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

Abstract (Summary)

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 .
Bibliographical Information:

Advisor:

School:North Carolina State University

School Location:USA - North Carolina

Source Type:Master's Thesis

Keywords:north carolina state university

ISBN:

Date of Publication: