Poisson Lie group
A Lie group with a Poisson structure
which is compatible with the group multiplication, i.e., the multiplication
is a Poisson mapping, where
is the product Poisson manifold (cf. Poisson algebra).
The corresponding infinitesimal object is a Lie bialgebra (see Quantum groups) , called the tangent Lie bialgebra of
. Here,
is the Lie algebra of the Lie group
and the linear mapping
is defined to be the linearization of
at the identity of the group; it is a Lie-algebra
-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
of
. 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 there corresponds a Lie bialgebra structure
on
, called the dual of
, and a Lie bialgebra structure on
, called the double of
. Therefore, each Poisson Lie group
has a dual
, and a double, with underlying manifold
. There are Poisson actions of
on
, and of
on
, called the dressing actions. The symplectic leaves of the Poisson manifold
are the orbits of the dressing action of
.
An element is called a solution of the classical (respectively, generalized) Yang–Baxter equation if the algebraic Schouten bracket
of
with itself vanishes (respectively, is ad-invariant). A solution
of the classical Yang–Baxter equation is also called a triangular
-matrix. A solution of the generalized Yang–Baxter equation defines a Lie bialgebra structure on
, and a Poisson Lie structure on any Lie group
with Lie algebra
. In particular, a triangular
-matrix defines both a left-invariant and a right-invariant Poisson structure on
whose difference is a Poisson Lie structure. An element
with an invariant symmetric part
is called a quasi-triangular
-matrix if it satisfies
, where
is its skew-symmetric part and
is the ad-invariant element in
, defined by
, for
. Thus, the skew-symmetric part of a quasi-triangular
-matrix defines a Poisson Lie structure on
. The equation
, which reduces to
when
, is variously called the classical Yang–Baxter equation, for
, or the modified Yang–Baxter equation, for
. It is usually written in the form
.
Examples are: trivial Poisson Lie groups, where , 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
; the simple Lie groups with the standard quasi-triangular
-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 |
[a7] | V. Chari, A. Pressley, "A guide to quantum groups" , Cambridge Univ. Press (1994) pp. Chapts. 1–3 |
[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 |
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