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Hello, I am interested in symplectic Lie groups, I will sketch out their definition. A symplectic Lie group is a given pair $(G,\omega)$, where $G$ is a Lie group and $\omega$ is a left invariant symplectic form on $G$.

The algebraic situation is as follows. If $\mathcal{G}$ is the Lie algebra of $G$ and $\omega$ is a non-degenerate $2$-cocycle on $\mathcal{G}$, that is $\omega\in\wedge^2\mathcal{G}^*$ such that for any $x,y,z\in\mathcal{G}$ $$\omega([x,y],z)+\omega([y,z],x)+\omega([z,x],y)=0$$ The left invariant $2$-form associated to $\omega$ is a symplectic form on $G$.

My questions are:

1) What are the Lie algebras that admit such $omega$? The algebra can not be semi-simple, according to an article by Chu, Symplectic homogeneous spaces.

2) What is the link with the existence of an invertible solution of the classical Yang-Baxter equation $[r,r]=0$ for $r\in\wedge^2\mathcal{G}$ where $[\,,\,]$ is the Schouten bracket?

Thanks for any help.

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If the Lie algebra is unimodular, then it follows from Chu's paper that it must be solvable, which is a stronger conclusion than "not semisimple". –  José Figueroa-O'Farrill Aug 1 '11 at 2:21
    
Such a Lie algebra is also a pre-Lie algebra. The pre-Lie product is obtained from the Lie bracket by adjunction with respect to $\omega$. You may have a look at various articles of Alberto Medina. –  F. C. Aug 1 '11 at 6:33
    
Thanks, José, a nice comment! Thanks, Frédéric. –  amine Aug 3 '11 at 8:03
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3 Answers 3

up vote 5 down vote accepted

Hello Amine,

One of the most general facts is that, if a Lie group admits a torsion free and (locally) flat connection (some authors call it an affine structure) which is invariant under left translations, then its Lie algebra properly contains (is not equal to) its derived ideal (Lie ideal generated by the Lie bracket of all vectors.) Equivalently, the dimension of such a Lie group is strictly greater than that of its commutator subgroup. As a particular case, this obstruction bans semisimple Lie groups from being endowed with such connections.

That is a result of J. Helmstetter, in "Radical d’une algebre symetrique a gauche. Ann. Inst. Fourier 29, no 4, (1979), 17-35.

Now, it is well known since the end of the 60s, that a left invariant symplectic structure in a Lie group provides the latter with a left invariant flat and torsion free connection, just as above (Koszul, Vinberg, Matsushima,...). If $\omega$ is the corresponding left invariant symplectic form, then $\nabla$ is provided by the formula $\omega (\nabla_xy,z):= - \omega (y, [x,y])$, where, of course, $\nabla$ stands for the needed connection and $x,y,z$ are any left invariant vector fields on the Lie group. Hence the same obtruction obviously applies, as well.

However, there are many nonsolvable symplectic Lie groups (Lie groups with a left invariant symplectic structure) and the most discussed amongst them, is certainly the ordinary group of affine transformations of the Euclidean space of dimension strictly greater than 1. Here the expression "affine transformations" relates to the ordinary ones, i.e. to transformations of the form F(x) = L(x) + v. Where L is a linear nonsingular map and v is a constant vector.

As for the relationship with the Classical Yang-Baxter Equation (CYBE), to the more general answer provided above by Nicola Ciccoli (D.V Alekseevsky and A.M. Perelomov, Journal of Geometry and Physics 22 (1997) 191-211, or, A. Lichnerowicz and A. Medina, Letters Math. Phy. 16, 225-335, (1988), ... ), I must add that invariant symplectic structures (1-1) correspond to invertible (skew-symmetric) solutions of the CYBE on Lie groups (same construction as explained by Nicola Ciccoli). For references on this part, you can have a look at, for example, A. Diatta and A. Medina, Manuscripta Math. 114 (2004), no. 4, 477–486.

Now, as for the question as to which Lie groups can bear left invariant symplectic structures and for more general results (including general properties, relationship with homogeneous bounded domains or Kahler Geometry, existence questions, construction procedures such as the so-called double extention and T*-extension,...) I can refer you to various papers by A. Medina, A. Lichnerowicz and A. Medina, A. Medina and Ph. Revoy, J.M. Dardie and A. Medina, M. Nguiffo Boyom, M. Bordemann, I. Bajo, S. Benayadi, D.V. Alekseevsky, E.B. Vinberg, S.G. Gindikin, I.I. Pjateckii-Sapiro, J.Dorfmeister, K.Nakajima, ... amongst the loads of authors who have worked on various aspects of the subject. The litterature is quite rich and yet, several interesting deep questions are still waiting for answers. However, there is a complete classification in low dimension up to dimension 4 and, at least partly, in dimension 6.

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Hello, André! Many thanks for your answer. So there are nonsolvable symplectic Lie groups! There is a nice thesis on the topic, by Hassène Siby "Géométrie des Groupes de Lie symplectiques" \url{tel.archives-ouvertes.fr/tel-00078872/en} where a complete classification of symplectic Lee groups is given, in dimension $4$, I have to compare it with the paper by Gabriela Ovando, "Four dimensional symplectic Lie algebras" \url{arxiv.org/abs/math/0407501v1} –  amine Aug 3 '11 at 8:19
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Given a solution $r$ of the classical Yang-baxter, identified with a map $\mathfrak g^*\to \mathfrak g$ one has that $ im r$ is a Lie subalgebra on which $r^{-1}$ gives a symplectic Lie algebra strutture and viceversa. This is simply explained in Alekseevsky Peremolov Poisson and symplectic structures on Lie algebras.

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The citation is D.K Alekseevsky A.M. Perelomov, Journal of Geometry and Physics 22 (1997) 191-211. The MathSciNet entry is ams.org/mathscinet-getitem?mr=1453001 which has links to the article (behind an Elsevier paywall). –  José Figueroa-O'Farrill Aug 1 '11 at 16:55
    
Sorry, that's D.V. Alekseevsky! –  José Figueroa-O'Farrill Aug 1 '11 at 16:56
    
Thanks Jose, i was out with the iPad and couldn't provide details... –  Nicola Ciccoli Aug 1 '11 at 18:11
    
Thanks, Nicola! Now, it's an elementary fact that a solution $r$ of the classical Yang-Baxter equation defines a Lie algebra homomorphism $r_\sharp : \mathcal{G}^*\rightarrow\mathcal{G}$ such that $$\omega(x,y)=r_\sharp(\alpha,\beta)$$ (where $r_\sharp(\alpha)=x$, $r_\sharp(\beta)=y$) is a well defined symplectic form on $\mathrm{Im}r$. –  amine Aug 3 '11 at 8:11
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Semi-simple Lie groups do not admit left invariant torsion free flat connections, the proof is 'easy', see:

H. Matsushima & K. Okamoto, Nonexistence of torsion free flat connections on a real semisimple Lie group}. http://projecteuclid.org/euclid.hmj/1206135198

A Lie group endowed with a complete left invariant torsion free flat connection is necessary solvable. See:

D. Segal, The structure of complete left-symmetric algebras. http://www.springerlink.com/content/k158432813523275/

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