Cohomologies of the Lie algebra of formal vector fields with coefficients in its adjoint space and variations of characteristic classes of foliations

1974 ◽  
Vol 8 (2) ◽  
pp. 99-112 ◽  
Author(s):  
I. M. Gel'fand ◽  
B. L. Feigin ◽  
D. B. Fuks
Keyword(s):  
Lie Algebra ◽  
Vector Fields ◽  
Characteristic Classes ◽  
Formal Vector ◽  
Adjoint Space

Cohomology of the Lie algebra of Hamiltonian formal vector fields

1973 ◽  
Vol 6 (3) ◽  
pp. 193-196 ◽  
Author(s):  
I. M. Gel'fand ◽  
D. I. Kalinin ◽  
D. B. Fuks
Keyword(s):  
Lie Algebra ◽  
Vector Fields ◽  
Formal Vector

COHOMOLOGY OF THE LIE ALGEBRA OF FORMAL VECTOR FIELDS

1970 ◽  
Vol 4 (2) ◽  
pp. 327-342 ◽  
Author(s):  
I M Gel'fand ◽  
D B Fuks
Keyword(s):  
Lie Algebra ◽  
Vector Fields ◽  
Formal Vector

scholarly journals The generalized centrally extended Lie algebraic structures and related integrable heavenly type equations

2020 ◽  
Vol 12 (1) ◽  
pp. 242-264
Author(s):  
O.Ye. Hentosh ◽  
A.A. Balinsky ◽  
A.K. Prykarpatski
Keyword(s):  
Vector Field ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Wide Class ◽  
Vector Fields ◽  
Arbitrary Dimension ◽  
Multidimensional Case ◽  
Coadjoint Orbits ◽  
Algebraic Structures ◽  
Adjoint Space

There are studied Lie-algebraic structures of a wide class of heavenly type non-linear integrable equations, related with coadjoint flows on the adjoint space to a loop vector field Lie algebra on the torus. These flows are generated by the loop Lie algebras of vector fields on a torus and their coadjoint orbits and give rise to the compatible Lax-Sato type vector field relationships. The related infinite hierarchy of conservations laws is analysed and its analytical structure, connected with the Casimir invariants, is discussed. We present the typical examples of such equations and demonstrate in details their integrability within the scheme developed. As examples, we found and described new multidimensional generalizations of the Mikhalev-Pavlov and Alonso-Shabat type integrable dispersionless equation, whose seed elements possess a special factorized structure, allowing to extend them to the multidimensional case of arbitrary dimension.

A note on the algebra of Poisson brackets

1984 ◽  
Vol 96 (1) ◽  
pp. 45-60 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Poisson Brackets ◽  
Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

scholarly journals Representations of the Lie algebra of vector fields on a sphere

2019 ◽  
Vol 223 (8) ◽  
pp. 3581-3593 ◽  
Author(s):  
Yuly Billig ◽  
Jonathan Nilsson
Keyword(s):  
Lie Algebra ◽  
Vector Fields
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