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By Korchmaros G.

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12 Let G and H be Lie groups and suppose H is abelian. A smooth map γ : G × G → H that satisfies γ(g1 g2 , g3 )γ(g1 , g2 ) = γ(g1 , g2 g3 )γ(g2 , g3 ) is called a smooth group 2-cocycle on G with values in H. A smooth 2-cocycle on G with values in H is called a 2-coboundary if there exists a smooth map λ : G → H such that γ(g1 , g2 ) = λ(g1 )λ(g2 )λ(g1 g2 )−1 . As before, the group 2-coboundaries correspond to the trivial group extensions, after a possible change of coordinates (more precisely, of the trivializing section for G → G).

In general, this field is not necessarily defined by a univalued Hamiltonian function on the whole of M . Even if we suppose that such a Hamiltonian function exists, it is defined only up to an additive constant. 2 The action of a Lie group G on M is called Hamiltonian if for every X ∈ g there exists a globally defined Hamiltonian function HX that can be chosen in such a way that the map g → C∞ (M ), associating to X the corresponding Hamiltonian HX , is a Lie algebra homomorphism of the Lie algebra g to the Poisson algebra of functions on M : H[X,Y ] = {HX , HY } .

The universal central extension of a semisimple Lie algebra g coincides with g itself: such algebras do not admit nontrivial central extensions. No abelian Lie algebra is perfect. Nevertheless, abelian Lie algebras can still have universal central extensions: for instance, the three-dimensional Heisenberg algebra is the universal central extension of the abelian algebra R2 . 24 I. 9 Let M be a finite-dimensional manifold. One can show that the Lie algebra Vect(M ) of vector fields on M is perfect.

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2-transitive abstract ovals of odd order by Korchmaros G.


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