Automorphism Group Of Compact Abelian Lie Group And Its Application In Weyl Group

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).