By Mark Adler

ISBN-10: 3642061281

ISBN-13: 9783642061288

ISBN-10: 366205650X

ISBN-13: 9783662056509

From the reports of the 1st edition:

"The target of this ebook is to give an explanation for ‘how algebraic geometry, Lie thought and Painlevé research can be utilized to explicitly clear up integrable differential equations’. … one of many major merits of this publication is that the authors … succeeded to provide the cloth in a self-contained demeanour with a variety of examples. consequently it may be extensively utilized as a reference ebook for lots of matters in arithmetic. In precis … an exceptional e-book which covers many fascinating matters in glossy mathematical physics." (Vladimir Mangazeev, The Australian Mathematical Society Gazette, Vol. 33 (4), 2006)

"This is an in depth quantity dedicated to the integrability of nonlinear Hamiltonian differential equations. The publication is designed as a educating textbook and goals at a large readership of mathematicians and physicists, graduate scholars and execs. … The e-book offers many beneficial instruments and strategies within the box of thoroughly integrable structures. it's a necessary resource for graduate scholars and researchers who prefer to input the integrability thought or to benefit interesting elements of integrable geometry of nonlinear differential equations." (Ma Wen-Xiu, Zentralblatt MATH, Vol. 1083, 2006)

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**Additional resources for Algebraic Integrability, Painlevé Geometry and Lie Algebras**

**Example text**

4) they define a foliation near m with two-dimensional leaves. On a neighborhood U of m we can then find functions z 1 , ... , Zn-2 such that dz1, ... , dzn-2 are independent on U and such that Xp 1 [zi] = Xq 1 [zi] = 0, by the description of the leaves of the foliation. It follows that Pl, Q1, z1, ... , Zn-2 are coordinates on U and that, on U, In order to show that {Zk, zl} is independent of Ql and Pl it suffices to show that {{zk, zl} ,pi} = 0 = {{zk, zt}, ql}, an easy consequence of the Jacobi identity.

M, {·, ·}) is called a regular Poisson manifold when Rkm {· , ·} = Rk {· , ·}, independently ofm EM. If we choose local coordinates (x 1 , ... , Xn) on a neighborhood of m then we see that Rkm {· , ·} equals the number of independent columns of the Poisson matrix X:= ({xi,Xj}) 1:1;;iJ:I;;n of{·,·} with respect to (x1, ... ,xn) at the point m, hence it is the rank of X at m. 18). Skew-symmetry of X implies that the rank of a Poisson structure at a point is always even. 12. 7 are regular (of respective rank 2r and 0).

19 a natural question to ask if the fundamental vector fields of the action are Hamiltonian. l. 7x for all X in a basis of g and then one extends by linearity). l. 7 is called the co-momentum map. Hamiltonian actions and their momentum maps have been studied mainly in the context of symplectic manifolds, and the theory of Hamiltonian actions, such as torus actions, on symplectic manifolds has become a subject of its own. , [19] and [73] and the references in these books. 41 below), and momentum maps that take values in a Lie group (see [64] and [16]).

### Algebraic Integrability, Painlevé Geometry and Lie Algebras by Mark Adler

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