Atiyah Classes Of Four-dimensional Lie Algebra Pairs Over R

Lie algebra is an important kind of non-associative algebras,and a Lie algebra can be considered as a Lie algebroid over a one-point manifold.Based on the theory of Atiyah classes of Lie algebroid pairs,the Atiyah class of a Lie algebra pair(L,A)is studied,where L is a Lie algebra,and A is a Lie subalgebra of L.In this thesis,we define the Atiyah classes of Lie algebra pairs in the light of the theory of Lie algebra modules and cohomology.In particular,a criterion for Atiyah class to be zero is proved.As an application,Atiyah classes of four-dimensional Lie algebra pairs over R are calculated.The thesis is organized as follows:In Chapter 1,some concepts and terminology of the research topics in this thesis are reviewed,including the basic theory of Lie algebras,Lie algebra modules and Lie algebra cohomology.In Chapter 2,the Atiyah classes of Lie algebra pairs are introduced.Furthermore,an equivalent condition for Atiyah class to be zero is obtained,which provides a theoretical foundation for the calculation in Section 3.In Chapter 3,Lie brackets between any two bases are calculated under the classification of four-dimensional Lie algebras over R.By applying the theory in Chapter 2,the methods of computing Atiyah classes of four-dimensional Lie algebra pairs over R are given.Moreover,all calculation results are listed in tables.