Cohomologies, deformations and identities of  Lie algebras of vector fields


Right-symmetric algebras satisfy the identity
$a\cdot (b\cdot c)-(a\cdot b)\cdot c=a\cdot (c\cdot b)-(a\cdot c)\cdot
b.$
Example: the Witt algebra $W_n$ (algebra of formal vector fields with $n$
variables)
is  right-symmetric under multiplication $u\partial_i\cdot
v\partial_j=v\partial_j(u)\partial_i. $  The author
developed a cohomology theory
for
right-symmetric algebras.
Connection between right-symmetric  and Lie algebra cohomologies
was established. It was used in calculation of ``hidden"
deformations and non-split
extensions of four Cartan series of Lie algebras of vector fields.
In particular, it is proved that Lie
algebras of divergence-free vector fields  and
Hamiltonian vector fields have new central extensions.
Virasoro algebra has one more hidden central extension.

All necessary notions will be introduced in the talk.
\bye
A.S. Dzhumadil'daev, {\em Virasoro Type Lie algebras and deformations},
Zeitschrift fuer Physik, Ser.C, {\bf 72}(1996), 509-517.

A.S. Dzhumadil'daev, {\em Cohomologies and deformations of
right-symmetric algebras,} J.Math.Sciences, {\bf 93}(1999), No. 6,
1836-1876. Preprint available math.RT/9807065.

A.S. Dzhumadil'daev, {\em Minimal polynomials for right-symmetric
algebras,}
J.Algebra, {\bf 225}(2000), 201-230.


With best regards,

Askar Dzhumadil'daev
