Poisson Lie group

A Lie group $ G $ with a Poisson structure $ P $ which is compatible with the group multiplication, i.e., the multiplication $ G \times G \rightarrow G $ is a Poisson mapping, where $ G \times G $ is the product Poisson manifold (cf. Poisson algebra).

The corresponding infinitesimal object is a Lie bialgebra (see Quantum groups) $ ( \mathfrak g, \gamma ) $, called the tangent Lie bialgebra of $ ( G,P ) $. Here, $ \mathfrak g $ is the Lie algebra of the Lie group $ G $ and the linear mapping $ \gamma : \mathfrak g \rightarrow {\Lambda ^ {2} \mathfrak g } $ is defined to be the linearization of $ P $ at the identity of the group; it is a Lie-algebra $ 1 $-cocycle with respect to the adjoint action (because of the compatibility condition) and a Lie cobracket, i.e., its transpose is a Lie bracket on the dual $ \mathfrak g ^ {*} $ of $ \mathfrak g $. Conversely, any Lie bialgebra can be integrated to a unique (up to isomorphism) connected and simply connected Poisson Lie group.

To each Lie bialgebra structure on $ \mathfrak g $ there corresponds a Lie bialgebra structure $ ( \mathfrak g ^ {*} , \gamma ^ {*} ) $ on $ \mathfrak g ^ {*} $, called the dual of $ ( \mathfrak g, \gamma ) $, and a Lie bialgebra structure on $ \mathfrak g \oplus \mathfrak g ^ {*} $, called the double of $ ( \mathfrak g, \gamma ) $. Therefore, each Poisson Lie group $ ( G, P ) $ has a dual $ ( G ^ {*} , P ^ {*} ) $, and a double, with underlying manifold $ G \times G ^ {*} $. There are Poisson actions of $ G $ on $ G ^ {*} $, and of $ G ^ {*} $ on $ G $, called the dressing actions. The symplectic leaves of the Poisson manifold $ ( G,P ) $ are the orbits of the dressing action of $ G ^ {*} $.

An element $ r \in \Lambda ^ {2} \mathfrak g $ is called a solution of the classical (respectively, generalized) Yang–Baxter equation if the algebraic Schouten bracket $ [r,r] $ of $ r $ with itself vanishes (respectively, is ad-invariant). A solution $ r \in \Lambda ^ {2} \mathfrak g $ of the classical Yang–Baxter equation is also called a triangular $ r $-matrix. A solution of the generalized Yang–Baxter equation defines a Lie bialgebra structure on $ \mathfrak g $, and a Poisson Lie structure on any Lie group $ G $ with Lie algebra $ g $. In particular, a triangular $ r $-matrix defines both a left-invariant and a right-invariant Poisson structure on $ G $ whose difference is a Poisson Lie structure. An element $ r \in \mathfrak g \otimes \mathfrak g $ with an invariant symmetric part $ s $ is called a quasi-triangular $ r $-matrix if it satisfies $ [a, a] = - [s, s] $, where $ a $ is its skew-symmetric part and $ [s, s] $ is the ad-invariant element in $ \Lambda ^ {3} g $, defined by $ [s, s] ( \xi, \eta, \zeta ) = \langle {\zeta, [s \xi, s \eta] } \rangle $, for $ \xi, \eta, \zeta \in \mathfrak g $. Thus, the skew-symmetric part of a quasi-triangular $ r $-matrix defines a Poisson Lie structure on $ G $. The equation $ [a, a] = - [s, s] $, which reduces to $ [r, r] = 0 $ when $ r = a \in \Lambda ^ {2} \mathfrak g $, is variously called the classical Yang–Baxter equation, for $ r = a + s $, or the modified Yang–Baxter equation, for $ a $. It is usually written in the form $ [r _ {12 } ,r _ {13 } ]+[r _ {12 } ,r _ {23 } ]+[r _ {13 } , r _ {23 } ] = 0 $.

Examples are: trivial Poisson Lie groups, where $ P = 0 $, with as dual Poisson Lie group the dual of the Lie algebra with its linear Poisson structure (also called the Kirillov–Kostant–Souriau Poisson structure), and with as double the cotangent bundle of $ G $; the simple Lie groups with the standard quasi-triangular $ r $-matrix; and the compact Lie groups with their Bruhat–Poisson structure.

It should be noted that the algebraic concept of a Lie bialgebra and the differential-geometric notion of a Poisson Lie group were first encountered as the classical limits of quantum objects participating in the quantum inverse scattering theory for integrable systems. (Thus, the classical Yang–Baxter equation is the classical limit of the quantum Yang–Baxter equation.) Conversely, the quantization problem is the problem of associating a quantum group to a given Poisson Lie group, i.e., of finding non-commutative deformations of its commutative algebra of functions, which is a Poisson–Hopf algebra (a Poisson algebra with a Hopf algebra structure such that the comultiplication is a morphism of Poisson algebras). Dually, quantum groups are also obtained as non-co-commutative deformations of the universal enveloping algebras of Lie bialgebras, which are co-commutative co-Poisson–Hopf algebras.

References

[a1]	V.G. Drinfeld, "Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang–Baxter equation" Soviet Math. Dokl. , 27 (1983) pp. 68–71 (In Russian)
[a2]	M.A. Semenov-Tian-Shansky, "Dressing transformations and Poisson group actions" Publ. RIMS Kyoto Univ. , 21 (1985) pp. 1237–1260
[a3]	V.G. Drinfeld, "Quantum groups" , Proc. Intern. Congress Mathematicians, Berkeley 1966 , 1 , Amer. Math. Soc. (1987) pp. 798–820
[a4]	Y. Kosmann-Schwarzbach, F. Magri, "Poisson–Lie groups and complete integrability" Ann. Inst. Henri Poincaré, Phys. Th. A , 49 (1988) pp. 433–460
[a5]	J.-H. Lu, A. Weinstein, "Poisson Lie groups, dressing transformations, and Bruhat decompositions" J. Diff. Geom. , 31 (1990) pp. 501–526
[a6]	S. Majid, "Matched pairs of Lie groups associated to solutions of the Yang–Baxter equation" Pacific J. Math. , 141 (1990) pp. 311–332
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[a8]	A.G. Reyman, M.A. Semenov-Tian-Shansky, "Integrable systems II" V.I. Arnold (ed.) S.P. Novikov (ed.) , Dynamical Systems VII , Springer (1994) pp. 116–259
[a9]	I. Vaisman, "Lectures on the geometry of Poisson manifolds" , Progress in Math. , 118 , Birkhäuser (1994) pp. Chapt. 10
[a10]	A.G. Reyman, "Poisson structures related to quantum groups" L. Castellani (ed.) J. Wess (ed.) , Quantum Groups and their Applications in Physics, Internat. School Enrico Fermi (Varenna 1994) , IOS , Amsterdam (1996) pp. 407–443