Root multiplicities of the indefinite type Kac-Moody algebras HCn(1)
Victor Kac and Robert Moody independently introduced Kac-Moody algebras around 1968. These Lie algebras have numerous applications 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 HCn(1). A well known construction allows us to view HCn(1) as the minimal graded Lie algebra with local part V direct sum g0 direct sum V', where g0 is the affine Kac-Moody algebra Cn(1). and V,V' are suitable g0-modules. From this viewpoint, root spaces of HCn(1) become weight spaces of certain Cn(1)-modules. Using a multiplicity formula due to Kang we reduce our problem to finding weight multiplicities in certain irreducible highest weight Cn(1)-modules. We then use crystal basis theory for the affine Kac-Moody algebras Cn(1) to find these weight multiplicities. With this strategy we calculate the multiplicities of some roots of HCn(1). In particular, we determine the multiplicities of the level two roots -2(alpha-1)-k(delta) of HCn(1) for 1 less than or equal to k less than or equal to 10. We also show that the multiplicities of the roots of HCn(1) of the form -l(alpha-1) -k(delta) are n for l equal to k and 0 for l greater than k. In the process, we observe that Frenkel's conjectured bound for root multiplicities does not hold for the indefinite Kac-Moody algebras HCn(1).
Advisor:Dr. Kailash Misra; Dr. Ernest Stitzinger; Dr. Naihuan Jing; Dr. Jacqueline Hughes-Oliver
School:North Carolina State University
School Location:USA - North Carolina
Source Type:Master's Thesis
Date of Publication:06/27/2003