M.Sc Thesis

M.Sc StudentDonin Dmitry
SubjectOn Laplace Operator and Homology of Some Lie Subalgebras
of the Loop Algebra sl2
DepartmentDepartment of Mathematics
Supervisors ASSOCIATE PROFESSOR Arie Juhasz


In the present Thesis we consider the problem of calculation of  the (co)homology space of  some  Lie subalgebras of the Loop algebra .
This problem has different approaches. The one of them, Which we consider in this Thesis is to define some Hermitian metric into the standard (co)chain  complex of the algebras.
If the standard (co)chain complex of the algebra can be graded in such a way that it splits into the direct sum of finite dimensional subcomplexes, then one can define the so-called  Laplace operator, which acts on this complex. In this way we naturally identify the homology and  cohomology spaces of a Lie algebra.

 It is well known and classical fact (Eilenberg) that the kernel of the Laplace operator coincides with The homology space. Therefore the calculation of the homology space reduces to the calculation of the  kernel of the Laplace operator.

 The technique of the Laplace operator has an advantage that sometimes the representing cycles also Can be obtained. Note that  usually one considers only the problem of calculationof the dimensions of the homology space of a certain algebra which is difficult and interesting by itself.

It is quite an obscure fact that there is a general analogy between the Loop algebra and the Witt al- gebra W. In particular these two algebras have the same dimensions of their (co)homology spaces. The problem of calculation of the (co) omology spaces of nilpotent subalgebras of the Witt algebra was posed by I. M. Gel'fand at the Mathematical Congress in Nice in 1970. He conjectured the formula for the dimensions of these algebras.

His conjecture was firstly proved by L. V. Goncharova (1973) and  later  (1985) by F. V. Weinstein. The approach of Weinstein allowed him to make the similar conc-lusion about the dimensions of nilpotent subalgebras of the Loop algebra. But the complete description

of the (co)homology spaces of these algebras is still unknown. In this Thesis we make clear the case for  some of these algebras.