Root multiplicities of the indefinite type Kac-Moody algebras HC?¹?n

by 1976- Williams, Vicky Lynn

WILLIAMS, VICKY. Root Multiplicities of the indefinite Kac-Moody algebras HC(1) n . (Under the direction of Kailash C. Misra) Victor Kac and Robert Moody independently introduced Kac-Moody algebras around 1968. These Lie algebras have numerous application in physics and mathematics and thus have been the subject of much study over the last three decades. Kac-Moody algebras are classified as finite, affine, or indefinite type. A basic problem concerning these algebras is finding their root multiplicities. The root multiplicities of finite and affine type Kac-Moody algebras are well known. However, determining the root multiplicities of indefinite type Kac-Moody algebras is an open problem. In this thesis we determine the multiplicities of some roots of the indefinite type Kac-Moody algebras HC(1) n . A well known construction allows us to view HC(1) n as the minimal graded Lie algebra with local part V ? g0 ? V ?,whereg0 is the affine Kac-Moody algebra C(1) n and V,V ? are suitable g0- modules. From this viewpoint root spaces of HC(1) n become weight spaces of certain C(1) n -modules. Using a multiplicity formula due to Kang we reduce our problem to finding weight multiplicities in certain irreducible highest weight C(1) n -modules. We then use crystal basis theory for the affine Kac-Moody algebras C(1) n to find these weight multiplicities. With this strategy we calculate the multiplicities of some roots of HC(1) n . In particular, we determine the multiplicities of the level two roots ?2??1 ? k? of HC(1) 2 for 1 ? k ? 10. We also show that the multiplicities of the roots of HC(1) n of the form ?l??1 ? k? are n for l = k and 0 for l>k. In the process, we observe that Frenkel’s conjectured bound for root multiplicities does not hold for the indefinite Kac-Moody algebras HC(1) n .