Graded Lie Algebras of Finite Characteristic

Abstract

We consider some examples of canonical transformations of the extended phase space, which map a completely integrable system into another completely integrable system. The proposed transformations are closed to the known Maupertuis-Jacobi mappings and to the Kepler change of the time. Using the Kepler transformation we construct new integrable systems related to the Toda lattices and the Stäckel family of integrable systems.

The classification of modular Lie superalgebras of type M

“…The natural filtrations of finite-dimensional modular Lie algebras of Cartan type were proved to be invariant in [2,3,9], in the infinite-dimensional case, the same conclusion was given in [1] by determining ad-nilpotent elements. In the situation of finite-dimensional modular Lie superalgebras of Cartan type, the invariances of the filtration structures for modular Lie superalgebras W , S , H , , M were obtained in [5,10,12,14] by means of investigating the minimal dimension of image spaces and ad-nilpotent elements, respectively.…”

Gradings of non-graded Hamiltonian Lie algebras

“…This is a simple Lie algebra if p > 2, see [40] or [58]. In characteristic two it is simple provided n\ > 1 and n 2 > 1, as one can prove along the lines of [58, Theorem 3.5 or Theorem 4.5 of Chapter 4].…”

Nottingham Lie Algebras With Diamonds of Finite Type

“…In [1,2] the positive parts of certain twisted loop algebras N = N q,µ 3 (for parameters q and µ 3 as in Theorem 2.4, with µ 3 lying in the prime field) of suitable central extensions of the finite-dimensional, simple, modular Hamiltonian Lie algebras H(2 : (1, n); ω 0 ) (see [23], [37], [24]) are constructed. (In two cases, one has to take the extension of such an algebra by an outer derivation.)…”