,Πgrp)(M,\Pi_{grp}): The intersection of the three leaves of Fig. 4 with RR is depicted in Fig. 9.

(a) a
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(b) b
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(c) c
Figure 9. Intersecting the red zone RR of the Poisson Lie group, drawn in red for η=1\eta=1, with the three symplectic leaves obtained for (A) c=1c=-1, (B) c=0c=0, and (C) c=1c=1, all drawn in green.

7. Geometric interpretation, green zones and red lines

There is a geometrical way of seeing from the pictures if a leaf SS or a region inside SS carries a non-degenerate induced metric. And this is the main reason, why we used the coordinates XX, YY, and TT for the pictures in this section—still keeping xx, yy, and zz for the calculations, as they are somewhat more convenient there. With the “causal coordinates” (X,Y,T)(X,Y,T), we identify MM with the 2++1-dimensional “Minkowski space” 3={(X,Y,T)}\mathbb{R}^{3}=\{(X,Y,T)\}.

Then the three symplectic leaves in Fig. ​1 receive the following reinterpretation. The one with T>0T>0 is the “future light cone”, the one with T<0T<0 the “past light cone”. They are separated by “present time”, the pointlike symplectic leaf at (0,0,0)(0,0,0).

Let us now recall some basic facts and terminology from Minkowski space MM: A vector vTMv\in TM is called time-like if its “length” (squared) is negative, g(v,v)<0g(v,v)<0, space-like if g(v,v)>0g(v,v)>0, and null, if g(v,v)=0g(v,v)=0. Vectors are time-like if they “essentially vertical” in our drawings, i.e. if they have an angle less than 45 degrees with respect to the the TT-axis, they are null, if this angle is precisely 4545 degrees, and space-like if they are “essentially horizontal” (angle bigger than 45 degrees). A curve is called time-like, space-like or null, if all its tangent vectors are of the corresponding nature.

A submanifold or part of a submanifold SS is of Riemannian or Euclidean nature—the induced metric gindS{g}_{\textnormal{ind}}^{S} is positive definite—if all curves lying inside SS are space-like. This is, for example, evidently the case for Fig. ​1A, but also applies to all the yellow parts in the leaves of Fig. ​4. Now, a submanifold SS, or a part of it, is pseudo-Riemannian or Lorentzian, if, at every of its points ss, it contains curves passing through ss that are spacelike and (other) such curves that are timelike. This applies, e.g., to the symplectic leaf depicted in Fig. ​2, but also to at least part of the light green regions in the leaves of Fig. ​4. Thus, for example, the leaf SS depicted in Fig. ​3 contains Euclidean regions—at least at the bottom and top of the hole visible in Fig. ​3A—as well as Lorentzian regions—like the left side of Fig. ​3A. Due to the signature change needed for this to happen, there must be lines, which we will call red lines, where the signature change happens and where thus necessarily the induced bilinear form gindS{g}_{\textnormal{ind}}^{S} must be degenerate.

For completeness we return also to the future and past light cones, see Fig. ​1B and Fig. ​8A: Through every point mm on them, there is a null-curve (consider the straight line connecting the origin with mm), there are also plenty of space-like curves. However, there is no time-like curve passing through mm. So the zero length of the null tangent vector can be explained there only by the fact that such a vector becomes an eigenvector with eigenvalue zero of the induced bilinear form gindSg_{\textnormal{ind}}^{S} at mm. We deal with entirely bad leaves in this case.

It is obvious from the above qualitative discussion, however, that leaves which are \mathcal{M}-singular everywhere (“bad leaves”) are very exceptional. More often they will be either good leaves—leaves which do not have an intersection with the red zone RR—or leaves of mixed nature, where Lorentzian and Euclidean regions are separated by red lines of \mathcal{M}-degenerate points.444More generally, one may want to call leaves SS where \mathcal{M}-degenerate points form a subset of measure zero “almost good leaves”: there some care will be needed when approaching the forbidden red walls, but within a good region, the conditions for Theorem 3.4 to hold true are satisfied.

Let us illustrate this by means of Example 4.2: Fig. ​2 shows a leaf SS that is endowed with a Lorentzian metric gSg^{S}. That it does not contain any \mathcal{M}-singular points is also confirmed by Fig ​10A, which shows SS (the green surface) together with RR (red surface): they do not intersect.

Refer to caption
(a) B
Refer to caption
(b) c
Figure 10. Intersecting the red zone RR of (3,g,Πqua)(\mathbb{R}^{3},g,\Pi_{qua}), drawn in red, with the two symplectic leaves, drawn in green, which are obtained for (A) c=1c=1 and (B) c=0c=0. While in (A) there are no intersections, in (B) there are (see also Fig. ​10 below).

This changes for the symplectic leaf SS^{\prime} of Fig. ​3: Fig ​10B shows that SS^{\prime} has non-empty intersections with RR. Keeping only SS^{\prime} and the parts of RR intersecting SS^{\prime}, we obtain Fig. ​10. We call the admissible, i.e. \mathcal{M}-regular parts of a symplectic leaf of interest a green zone and each intersection of the leaf with the red zone a red line.555The “red lines” might be called “red domain walls” for higher dimensional almost regular leaves.

Refer to caption
Figure 11. The green zones and red lines for the symplectic leaf corresponding to c=0c=0 in the example with quadratic brackets. The dark green region between the two red lines is of Euclidean signature, the light green regions left and right of the lines are of Lorentzian signature.

In the class of examples discussed in this paper, there is a nice way of characterizing the red lines. Let us fix a symplectic leaf ScS_{c} corresponding to the value cc of the Casimir function (4.4). Then we see that on the symplectic leaf ScS_{c}, we can express xyxy as a function of zz only: xy=exp[P(z)][cQ(z)]xy=\exp[-P(z)]\cdot\left[c-Q(z)\right]. Plugging this into (6.7), we obtain a function Fc=f|Sc:F_{c}=f|_{S_{c}}\colon\mathbb{R}\to\mathbb{R}, which takes the form

Fc=U2+2(1+UV)eP(cQ)+V2e2P(cQ)2.F_{c}=U^{2}+2(1+UV)e^{-P}(c-Q)+V^{2}\,e^{-2P}(c-Q)^{2}. (7.1)

It is precisely the zeros zredz_{red} of this function, Fc(zred)F_{c}(z_{red}), that determine the red lines on the leaf ScS_{c}; they consist of those points on ScS_{c} where the XX-coordinate takes one of the values of the zeros of the function FcF_{c},

Xz=zred.X\equiv z=z_{red}. (7.2)

It is remarkable that, for every choice of UU and VV, the intersection of the red zone with any symplectic leaf has a constant value for XX.

Note that the intersection of the planes (7.2) with a symplectic symplectic leaf SS gives the red lines only if S=ScS=S_{c}. Such as the zeros of the function hch_{c} determine the topological nature of the (corresponding) symplectic leaf ScS_{c} of (M,Π)(M,\Pi), the zeros of the function FcF_{c} determine the red lines and green zones of this leaf in (M,g,Π)(M,g,\Pi). Moreover, if Fc(z)>0F_{c}(z)>0, then for this value of zz or XX the induced metric has Lorentzian signature, and if Fc(z)<0F_{c}(z)<0, it is Euclidean.

Let us return to the example of quadratic brackets for an illustration again: The function (7.1) becomes Fc=9z42z36z2+2z+1+2cF_{c}=9z^{4}-2z^{3}-6z^{2}+2z+1+2c. Its graph is drawn in Fig. ​12 for the two values of cc corresponding to the leaves depicted in Fig. ​2 (c=1c=1) and Fig. ​3 (c=0c=0).

Refer to caption
Figure 12. The function FcF_{c} for the quadratic Poisson structure, orange color for F0F_{0} and blue color for F1F_{1}.

We see from this diagram that for c=1c=1 there are no zeros of this function and that it is strictly positive. This implies that the corresponding leaf S0S_{0}, see Fig. ​2, is a good leaf and that the induced metric is of Lorentzian signature everywhere. Certainly, we found this before already by other means, but we see that it can be deduced very easily from just the inspection of the graph of the one-argument function F1F_{1}.

Likewise, we see that F0F_{0} has two zeros. They are located at the values zred,10.77z_{red,1}\approx-0.77 and zred,20.30z_{red,2}\approx-0.30. This fixes the location of the two red lines on the leaf S0S_{0} depicted in Fig. ​11. Between these two values of zXz\equiv X the function F0F_{0} is negative and thus the region between the red lines, depicted as dark green in Fig. ​11, is Riemannian. For values of XX smaller than zred,1z_{red,1} or bigger than zred,2z_{red,2}, F0F_{0} is positive and therefore those green zones, depicted by light green in Fig. ​11, carry an induced metric of Lorentzian signature.

8. Generalized double bracket vector field as gradient vector field in green zones

Now we turn to determining explicitly the geometric data on a regular symplectic leaf SS, i.e. the induced metric gindSg_{\textnormal{ind}}^{S} and the symplectic form ωS\omega^{S}. The leaves result from imposing C(x,y,z)=cC(x,y,z)=c. Taking the differential of this equation, one has

xdy+ydx+Wdz0.x\,\mathrm{d}y+y\,\mathrm{d}x+W\mathrm{d}z\approx 0. (8.1)

Here and in what follows we use \approx for all equations valid on SS only. Then, when using the coordinates (x,z)(x,z) on SS, one has

dyydx+Wdzx.\mathrm{d}y\approx-\frac{y\,\mathrm{d}x+W\mathrm{d}z}{x}\,. (8.2)

We see that these coordinates cover regions of the leaf where x0x\neq 0. Likewise, when using (y,z)(y,z) on the leaf, we get expressions valid for y0y\neq 0. Finally, the coordinates (x,y)(x,y) are good coordinates on SS whenever W(x,y,z)0W(x,y,z)\neq 0. Since the simultaneous vanishing of xx, yy and WW correspond precisely to the singular pointlike leaves, we see that we can cover all of SS with these three coordinate charts. We will, henceforth, content ourselves with the coordinate chart (x,z)(x,z). For the other two charts one obtains similar expressions.

Plugging (8.2) into the embedding metric (6.1), we see that the induced metric becomes

gindS2yxdx22Wxdxdz+dz2g_{\textnormal{ind}}^{S}\approx-\frac{2y}{x}\mathrm{d}x^{2}-\frac{2W}{x}\mathrm{d}x\mathrm{d}z+\mathrm{d}z^{2} (8.3)

in the chart with x0x\neq 0. Here yy is understood as the following function of xx, zz, and the parameter cc:

yeP(z)(cQ(z))x=:y(x,z),y\approx\frac{e^{-P(z)}\,\left(c-Q(z)\right)}{x}=:y(x,z)\,, (8.4)

where PP and QQ are the functions defined in Lemma 4.1. The coordinate yy also enters WW, so in (8.3)

WU(z)+2xy(x,z)V(z).W\approx U(z)+2x\,y(x,z)\,V(z). (8.5)

The apparent singularity of gindSg_{\textnormal{ind}}^{S} results from the fact that the coordinate system (x,z)(x,z) on SS breaks down for x=0x=0. We remark in parenthesis that this corresponds to the plane Y=TY=T in the coordinate system (4.7)\eqref{XYT}. On the other hand, there is an inherent problem with gindSg_{\textnormal{ind}}^{S} when a point on the leaf SS lies inside the red zone RR, cf. (6.6): Calculating the determinant of the induced metric, we find

detgindSfS(x,z)x2,\det g_{\textnormal{ind}}^{S}\approx-\frac{f^{S}(x,z)}{x^{2}}\,, (8.6)

where fS(x,z):=f(x,y(x,z),z)f^{S}(x,z):=f(x,y(x,z),z) with the function ff as defined in (6.4). So, as expected, detgindS\det g_{\textnormal{ind}}^{S} vanishes on the red lines. This corresponds to

kergindS|mVect(xx|m+Wz|m)mRS.\ker\sharp_{g_{\textnormal{ind}}^{S}}|_{m}\approx\mathrm{Vect}(x\partial_{x}|_{m}+W\partial_{z}|_{m})\qquad\qquad\forall m\in R\cap S\,. (8.7)

On the other hand, from the second Poisson bracket in (4.3), we deduce that

ωSdxdzx.\omega^{S}\approx\frac{\mathrm{d}x\wedge\mathrm{d}z}{x}\,. (8.8)

Certainly, ωS\omega^{S} is well-defined on all of the leaf SS. The apparent singularity at x=0x=0 is a coordinate singularity.

This brings us to the position of determining the double-bracket metric (3.1). A direct calculation yields

τDB2ydxdx+2Wdxdzxdzdzx(2xy+W2),\tau_{DB}\approx\frac{2y\mathrm{d}x\mathrm{d}x+2W\,\mathrm{d}x\mathrm{d}z-x\mathrm{d}z\mathrm{d}z}{x(2xy+W^{2})}\,, (8.9)

with yy and WW as in (8.4) and (8.5) certainly. We remark that

τDB1fSgindS.\tau_{DB}\approx-\frac{1}{f^{S}}\;g_{\textnormal{ind}}^{S}\,. (8.10)

So evidently this tensor is not well-defined on the red lines, i.e. for the points mRSm\in R\cap S—and it also does not have a continuous continuation into such points. This is in contrast to gindSg_{\textnormal{ind}}^{S}, which is a well-defined tensor on all of SS; it just does not define a pseudo-metric on RSR\cap S, since there it has a kernel, see (8.7). In the green zones, however, i.e. on S\(RS)S\backslash(R\cap S), both, gindSg_{\textnormal{ind}}^{S} and τDB\tau_{DB}, define a (pseudo) metric.

The second main ingredient in Theorem 3.4 is the generalized double bracket vector field G\partial_{\mathcal{M}}G, which is defined on all of M=3M=\mathbb{R}^{3} certainly. The easiest way to determine it for a function GC(3)G\in C^{\infty}(\mathbb{R}^{3}) at this point is—see (2.5)—to apply the negative of the matrix (6.2) to the vector [dG]=(G,x,G,y,G,z)[\mathrm{d}G]=(G,_{x},G,_{y},G,_{z}), where the comma denotes the derivative with respect to the corresponding coordinate. This leads to

G=\displaystyle\partial_{\mathcal{M}}G=\> (x2G,x+[xy+W2]G,yxWG,z)x+\displaystyle\left(-x^{2}G,_{x}+\left[xy+W^{2}\right]G,_{y}-xWG,_{z}\right)\partial_{x}\,+
(y2G,y+[xy+W2]G,xyWG,z)y+\displaystyle\left(-y^{2}G,_{y}+\left[xy+W^{2}\right]G,_{x}-yWG,_{z}\right)\partial_{y}\,+ (8.11)
(xWG,xyWG,y+2xyG,z)z.\displaystyle\left(-xWG,_{x}-yWG,_{y}+2xyG,_{z}\right)\partial_{z}\,.

To see how it arises from more geometrical quantities, we also display the three elementary Hamiltonian vector fields corresponding to the canonical coordinates (x,y,z)3(x,y,z)\in\mathbb{R}^{3},

Xx\displaystyle X_{x} =Wyxz,\displaystyle=W\partial_{y}-x\partial_{z}\,,
Xy\displaystyle X_{y} =Wx+yz,\displaystyle=-W\partial_{x}+y\partial_{z}\,, (8.12)
Xz\displaystyle X_{z} =xxyy.\displaystyle=x\partial_{x}-y\partial_{y}\,.

Now, g\flat_{g} applied to any vector field ax+by+cza\partial_{x}+b\partial_{y}+c\partial_{z} yields the 1-form bdx+ady+cdzb\mathrm{d}x+a\mathrm{d}y+c\mathrm{d}z, and since πdx=Xx\sharp_{\pi}\mathrm{d}x=X_{x} etc, we easily find the image of (8) under πg\sharp_{\pi}\circ\flat_{g} to be

π(g(Xx))\displaystyle\sharp_{\pi}\left(\flat_{g}(X_{x})\right) =WXxxXz,\displaystyle=WX_{x}-xX_{z}\,,
π(g(Xy))\displaystyle\sharp_{\pi}\left(\flat_{g}(X_{y})\right) =WXy+yXz,\displaystyle=-WX_{y}+yX_{z}\,, (8.13)
π(g(Xz))\displaystyle\sharp_{\pi}\left(\flat_{g}(X_{z})\right) =xXyyXx.\displaystyle=\>\;\>\,xX_{y}-yX_{x}\,.

Plugging (8) into (8), we obtain—cf. (2.6)—the three fundamental double bracket vector fields x\partial_{\mathcal{M}}x, y\partial_{\mathcal{M}}y, and z\partial_{\mathcal{M}}z, respectively.

The vector field G\partial_{\mathcal{M}}G is defined on 3\mathbb{R}^{3}, but it is tangent to the symplectic leaves and thus also to the singled-out leaf SS. We thus can view the restriction G|S\partial_{\mathcal{M}}G|_{S} of G\partial_{\mathcal{M}}G to SS as a vector field on SS. To see what section in TSTS it is in our coordinate system (x,z)(x,z) on SS chosen above, we first express GΓ(3)\partial_{\mathcal{M}}G\in\Gamma(\mathbb{R}^{3}) in a coordinate system adapted to the symplectic leaves. We choose x~:=x\widetilde{x}:=x, y~:=C(x,y,z)\widetilde{y}:=C(x,y,z), and z~:=z\widetilde{z}:=z; this is a good coordinate system on 3\({0}×2\mathbb{R}^{3}\backslash(\{0\}\times\mathbb{R}^{2}). Under such a coordinate change, x\partial_{x} does not simply become x~\partial_{\widetilde{x}}, for example, despite the fact that x=x~x=\widetilde{x}, but one rather has

x=x~+C,yy~.\partial_{x}=\partial_{\widetilde{x}}+C,_{y}\partial_{\widetilde{y}}\,.

However, since the vector field G\partial_{\mathcal{M}}G is tangent to the symplectic leaf SS when restricted to it, all contributions proportional to y~\partial_{\widetilde{y}} will cancel out, while the ones proportional to x\partial_{x} and z\partial_{z} remain unchanged. Thus, in this new coordinate system, the vector field (8.11) takes the same form after replacing untilded coordinates by tilded ones and, at the same time, simply dropping the second line on the right-hand side.

Therefore, after the dust clears and using again the coordinates x=x~x=\widetilde{x} and z=z~z=\widetilde{z} as coordinates on the leaf SS, we obtain

G|S(x2G,x+[xy+W2]G,yxWG,z)x+(xWG,xyWG,y+2xyG,z)z.\partial_{\mathcal{M}}G|_{S}\approx\left(-x^{2}G,_{x}+\left[xy+W^{2}\right]G,_{y}-xWG,_{z}\right)\partial_{x}+\left(-xWG,_{x}-yWG,_{y}+2xyG,_{z}\right)\partial_{z}\,. (8.14)

Here, the derivatives of GG are taken before restricting to SS, and, as before, the functions yy and WW are given by (8.4) and (8.5), respectively.

We are finally ready to combine the two main ingredients (8.9) and (8.14). After a somewhat tedious calculation and the cancellation of several terms, one obtains from this:

τDB(G|S,)G,xdx+G,y(yxdx+Wxdz)G,zdz.\tau_{DB}\left({\partial_{\mathcal{M}}G}|_{S},\cdot\right)\approx-G,_{x}\mathrm{d}x+G,_{y}\left(\frac{y}{x}\mathrm{d}x+\frac{W}{x}\mathrm{d}z\right)-G,_{z}\mathrm{d}z\,. (8.15)

Upon usage of (8.2), the term in the brackets following G,yG,_{y} is recognized to be precisely dy-\mathrm{d}y. This implies

τDB(G|S,)d(G|S),\tau_{DB}\left({\partial_{\mathcal{M}}G}|_{S},\cdot\right)\approx-\mathrm{d}\left(G|_{S}\right)\,, (8.16)

which is equivalent to Equation (3.2).

In the above manipulations, it is understood that one is within the green zone of SS since otherwise τDB\tau_{DB} would not be defined (cf. (8.10) and the discussion following it). On the other hand, we see that according to the right-hand side of (8.16), the left-hand side has a continuous continuation to \mathcal{M}-singular points. This can be explained as follows: Recall that G\partial_{\mathcal{M}}G is a well-defined vector field on all of 3\mathbb{R}^{3} and thus also the right-hand side of (8.14) is defined on RSR\cap S if only x0x\neq 0 (applicability of our coordinate patch). Using that on RR we can replace 2xy-2xy by W2W^{2}, see (6.4) and (6.6), we find that in our chart (x,z)(x,z) on SS:

G|SR(xG,x+yG,y+WG,z)(xx+Wz).\partial_{\mathcal{M}}G|_{S\cap R}\approx\left(xG,_{x}+yG,_{y}+WG,_{z}\right)\>\left(x\partial_{x}+W\partial_{z}\right)\,. (8.17)

On RR, the image of \sharp_{\mathcal{M}} becomes one-dimensional and, upon restriction to a leaf SS coincides with the kernel of τDBS\tau_{DB}^{S} there, see (8.7). So, while τDB\tau_{DB} simply blows up on RR, see (8.10), it is not simply that the vector field G\partial_{\mathcal{M}}G would go to zero there to compensate for the singularity in τDB\tau_{DB}. Instead, approaching a point mm in the red zone, it more and more turns into the direction of the non-zero kernel of fSτDBf^{S}\,\tau_{DB} at such an mm.


Conclusions. In this paper we have generalized the double vector field construction defined originally only on semi-simple Lie algebras to general pseudo-Riemannian Poisson manifolds. The generalized double bracket vector field has some useful properties. It is tangent to all the symplectic leaves. Moreover, when restricted to such a leaf, it proves to be of gradient type. This happens with respect to a metric on the leaf which generalizes the normal metric on adjoint orbits in compact semi-simple Lie algebras.

In the case of a Poisson manifold with metric of indefinite signature, the generalization works only on regions where the metric induced on the symplectic leaves is non-degenerate. These regions are called green zones and leaves for which the induced metric is non-degenerate everywhere are called good leaves. We have provided a characterization of such regions which avoids determining the induced metric explicitly, which can become quite complicated. All this have been illustrated for a wide class of Poisson structures on 3\mathbb{R}^{3}, which we have discussed in full detail.


Acknowledgements. Z. Ravanpak acknowledges a scholarship “Cercetare postdoctorală avansată” funded by the West University of Timişoara, Romania, and the financial support from the Spanish Ministry of Science and Innovation under grants PID2022-137909NB-C22. The authors are grateful to Thomas Strobl for valuable discussions and helpful suggestions to improve significantly the presentation of the paper.

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