Contractions: Nijenhuis and Saletan tensors for general algebraic structures
نویسندگان
چکیده
In this note we study generalizations in many directions of the contraction procedure for Lie algebras introduced by Saletan [Sa]. We consider products of arbitrary nature, not necessarily Lie brackets, and we generalize to infinite dimension, considering a modification of the approach by Nijenhuis tensors to bilinear operations on sections of finite-dimensional vector bundles. We apply our general procedure to Lie algebras, Lie algebroids, and Poisson brackets. We present also results on contractions of n-ary products and coproducts. Supported by KBN, grant No. 2 P03A 031 17. Supported by PRIN SINTESI.
منابع مشابه
Courant algebroid and Lie bialgebroid contractions
Contractions of Leibniz algebras and Courant algebroids by means of (1,1)-tensors are introduced and studied. An appropriate version of Nijenhuis tensors leads to natural deformations of Dirac structures and Lie bialgebroids. One recovers presymplectic-Nijenhuis structures, PoissonNijenhuis structures, and triangular Lie bialgebroids as particular examples. MSC 2000: Primary 17B99; Secondary 17...
متن کاملNijenhuis Integrability for Killing Tensors
The fundamental tool in the classification of orthogonal coordinate systems in which the Hamilton–Jacobi and other prominent equations can be solved by a separation of variables are second order Killing tensors which satisfy the Nijenhuis integrability conditions. The latter are a system of three non-linear partial differential equations. We give a simple and completely algebraic proof that for...
متن کاملThe Variety of Integrable Killing Tensors on the 3-Sphere
Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton–Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing tensors on the sphere S and give a set of isometry invariants for the integrability of a Killing tensor. We describe explicitly the space of solutions as well as i...
متن کاملCourant-Nijenhuis tensors and generalized geometries
Nijenhuis tensors N on Courant algebroids compatible with the pairing are studied. This compatibility condition turns out to be of the form N + N = λI for irreducible Courant algebroids, in particular for the extended tangent bundles T M = TM ⊕ TM . It is proved that compatible Nijenhuis tensors on irreducible Courant algebroids must satisfy quadratic relations N −λN + γI = 0, so that the corre...
متن کاملQuantum Bi-Hamiltonian Systems
We define quantum bi-Hamiltonian systems, by analogy with the classical case, as derivations in operator algebras which are inner derivations with respect to two compatible associative structures. We find such structures by means of the associative version of Nijenhuis tensors. Explicit examples, e.g. for the harmonic oscillator, are given.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008