In this paper, we principally introduce two important results about compact abelian Lie group. In part2, we first give that for any compact abelian Lie group G, we have G=U(1)k×H, where H is a finite abelian group, then we proof the structure of automorphism group of G. In part3, for PU(n), a special compact abelian Lie group, we first consider the MAD subgroup of PU(n), then, we define a anti-symmetric pairing of MAD subgroup of PU(n), last, we consider the structure of Weyl group of MAD subgroup of PU(n).And we will prove the following two theorems in detail.Theoreml Assume G is a compact abelian lie group, one has G=G0×G1, then Aut(G)≈G1k×(GL(k,Z)×Aut(G1))Theorem2Assume that K is a MAD-subgroup of G=PU(n),let K0be the identity component subgroup of G and H=K/K0,thenWG(K)=Hom(H,K0)×(SP(H)×St). |