Poisson-Lie groups and inequalities

Thesis

Reference

Poisson-Lie groups and inequalities

DAVYDENKOVA, Irina

Abstract

In this thesis, we establish a new link between Poisson Geometry and Combinatorics. We introduce the notion of tropicalization for Poisson structures on R^n with coefficients in Laurent polynomials. To such a Poisson structure we associate a polyhedral cone and a constant Poisson bracket on this cone. There is a version of this formalism applicable to C^n viewed as a real Poisson manifold. In this case, the tropicalization gives rise to a completely integrable system with action variables taking values in a polyhedral cone and angle variables spanning a torus. The main example considered in the thesis is the canonical Poisson bracket on the dual Poisson-Lie group G^* for G=U(n) in the cluster coordinates of Fomin-Zelevinsky defined by a certain choice of solid minors. We prove that the corresponding integrable system is isomorphic to the Gelfand-Zeitlin completely integrable system of Guillemin-Sternberg and Flaschka-Ratiu.

DAVYDENKOVA, Irina. Poisson-Lie groups and inequalities. Thèse de doctorat : Univ.

Genève, 2014, no. Sc. 4638

URN : urn:nbn:ch:unige-344710

DOI : 10.13097/archive-ouverte/unige:34471

Available at:

http://archive-ouverte.unige.ch/unige:34471

Disclaimer: layout of this document may differ from the published version.

UNIVERSIT´E DE GEN`EVE FACULT´E DES SCIENCES Section des Math´ematiques Professeur Anton Alekseev

Poisson-Lie Groups and Inequalities

TH`ESE

pr´esent´ee `a la Facult´e des Sciences de l’Universit´e de Gen`eve pour obtenir le grade de Docteur `es sciences, mention interdisciplinaire

par

Irina Davydenkova

de

Saint-P´etersbourg (Russie)

Th`ese No. 4638

Contents

R´esum´e 2

1 Introduction 5

2 Poisson structure and Poisson-Lie groups 7 2.1 Log-canonical Poisson brackets . . . 7 2.2 Poisson-Lie groups . . . 9

3 Lindstr¨om Lemma 11

Article 16

A Example: tropicalization of Poisson-Lie group U^∗(3) 51

R´ esum´ e

Dans cette th`ese, nous ´etablissons un nouveau lien entre la g´eom´etrie de Poisson et la combinatoire. Consid´erons des coordonn´ees {x1, . . . , xn} sur une vari´et´e de Poisson et supposons que les crochets de Poisson des fonctions {xi, xj} sont des polynˆomes de Laurent en x1, . . . , xn. Ensuite, nous associons `a ces donn´ees un cˆoneC ⊂V ∼=Rⁿ poly´edrique et le crochet de Poisson constantπ_∞surC. Ce crochet de Poisson constant est d´etermin´e par la partie log-canonique du crochet de Poisson original, c’est-`a-dire par les contributions de la formeCxixj

dans {xi, xj}. Nous appelons les donn´ees combinatoires (C, π_∞) la tropicalisation de la structure de Poisson d’origine.

Il existe une version de notre construction qui permet de travailler avec des valeurs complexes des fonctions de coordonn´ees zi (qui sont accompagn´ees par leurs conjugu´es zi). Dans ce cas, la tropicalisation est `a nouveau donn´ee par une paire (C, π<sub>∞</sub>), o`u C ⊂ V ∼= Rⁿ est un cˆone poly´edrique, n est le nombre total de fonctions (r´eelles et complexes) de coordonn´ees, etπ_∞est un crochet de Poisson surC×T^m. Ici, T^m est un tore r´eel de dimension m ´egal au nombre des fonctions de coordonn´ees complexes. En fait, la paire (C, π_∞) donne lieu `a un syst`eme compl`etement int´egrable avec les variables d’action propos´ees par des fonctions lin´eaires sur C et les variables d’angle par les les fonctions naturelles de coordonn´ees sur T^m.

L’exemple principal consid´er´e dans la th`ese est le groupe de Poisson-Lie dual U^∗(n) ´equip´e de la structure standard de Poisson-Lie propos´ee par Semenov-Tian-Shansky et Lu-Weinstein.

Nous avons choisi le syst`eme particulier de coordonn´ees sur U<sup>∗</sup>(n) d´efinies par Fomin-Zelevinsky dans le cadre de la th´eorie de la positivit´e totale. Le r´esultat principal de la th`ese affirme que la tropicalisation de la structure de Poisson sur U^∗(n) avec les coordonn´ees de Fomin-Zelevinsky est isomorphe au syst`eme int´egrable de Gelfand-Zeiltin de Guillemin et Sternberg. Ce dernier est un

syst`eme int´egrable d´efini sur l’espace des matrices hermitiennes avec le crochet de Poisson de Kirillov-Kostant-Souriau avec des variables d’action propos´ees par les valeurs propres de sous-matrices principales La preuve du r´esultat principal repose sur deux techniques.

D’une part, nous utilisons les propri´et´es de l’application tropicale de Gelfand-Zeiltin d´evelopp´ee par Alekseev, Podkopaeva et Szenes.

Cette technique permet de montrer que le cˆone de Gelfand-Zeiltin est contenu dans le cˆone C qui nous sommes int´eresse. D’autre part, par une ´etude attentive de la combinatoire du crochet de Poisson sur U^∗(n), on trouve toutes les in´egalit´es qui d´efinissent le cˆone de Gelfand-Zeiltin parmi les in´egalit´es qui d´efinissent le cˆone C. Pour obtenir le crochet de Poisson π_∞, nous ´etudions en d´etail la version simplifi´ee U₀^∗(n) de U^∗(n) qui capture les propri´et´es de la partie log-canonique du crochet de Poisson.

La th`ese se compose des parties suivantes. Nous commen¸cons par deux chapitres d’introduction. Dans le premier, nous rappelons les d´efinitions de base des crochets de Poisson log-canonique et des groupes de Poisson-Lie et donnons quelques exemples. Dans le second, nous discutons du lemme de Lindstr¨om qui lie des r´eseaux planaires et des matrices. La partie centrale de la th`ese est un article commun avec A. Alekseev intitul´e “Inequalities from Poisson brackets”. Dans cette partie, nous introduisons et ´etudions la notion de tropicalisation et calculons la tropicalisation de U^∗(n). La th`ese se termine par un appendice o`u nous donnons des calculs d´etaill´es pour le premier exemple non-trivial de U^∗(3). Ces calculs explicites ont donn´e lieu `a l’une des preuves cl´es dans la partie principale de la th`ese

Acknowledgements

First of all my most sincere thanks go to my advisor Prof. Anton Alekseev without whose guidence this thesis would not have been possible. I have been fortunate to have his care and support through all these years, and I have truly learnt a lot from him.

I am thankful to Prof. Marcos Mari˜no and Prof. Tudor Ratiu for kindly agreeing to be on my thesis committee.

This work has been supported by the grant of the Swiss National Science Foundation PDFMP2 141756 in the framework of the ProDoc program “Geometry, Algebra and Mathematical Physics”.

I would also like to give my thanks to

- my master thesis advisor Andrey Bytsko and all my teachers from high-school to university who believed in me and kept my interest in science alive,

- my friend and colleague Masha Podkopaeva for fruitful discussions,

- my ancient greek teacher Christiane and my colleague Samuel for helping with the French part,

- Pavol, Nikita, Stefan, Yves and other people from the mathematical department of Geneva for friendly and helpful environment and making past five years pleasant,

- my non-mathematical friends for being in my life and encouraging me,

- my mother for her support and coming to Geneva to cook for me when the work on this thesis was most intense,

- my husband Pavel for all his help and always being there for me, - God for creating the universe with such interesting mathematical

problems in it and making all this possible.

Chapter 1

Introduction

In this thesis, we establish a new link between Poisson Geometry and Combinatorics. Consider a coordinate chart{x1, . . . , xn}on a Poisson manifold and assume that the Poisson brackets of coordinate functions {x_i, x_j} are Laurent polynomials in x₁, . . . , x_n. Then, to these data we associate a polyhedral cone C ⊂ V ∼= Rⁿ and a constant Poisson bracket π_∞ on C. This constant Poisson bracket is determined by the log-canonical part of the original Poisson bracket, that is by the contributions of the formCxixj in{xi, xj}. We call the combinatorial data (C, π_∞) the tropicalizationof the original Poisson structure.

There is a version of our construction which allows for complex-valued coordinate functions z_i (which are then accompanied by their complex conjugates z_i). In this case, the tropicalization again is given by a pair (C, π_∞), where C ⊂ V ∼= Rⁿ is a polyhedral cone, n is the total number of coordinate functions (both real and complex-valued), and π_∞ is a Poisson bracket onC ×T^m. Here T^m is a real torus of dimension m equal to the number of complex-valued coordinate functions. In fact, the pair (C, π_∞) gives rise to acompletely integrable system with action variables given by linear functions on C and angle variables the natural coordinate functions on T^m.

The main example considered in the thesis is the dual Poisson Lie group U^∗(n) equipped with the standard Poisson-Lie structure due to Semenov-Tian-Shansky [7] and Lu-Weinstein [5]. We choose the particular coordinate system onU^∗(n) defined by Fomin-Zelevinsky in the framework of the theory of total positivity [2]. The main result of the thesis states that the tropicalization of the Poisson structure on U^∗(n) with Fomin-Zelevinsky coordinates is isomorphic to the

is an integrable system defined on the space of Hermitian matrices with the linear Kirillov-Kostant-Souriau Poisson bracket with action variables given by the eigenvalues of principal submatrices.

The proof of the main result relies on two techniques. On the one hand, we are using the properties of the tropical Gelfand-Zeiltin map developed by Alekseev-Podkopaeva-Szenes [1]. This technique allows to show that the Gelfand-Zeiltin cone is contained in the cone C that we are interested in. On the other hand, by careful study of combinatorics of the Poisson bracket onU^∗(n) we find all the defining inequalities of the Gelfand-Zeiltin cone among the defining inequalities of the cone C. To obtain the Poisson bracket π_∞, we study in detail the simplified versionU₀^∗(n) of U^∗(n) which captures the properties of the log-canonical part of the Poisson bracket.

The thesis consists of the following parts. We start with two introductory chapters. In the first one, we recall the basic definitions of log-canonical Poisson brackets and of Poisson-Lie groups and give some examples. In the second one, we discuss the Lindstr¨om Lemma relating planar networks and matrices. For convenience of the reader, we give a detailed proof of this classical result. The core part of the thesis is a joint paper with A. Alekseev entitled “ Inequalities from Poisson brackets”. In this part, we introduce and study the notion of tropicalization and compute the tropicalization of U^∗(n). The thesis ends with an appendix where we give detailed computations for the first non-trivial example of U^∗(3). These explicit computations gave rise to one of the key proofs in the main part of the thesis.

Chapter 2

Poisson structure and Poisson-Lie groups

2.1 Log-canonical Poisson brackets

Let M be a smooth manifold equipped with a section π ∈ Γ(M,∧²T M) of the second exterior power of the tangent bundle. Such sections are also called bi-vectors. In a coordinate chart {x1, . . . , xn}, the expression for π is of the form

π= 1 2

X

i,j

πi,j(x) ∂

∂xi ∧ ∂

∂xj

.

Hereπ_i,j(x) is a skew-symmertic matrix which depends on coordinates x1, . . . , xn. A bi-vector π defines a skew-symmetric bracket on the space of smooth functions C^∞(M). This bracket is given by

{f, g}=hdf ⊗dg, πi,

where h·,·i is the natural pairing. In local coordinates, we have {f, g}= 1

2 X

i,j

πi,j

∂f

∂xi

∂g

∂xj

.

One say that the bi-vectorπdefines a Poisson structure onM if the bracket{f, g}on smooth functions is a Lie bracket. This is equivalent to saying that the Schouten bracket of π with itself (for definition, see [8, 3]) vanishes

Example 2.1. Letπi,j be a an arbitrary skew-symmetricn bynmatrix (independent of x’s). Then, it defines a Poisson structure onRⁿ. Example 2.2. Let πi,j(x) = Pn

k=1f_i,j^k xk. Denote the dual basis of the space V^∗ = (Rⁿ)^∗ ∼= Rⁿ by e^∗₁, . . . , e^∗_n. Define a skew-symmetric bilinear map [·,·] :V^∗×V^∗ →V^∗ by formula

[e^∗_i, e^∗_j] = Xn

k=1

f_i,j^ke^∗_k.

Then, π defines a Poisson structure on Rⁿ if and only if [·,·] is a Lie bracket.

Proposition 2.3. Let πi,j be a skew-symmetric n byn matrix. Then, the bi-vector with components {xi, xj} = πi,jxixj defines a Poisson structure on Rⁿ.

Remark 2.4. Such Poisson structures are called log-canonical. They will play an essential role in this thesis.

Proof. We need to show that the Jacobi identity holds for tensor π, that is

{xi,{xj, xk}}+{xj,{xk, xi}}+{xk,{xi, xj}}= 0.

Let us perform an explicit calculation and use the fact that the tensor π is antisymmetric:

{xi,{xj, xk}}+{xj,{xk, xi}}+{xk,{xi, xj}}=

=πjk{xi, xjxk}+πki{xj, xixk}+πij{xk, xixj}=

= (πjkπij +πjkπik+πkiπji+πkiπjk +πijπki+πijπkj)xixjxk= 0

Remark 2.5. Let’s restrict a log-canonical Poisson bracket fromRⁿ to Rⁿ+. Then, one can make the change of variables xi = exp(ξi), ξi = log(xi). In the new variables, the bi-vector is constant,

{ξi, ξj}= ∂ξi

∂x_i

∂ξj

∂x_j {xi, xj}= 1

x_ix_j πi,jxixj =πi,j. Hence, by Example 2.1, it defines a Poisson structure on Rⁿ+.

Let (M, π) and (N, σ) be two Poisson manifolds. A smooth map f : M → N is called a Poisson map if f_∗πx =σy for all x ∈ M such that f(x) = y ∈ N. If the Poisson map f is surjective, the bi-vector σ is uniquely defined by the bi-vector π and the map f. In this case, one says that the bi-vector π descends along the mapf.

The map f : (M, π) → (N, σ) being Poisson is equivalent to the equality

{u, v}N ◦f ={u◦f, v◦f}M

for every pair of smooth functions u, v ∈C^∞(N).

Example 2.6. LetM =R²+with coordinatesx, y and the log-canonical Poisson bracket {x, y} = xy. Then, the map f : M → M defined by formulas

x⁰ = 1

y, y⁰ = x 1 +y is a Poisson map. Indeed,

{x⁰, y⁰}= ∂x⁰

∂y

∂y⁰

∂x {y, x}= 1

y²(1 +y)xy=x⁰y⁰.

2.2 Poisson-Lie groups

Let G be a Lie group equipped with a Poisson structureπ. One says that a pair (G, π) is a Poisson-Lie group if the multiplication map G×G→G is a Poisson map. Among other things, this implies that the bi-vectorπ vanishes at the group unit, πe= 0, and that the group inversion Inv_G:G→Gis an anti-Poisson map. That is, Inv_∗π =−π.

Example 2.7. Consider the group of upper triangular 2 by 2 matrices with unit determinant

G=

g =

a b 0 a⁻¹

; a∈R^∗, b∈R

.

The log-canonical Poisson bracket {a, b} = ab makes G into a Poisson-Lie group. The product map is given by formula

a1 b1

0 a⁻₁¹

a2 b2

0 a⁻₂¹

=

a1a2 a1b2+b1a⁻¹₂ 0 a⁻₁¹a⁻₂¹

.

We directly check that it defines a Poisson map. Indeed,

{a₁a₂, a₁b₂+b₁a⁻₂¹}=a²₁{a₂, b₂}+{a₁, b₁}=

=a²₁a2b2+a1b1 =a1a2(a1b2+b1a⁻¹₂ ).

Example 2.8. The next example is the standard Poisson structure on the group GL(2), see [4]. We use the parametrization in terms of matrix elements

g =

a b c d

,

and define the Poisson brackets as follows:

{a, b}= ¹₂ab, {a, c}= ¹₂ac, {a, d}=cb, {b, c}= 0, {b, d}= ¹₂bd, {c, d}= ¹₂cd.

Note that the determinant det(g) = ad− bc is a Casimir function.

Hence, by putting det(g) = 1 on can define a Poisson-Lie structure on the group SL(2).

Example 2.9. The dual Poisson-Lie group SL^∗(2) plays an important role in this thesis. Its elements are pairs (g, f), where g is an upper triangular matrix, f is a lower triangular matrix, and the diagonals of f and g are opposite to each other:

g =

a b 0 a⁻¹

, f =

a⁻¹ 0

c a

.

The group multiplication is defined by (g1, f1)(g2, f2) = (g1g2, f1f2) and the Poisson bracket is given by (see [6]):

{a, b}= 1

2ab, {a, c}=−1

2ac, {b, c}=−a²+a⁻².

It is instructive to check the Poisson-Lie property for the bracket{b, c}:

{a1b2+b1a⁻₂¹, c1a⁻₂¹+a1c2}=a⁻₂²{b1, c1}+a²₁{b2, c2}=

=a⁻²₂ (−a²₁+a⁻²₁ ) +a²₁(−a²₂+a⁻²₂ ) = −(a₁a₂)²+ (a₁a₂)⁻². Here we have used the fact that {a1b2, c1a⁻₂¹}={b1a⁻₂¹, a1c2}= 0.

Chapter 3

Lindstr¨ om Lemma

The Lindstr¨om lemma in its present form has been first proven by Fomin and Zelevinsky in [2]. We present the proof here in full detail, and then illustrate the statement of the Lemma by an example.

For convinience of the reader, let us start by recalling the definition of a planar network.

Definition 3.1. A planar networkΓ is a finite planar oriented graph satisfying the conditions listed below:

• It is contained between two vertical straight lines on the plane.

• The edges of the graph are segments of straight lines, with non-zero horizontal projections, and they are oriented from left to right.

• There are exactlyn sources andn sinks on the vertical lines. The number n is called the type of a planar network.

Let VΓ and EΓ be the sets of vertices and edges of the network Γ, respectively. A map w : EΓ → C is called a weighting of a planar network. A weight matrix is associated to Γ in a following way:

M(Γ, w)ij = X

γ∈PΓij

Y

e∈γ

w(e),

wherePΓij is the set of paths in Γ starting in the source iand ending in the sink j, e∈γ are the edges of a path γ.

We call a collection ofk paths with no common vertices a k–path in Γ.

Lemma 3.2 (Lindstr¨om Lemma). A minor M(Γ, w)IJ of the weight matrix of a planar network Γ is equal to the sum of weights of all k–paths γ ∈PΓIJ connecting the sources labeled by the set I with the sinks labeled by the set J (I andJ being ordered sets of cardinalityk), that is:

M(Γ, w)IJ = X

γ∈PΓ_IJ

Y

e∈γ

w(e).

Proof. It is enough to prove the satement of the lemma for the determinant of the whole weight matrix M(Γ, w):

detM(Γ, w) = X

γ∈PΓ

Y

e∈γ

w(e).

By definition of the determinant we have:

detM(Γ, w) = X

σ∈Sn

(−1)^σ Yn i=1

M_i,σ(i).

Each matrix element Mi,σ(i) is in turn the sum of weights of all paths connecting the source i with the sink j. Thus we can rewrite the formula for the determinant in the following way:

detM(Γ, w) = X

σ∈Sn

X

γ∈PΓ

(−1)^σ Y

e∈γ

w(e).

It is clear that for the collection of paths that is vertex disjoint, σ is the identity permutation, thus (−1)^σ = 1. We need to show that all the other terms cancel out. Let us consider the rightmost point of intersection of two paths (by definition of a planar network, one can assume that no two vertices belong to the same vertical line). One can switch the parts of these paths which lie to the right of the intersection point. Clearly, such an operation does not change the weight of the collection of paths under consideration, but it does change the sign of the associated permutation. Thus each term corresponding to a non-disjoint collection has a partner with an opposite sign, and they cancel out.

Example 3.3. To illustrate the above Lemma let us consider a planar network Γ_ex represented on the Figure 3.1.

1 2 3

1 2 3

a b

c e

g f

1 1 1

1 1 1 d 1

1

Figure 3.1: Planar network Γex and Lindstr¨om Lemma presentation of the minor M_{2,3;1,2} of its weight matrix

The weight matrix is of the form:

M =



 b bcd 0 ab+abcde+de d+abcd 0

f gde f gd f





Let us consider the minor M_{2,3;1,2} of this weight matrix. A direct calculation gives:

M_{2,3;1,2} =abf gd+abcdef gd+def gd−f gded−f gdeabcd=

=abf gd (3.1) The same result can be obtained by calculating the weight of the disjoint collection of paths starting in the sources 2,3 and ending in the sinks 1,2 (see Figure 3.1).

There are four possible non-disjoint collections of paths between these sources and sinks coming in two pairs presented on the Figure 3.2. They correspond to the four terms in (3.1) which cancel out. Following the proof of the lemma, two paths in one pair differ by exchanging the parts to the right of the intersection point. This gives rise to the contributions which differ by a sign.

1 2 3

1 2 3

1 2 3

1 2 3 1

2 3

1 2 3

1 2 3

1 2 3

Figure 3.2: Non-disjoint collections of paths connecting sources 2,3 to sinks 1,2

Bibliography

[1] A. Alekseev, M. Podkopaeva, A. Szenes, The Horn problem and planar networks, preprint arXiv:1207.0640

[2] S. Fomin, A. Zelevinsky, Total positivity: tests and parametrizations, Math. Intelligencer22 (2000), no. 1, 23–33 [3] C. Laurent-Gengoux, A. Pichereau, P. Vanhaecke, Poisson

Structures, Springer (2012)

[4] Jiang-Hua Lu, Multiplicative and Affine Poisson Structures on Lie groups, Ph.D. Thesis, University of California, Berkeley, (1990) [5] J. H. Lu, A. Weinstein, Poisson-Lie groups, dressing

transformations and Bruhat decompositions, J. Differential Geom. 31 (1990), no.2, 501–526

[6] M. A. Semenov-Tyan-Shanskii, What is a classical r-matrix?, Functional Analysis and Its Applications 17(1983) no. 4, 259–272 [7] M. A. Semenov-Tian-Shanskii, Dressing transformations and Poisson Lie group actions, Publ. Res. Inst. Math. Sci 21 (1985), 1237–1260.

[8] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkh¨auser Basel (1994)

Inequalities from Poisson brackets

A. Alekseev^∗, I. Davydenkova ^† February 17, 2014

Abstract

We introduce the notion of tropicalization for Poisson structures on Rⁿwith coefficients in Laurent polynomials. To such a Poisson structure we associate a polyhedral cone and a constant Poisson bracket on this cone. There is a version of this formalism applicable to Cⁿ viewed as a real Poisson manifold. In this case, the tropicalization gives rise to a completely integrable system with action variables taking values in a polyhedral cone and angle variables spanning a torus.

As an example, we consider the canonical Poisson bracket on the dual Poisson-Lie group G^∗ for G = U(n) in the cluster coordinates of Fomin-Zelevinsky defined by a certain choice of solid minors. We prove that the corresponding integrable system is isomorphic to the Gelfand-Zeitlin completely integrable system of Guillemin-Sternberg and Flaschka-Ratiu.

1 Introduction

Log-canonical coordinates on Poisson manifolds play an important role in Poisson Geometry. In particular, they have proved to be useful in the theory of cluster varieties (see e.g. [5]). Log-canonical coordinates are characterized by the fact that for two coordinate functions, say xand y, their Poisson bracket is of the form

{x, y}=c xy.

If x andy take real positive values, one can define new coordinates ξ = log(x) and η = log(y) so as the Poisson bracket of ξ and η is constant,

{ξ, η}=c.

∗Anton.Alekseev@unige.ch, Universit´e de Gen`eve, 2-4 rue du Li`evre, c.p. 64, 1211 Gen`eve 4 (Switzerland)

†Irina.Davydenkova@unige.ch, Universit´e de Gen`eve, 2-4 rue du Li`evre, c.p. 64, 1211 Gen`eve 4 (Switzerland)

In this paper, we consider Poisson brackets of more general type.

For coordinate functions (that we denote again byxandy) we now have

{x, y}=c xy+p(x, y, . . .), (1) where p(x, y, . . .) is a Laurent polynomial in x, y and (possibly) other coordinate functions. To a Poisson bracket of this type, we assign its tropicalization which is a pair (C,{·,·}∞) where C is a polyhedral cone and {·,·}∞ is a constant Poisson bracket on C.

Recall that the tropical calculus is a semi-ring structure on R where addition is replaced by the maximum function and multiplication is replaced by addition

ξ +_trop η = max(ξ, η), ξ ·^trop η =ξ+η.

One can obtain this semi-ring structure as a t → +∞ limit of the standard semi-ring structure on R+ under the map x 7→ ξ = t⁻¹log(x). Indeed,

t→lim+∞ t⁻¹log e^tξ+e^tη

= max(ξ, η), lim

t→+∞ t⁻¹log e^tξ·e^tη

= ξ+η.

Returning to tropicalization of Poisson brackets, we consider an example

{x, y}=c xy+a x+b y.

Let t∈R+ be a real positive parameter, and let ξ =t⁻¹log(x), η= t⁻¹log(y). In coordinatesξ, ηthe Poisson bracket acquires the form

{ξ, η}t =t⁻² c+ae^−tη+be^−tξ .

We require that the log-canonical contribution (t⁻²c on the right hand side) is dominant fort→ +∞. This yields two inequalities

ξ >0, η >0

which define the cone C. By rescaling the bracket by a factor of t², we obtain an expression which has a well-defined limit onC when t tends to infinity,

{ξ, η}∞ := lim

t→+∞t²{ξ, η}^t =c.

The resulting Poisson bracket {·,·}∞ is constant.

There is a version of this formalism adapted to complex

this case, we use the change of variables z_i = exp(tζ_i +iϕ_i) with parameter t → +∞. The result of the tropicalization procedure is again an open polyhedral coneCand a constant Poisson structure on C×Tⁿ. Under this constant Poisson structure, coordinatesζ_iPoisson commute with each other. That is, we obtaina completely integrable systemwith ζ_i’s as action variables andϕ_i’s as angle variables.

As an example, we consider the Poisson bracket on the dual Poisson-Lie group G^∗ for G = U(n). This Poisson bracket was defined in [11] and [10]. As a coordinate system we use solid minors ∆^(k)_l from the total positivity theory [4]. Theorem of Kogan-Zelevinsky [9] shows that these minors provide log-canonical coordinates on the Poisson-Lie group G. For the Poisson-Lie group G^∗, the corresponding Poisson bracket is no longer log-canonical, but it admits the form (1).

The main result of this paper is the following theorem:

Theorem 1. The tropicalization of the Poisson bracket on the Poisson-Lie group U(n)^∗ is isomorphic to the Gelfand-Zeitlin completely integrable system.

Our interest is motivated by the following observations: by the Ginzburg-Weinstein Isomorphism Theorem [6], the Poisson manifold (G^∗, π_G^∗) is isomorphic to (g^∗, π_KKS), where π_KKS is the Kirillov-Kostant-Souriau Poisson bracket on g^∗. Since π_KKS is a linear Poisson structure, the scaling transformation x7→t⁻¹x, π_t = t π_KKS is a Poisson isomorphism. This implies that (G^∗, π_G^∗) is Poisson isomorphic to (G^∗, tπ_G∗) for allt > 0.

The tropicalization procedure described in the paper assigns a limiting object at t = +∞ to the family (G^∗, tπ_G^∗). Theorem 1 shows that in the case of G = U(n) this object is isomorphic to the Gelfand-Zeitlin completely integrable system. Flaschka-Ratiu [3] discovered a Gelfand-Zeitlin type integrable system on G^∗, and in [1] it was shown that the Flaschka-Ratiu system is isomorphic to the Gelfand-Zeitlin system. Hence, Theorem 1 provides a t = +∞ extension of the Ginzburg-Weinstein Isomorphism in the case of G= U(n).

The structure of the paper is as follows: in Section 2 we introduce a notion of tropicalization and associate a polyhedral cone and a constant Poisson bracket on this cone to a certain type of Poisson structures. First, we consider real positive Poisson manifolds, then

we allow for complex coordinate functions and introduce a notion of linear scaling. In Section 3 we consider Poisson structures on the group of upper triangular matrices and on its close relative G^∗₀. Finally, in Section 4 we apply the machinery developed in Section 2 to the Poisson structure on the dual Poisson-Lie group U(n)^∗ to obtain the isomorphism with the Gelfand-Zeitlin completely integrable system.

Acknowledgements. Research of A.A. was supported in part by the grant MODFLAT of the European Research Council and by the grants 200020–140985 and 200020–141329 of the Swiss National Science Foundation. Research of I.D. was supported in part by the grant PDFMP2–141756 of the Swiss National Science Foundation. We are grateful to M. Podkopaeva and A. Szenes for useful discussions.

2 Log-canonical Poisson brackets and Tropicalization

2.1 Real positive Poisson manifolds

LetM be a real Poisson manifold andU ⊂M be a coordinate chart with positive coordinate functions {x₁, x₂, . . . , x_N}, x_i ∈R+.

We say that the Poisson bivector π on M is log-canonical with respect to the coordinate chart U if it has the form

π= 1 2

X

i,j

π_i,jx_ix_j ∂

∂x_i ∧ ∂

∂x_j.

That is, the Poisson brackets of coordinate functions are given by formula

{x_i, x_j}=π_i,jxⁱx^j,

where no summation over repeating indices is assumed.

The main object of our study will be Poisson brackets of the form {x_i, x_j}=π_i,jxⁱx^j +p_i,j(x), (2) where p_i,j(x) are Laurent polynomials in variables x₁, . . . , x_N.

The procedure oftropicalization will associate two combinatorial objects to a Poisson bracket of type (2): an open polyhedral cone

LetV =R^N with elements (ξ₁, . . . , ξ_N)∈V and let {e₁, . . . , e_N} be the corresponding dual basis in V^∗. For every pair (i, j) with 1≤i < j≤N consider the decomposition

p_i,j = X

I∈Fi,j

c_Ix^I,

where I = (i₁, . . . , i_N) is a multi-index, x^I = xⁱ₁¹. . . xⁱ_N^N, and F_i,j is the set of multi-indices for which the coefficients c_I are non-vanishing. Put n_i,j =e_i+e_j ∈V^∗, denote

n(I) = XN

r=1

i_re_r

and let Ci,j ⊂V be the convex cone defined as follows Ci,j ={ξ ∈V; hn_i,j−n(I), ξi>0∀ I∈F_i,j}. In more detail, the cone Ci,j is defined by the inequalities

ξ_i+ξ_j >

XN r=1

i_rξ_r

for all I ∈F_i,j. We define the cone C(π;x)⊂V as the intersection of the cones C^i,j for all pairs (i, j):

C(π;x) =∩^i<j C^i,j.

Example 1. Letπ be a log-canonical Poisson bracket in coordinates x₁, . . . , x_N. Then the setF_i,j is empty for alli, j andC^i,j =V which yields C(π;x) =V.

Example 2. LetN = 2 and let π be the Poisson bracket defined by formula

{x₁, x₂}=x₁(1 +x²₂).

In this case, we obtain two inequalities,

ξ₁+ξ₂ > ξ₁, ξ₁+ξ₂ > ξ₁+ 2ξ₂.

They contradict each other, and in this case the cone C(π, x) is empty.

The next step is to introduce the followingscaling transformation:

let t ∈ R+ be a parameter, make a change of variables x_i = exp(tξ_i), ξ_i = ¹_t ln(x_i) and scale the Poisson bivector as follows, π_t = t²π. If the Poisson bracket is log-canonical, it will become constant in variables ξ₁, . . . , ξ_N

{ξ_i, ξ_j}^t = 1

t² {ln(x_i),ln(x_j)}^t =π_i,j.

Note that the right hand side does not depend ont. This observation motivates the following definition: let π be a Poisson bracket of the form (2). Then, a constant Poisson bracket on the cone C(π;x) denoted byπ_∞ and given by the following formula can be associated to it

π_∞ = 1 2

X

i,j

π_i,j ∂

∂ξ_i ∧ ∂

∂ξ_j thus {ξ_i, ξ_j}∞ =π_i,j.

Example 3. LetN = 2 and

{x₁, x₂}=x₁x₂+x²₁+x₂

which implies F_1,2 = {(2,0); (0,1)}. The set F_1,2, the vector n_1,2 and the cone C are represented on the Figure 3.

Figure 1: The setF1,2and the coneC

It is easy to see that the Poisson bracket π_∞ is of the form {ξ₁, ξ₂}∞ = 1.

Proposition 1. Let π be a Poisson bracket of type (2). Then, in coordinates ξ we have

t²π→^t→+∞ π_∞.

Proof. Let 1≤i < j≤N, and compute

{x_i, x_j}t ={e^tξi, e^tξj}t =t²e^t(ξi+ξj){ξ_i, ξ_j}t.

That is, for the bracket {ξ_i, ξ_j}t we obtain the following expression

{ξ_i, ξ_j}^t =t⁻²e^{−t(ξi+ξj)}{x_i, x_j}^t =

=e^{−t(ξi+ξj)}



π_i,je^t(ξi+ξj)+ X

I∈Fi,j

c_Ie^tPikξk



. Forξ ∈ C(π;x) we haveξ_i+ξ_j > P

i_kξ_k for allI ∈F_i,j. Hence, the right hand side tends to π_i,j when t→+∞.

2.2 Complex coordinates and linear scaling

In this Section, we shall allow for complex valued coordinate functions. The coordinate chart U will carry coordinates of the form {x₁, . . . , x_k, z₁, . . . , z_l}, where x₁, . . . , x_k are real positive and z₁, . . . , z_l are complex valued non-vanishing functions. Then, the real dimension of M is 2l+k, and we also get complex conjugates of the coordinate functions ¯z₁, . . . ,z¯_l on U.

A Poisson bracketπ is log-canonical in the coordinate chartU is it is of the form

{x_i, x_j}=π_i,jx_ix_j, {x_i, z_a}=π_i,ax_iz_a, {x_i,z¯_a}=π_i,¯ax_iz¯_a, {z_a, z_b}=π_a,bz_az_b, {z¯_a,z¯_b}=π_¯a,¯bz¯_az¯_b, {z_a,z¯_b}=π_a,¯bz_az¯_b. Since the bivector π is supposed to be real, we have the following reality conditions imposed on the components of π:

π_i,¯a= π_i,a , π_¯a,¯b =π_a,b , π_a,¯b =−π_b,¯a .

More generally, we shall consider Poisson brackets of the form

π=π₀+π^′, (3)

where π₀ is a log-canonical Poisson bracket and π^′ is a bivector with coefficients in Laurent polynomials in variablesx, z and ¯z. Let V = R^k+l with elements (ξ₁, . . . , ξ_k, ζ₁, . . . , ζ_l). Denote the dual basis in V^∗ by e_i, i= 1, . . . , k and f_a, a= 1, . . . , l. Similarly to the

previous Section, we define the cones C^i,j for i < j, C^a,b for a < b andC^i,a. For example, we have

{x_i, z_a}=π_i,ax_iz_a+p_i,a(x, z),

where p_i,a(x, z) is a Laurent polynomial in variables x, z and ¯z. It can be written in the form

p_i,a(x, z) = X

I,J,K∈Fi,a

c_I,J,Kx^Iz^Jz¯^K,

where I, J and K are multi-indices, and F_i,a is the finite set where the coefficients c_I,J,K are non-vanishing. Denote n_i,a= e_i+f_a∈V^∗ and

n(I, J, K) = Xk

r=1

i_re_r+ Xl

s=1

(j_s+k_s)f_s for (I, J, K)∈G_i,a. The cone C^i,a is defined as follows

Ci,a ={η = (ξ, ζ)∈V; hn_i,a−n(I, J, K), ηi>0 ∀I, J, K ∈F_i,a}, That is we have the inequalities

ξ_i+ζ_a>

Xk r=1

i_rξ_r+ Xl

s=1

(j_s+k_s)ζ_s.

We define the coneC(π;x, z) as the intersection of the conesC^i,j,C^i,a andC^a,b.

We shall assume in addition the following reality conditions on the log-canonical part of the bivector π:

π_i,j = 0, Reπ_i,a = 0, Reπ_a,b = 0, Reπ_a,¯b = 0. (4) Under these assumptions, a log-canonical bivector admits the following linear scaling. Again, let t ∈ R+ be a parameter.

We introduce new coordinates on U via x_i = exp(tξ_i), z_a = exp(tζ_a+ iϕ_a). Consider the scaled Poisson bracket π_t = tπ in new coordinates. It yields the following Poisson brackets:

{ξ_i, ξ_j}^t = 0, {ξ_i, ζ_a}^t = 0, {ξ_i, ϕ_a}t= Imπ_i,a, {ζ_a, ζ_b}t = 0, {ζ_a, ϕ_b}t = ¹₂Im (π_a,b−π_a,¯b), {ϕ_a, ϕ_b}t = 0.

(5) As before, this bracket does not depend ont, and we can denote it byπ . It is defined on the product C(π;x, z)×T^l, where (ξ, ζ)∈

Remark 1. Log-canonical Poisson brackets without reality conditions (4) do not allow for a linear scaling limit. Instead, one can consider the limit of π (no powers of t added) in coordinates (ξ, ζ, ϕ). It yields constant Poisson brackets between the angle variables {ϕ_a, ϕ_b} while ξ’s and ζ’s become Casimir functions in the limit.

Remark 2. Note that log-canonical Poisson brackets with reality conditions (4) naturally give rise to completely integrable systems.

Indeed, variables ξ_i and ζ_a Poisson commute. Assuming that the rank of the bracket π is equal to 2l (which is the maximal possible rank), this is a maximal family of Poisson commuting functions.

The dual angles areϕ_a’s. They are spanning the Liouville tori. The variables (ξ, ζ, ϕ) are in fact action-angle variables for the resulting completely integrable system.

Example 4. Letk = 1, l= 1 and consider the Poisson bracket of the form

{x, z}=ixz, {x,z¯}=−ixz,¯ {z,z¯}=i(x²−x⁻²).

The set G_1,¯1 and the cone C are represented at the Figure 4:

Figure 2: The setF1,¯1and the coneC

After changing variablesx=e^tξ, z =e^tζ+iϕ,z¯=e^tζ−iϕ and applying thet→+∞ limit we obtain the following constant Poisson bracket on C ×S¹

{ξ, ζ}∞= 0, {ξ, ϕ}∞ = 1, {ζ, ϕ}∞= 0.

Proposition 2. Let π be a Poisson bracket of type (3). Then, in coordinates (ξ, ζ, ϕ)we have

tπ→^t→+∞ π_∞.

for (ξ, ζ, ϕ)∈ C(π;x, z)×T^l

Proof. We shall give a proof for the Poisson bracket {ξ_i, ϕ_a}∞, the calculation for other Poisson brackets between coordinates is similar and will be omitted. Consider

{x_i, z_a}t ={e^tξi, e^tζa+iϕa}t =

=t²e^{t(ξi+ζa)+iϕa}{ξ_i, ζ_a}t+ite^{t(ξi+ζa)+iϕa}{ξ_i, ϕ_a}t. Thus, for the bracket {ξ_i, ϕ_a}t we obtain the following expression

{ξ_i, ϕ_a}^t =t⁻¹e^{−t(ξi+ζa)}Im

e^−iϕa{x_i, z_a}^t

=

=t⁻¹e^{−t(ξi+ζa)}Im

te^−iϕa(π_i,ae^tξie^tζa+iϕa) +te^−iϕap_i,a(x, z)

=

= Im



π_i,a+e^{−t(ξi+ζa)}e^−iϕa X

I,J,K∈Gi,a

c_I,J,Kx^Iz^Jz¯^K



.

Let I, J, K ∈G_i,a and consider the expression x^Iz^Jz¯^K = exp

Xk r=1

i_rtξ_r+ Xl

s=1

j_s(tζ_s+iϕ_s) + Xl

t=1

k_t(tζ_t−iϕ_t)

! .

For (ξ, ζ)∈ C(π;x, z), we have ξ_i+ζ_a>

Xk r=1

i_rξ_r+ Xl

s=1

(j_s+k_s)ζ_s

for all I, J, K ∈G_i,a. Hence, the exponential e^t(ξi+ζa) dominates all the expressionsc_I,J,Kx^Iz^Jz¯^K ande^{−t(ξi+ζa)}c_I,J,Kx^Iz^Jz¯^K tends to zero when t→+∞, as required.

3 Poisson brackets on Poisson-Lie groups B

and G

In this Section we recall the definitions of Poisson brackets and of log-canonical coordinates on the group of upper triangular invertible

∗

3.1 Poisson-Lie group of upper triangular matrices

Letg= gl(n,C), and letr∈g⊗gbe the standard classicalr-matrix given by formula

r = 1 2

X

i

e_i,i⊗e_i,i+X

i<j

e_i,j⊗e_j,i,

where e_i,j is the elementary matrix with the only non-vanishing matrix entry equal to 1 at the intersection of the i’th row and j’th column. Sometimes it is convenient to split the r-matrix into two parts,

r₀= 1 2

X

i

e_i,i⊗e_i,i , r^′=X

i<j

e_i,j⊗e_j,i.

The group B₊ of invertible upper-triangular matrices carries a Poisson structure given by formula

{g¹, g²}= [r, g¹g²] =rg¹g²−g¹g²r, (6) where we are using the Saint–Petersburg notation g¹ = g⊗1, g² = 1⊗g.

Remark 3. To illustrate the usage of this notation convention, consider a simpler bracket {g¹, g²} = r₀g¹g². In terms of more standard notation, this bracket looks as

{f, h}=h∇^Lf ⊗ ∇^Lh, r₀i,

where f and hare two functions onB₊,∇^L is defined as (∇^Lf)_g(x) = d

dtf e^txg)|^t=0

forx∈b+= Lie(B₊), and the pairingh·,·iis induced by the natural pairing between b₊ andb^∗₊.

The writing {g¹, g²} = r₀g¹g² encodes the following (non skew-symmetric) brackets of the matrix elements of g:

{g_ij, g_st}= δ_isg_ijg_st.

This formula is obtained by taking the matrix element (i, j) in the first factor of the tensor product (the matrix g¹), and the matrix element (s, t) in the second factor (the matrix g²). Note that the brackets of matrix elements completely determine the Poisson bracket on B₊.

The Jacobi identity for the bracket (6) is a corollary of the classical Yang-Baxter equation for the element r:

[r^1,2, r^2,3] + [r^1,2, r^1,3] + [r^1,3, r^2,3] = 0. (7) The group multiplication B₊×B₊→ B₊ is a Poisson map making B₊ into a Poisson-Lie group.

Following Kogan-Zelevinsky [9], we introduce a log-canonical coordinate chart onB₊ in the following way. Letn≥k ≥l≥1 and denote by ∆^(k)_l the solid minor of the matrixgof sizelformed by the intersection of rows with consecutive numbers n−k+ 1, . . . , n−k+l and the last l columns (see Figure 3.1). These n(n+ 1)/2 minors define coordinates on an open dense subset inB₊. Hence, a smooth Poisson bracket on B₊ is completely characterized by the brackets between ∆^(k)_l ’s.

Figure 3: minor ∆^(k)_l

Theorem 2. The Poisson bracket of functions ∆^(k)_l , n≥k≥l ≥1 is log-canonical, and it has the form

{∆^(k)_l ,∆^(p)_q }= 1

2ε(k−p)(C−R)∆^(k)_l ∆^(p)_q , (8) whereRis the number of common rows, C is the number of common columns of the two minors, and ε(x) is the sign function (that is, ε(x) = 1 for x >0, ε(x) =−1 for x <0, and ε(0) = 0).

Proof. To prove the theorem we shall use formula (24) (see Appendix

reads

{g_IJ, g_ST}=X

u<v

χ_I(u)χ_S(v)g_σu,v(I),Jg_σv,u(S),T−

−X

u<v

χ_J(v)χ_T(u)g_I,σv,u(J)g_S,σu,v(T)+ 1

2(|I∩S| − |J ∩T|)g_IJg_ST, whereχ_I is the characteristic function of the setI(that is,χ_I(k) = 1 for k ∈I and χ_I(k) = 0 for k /∈I), and σ_v,u(I) is the set obtained fromI by replacing v with u.

Consider the second term on the right hand side. In our situation, either J ⊂ T or T ⊂ J (or J = T). Hence, one of these subsets necessarily contains both u and v. After the replacement the corresponding matrix will contain two identical columns and its determinant (eitherg_I,σv,u(J)org_S,σu,v(T)) will vanish. Therefore, this term always vanishes.

In the first term on the right hand side, non trivial contributions come from the terms with u ∈ I\(I ∩S) and v ∈ S\(I ∩S). If p≥k, this implies v < u whereas the summation is over the range of u < v. Hence, in this case the first term in the sum vanishes as well.

By definition,R=|I∩S|andC =|J∩T|which yields fork ≥p {∆^(k)_l ,∆^(p)_q }= 1

2(C−R)∆^(k)_l ∆^(p)_q .

The statement of the theorem follows by skew-symmetry of the Poisson bracket.

Example 5. Letn= 2. In this case, for g =

g₁₁ g₁₂ 0 g₂₂

we have three coordinate functions on B₊

∆⁽¹⁾₁ =g₂₂, ∆⁽²⁾₁ =g₁₂, ∆⁽²⁾₂ =g₁₁g₂₂. Their Poisson brackets read

{∆⁽¹⁾₁ ,∆⁽²⁾₂ }= 0, {∆⁽²⁾₁ ,∆⁽²⁾₂ }= 0, {∆⁽¹⁾₁ ,∆⁽²⁾₁ }=−1

2∆⁽¹⁾₁ ∆⁽²⁾₁ .

Note that the determinant ofgis a Casimir function. Putting ∆⁽²⁾₂ = 1, we obtain a Poisson algebra with generators ∆⁽¹⁾₁ and ∆⁽²⁾₁ and the log-canonical Poisson bracket {∆⁽¹⁾₁ ,∆⁽²⁾₁ }= −¹₂∆⁽¹⁾₁ ∆⁽²⁾₁ .

One can also consider the groupB₋ of lower triangular matrices with Poisson bracket

{f¹, f²}= [r, f¹f²].

The matrix elements of the inverse matrixf⁻¹have Poisson brackets of the same type (up to sign):

{(f⁻¹)¹,(f⁻¹)²}=−[r,(f⁻¹)¹(f⁻¹)²].

We shall denote by Λ^(k)_l the solid minor of the matrix f⁻¹ formed by the columns with labels n−k+ 1, . . . , n−k+l and the last l rows (see Figure 3.1).

Figure 4: minor Λ^(k)_l

Similarly to Theorem 2, one can show that Λ^(k)_l ’s are log-canonical coordinates on an open dense chart in B₋ and that their Poisson brackets are of the form

{Λ^(k)_l ,Λ^(p)_q }= 1

2ε(k−p)(R−C)Λ^(k)_l Λ^(p)_q .

Remark. We could have chosen some solid minors of f as log-canonical coordinates on B₋. Our choice of solid minors of f⁻¹ is dictated by convenience of calculations in the next section.