Heat Kernel On Compact Lie Groups And Cones

The purpose of this thesis is twofold.First,we survey the computation of heat kernel on compact Lie group with bi-invariant metric.We select S1 and S3 as examples.In perspective of analysis and algebra,we calculate the formula for heat kernels with standard metric,and find that they are the same in conclusion.The second is to study heat kernel on a cone.By using representation theory and the proof of heat kernel on S3,we will construct heat kernel on four-dimensional cone C4 with standard metric.We will use two methods to calculate the heat kernel on S1 and S3 with standard metric.The first method is analytic.We find eigenvalues and eigenfunctions of Laplace operator on S1 by solving equation directly.With the help of Sturm-Liouville theorem,we obtain the heat kernel on S1.Then we find the heat kernel on S3 by using recursion formula on spheres.The second method is algebraic.We find all of the irreducible unitary representations of S1 and S3 using the space of complex homogeneous polynomials on as representation space.As a result,we have eigenvalues and eigenfunctions of Laplace operator with respect to bi-invariant metric.With the help of Sturm-Liouville theorem,we obtain the formula of heat kernels on S1 and S3.Finally,we compare two results and find that they are the same.In the end,we will study the heat kernel on C4 with standard metric.In inspired of Sturm-Liouville theorem,we construct the heat kernel on C4 by using representation theory and heat kernel on S3.Finally,we check that it is indeed the heat kernel with respect to standard metric.