Root multiplicities of the indefinite type Kac-Moody algebras HC[subscript n]1

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Title: Root multiplicities of the indefinite type Kac-Moody algebras HC[subscript n]1
Author: Williams, Vicky Lynn
Advisors: Dr. Kailash Misra, Committee Chair
Dr. Ernest Stitzinger, Committee Member
Dr. Naihuan Jing, Committee Member
Dr. Jacqueline Hughes-Oliver, Committee Member
Abstract: 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 HC[subscript n]⁽¹⁾. A well known construction allows us to view HC[subscript n]⁽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 C[subscript n]⁽¹⁾. and V,V' are suitable g₀-modules. From this viewpoint, root spaces of HC[subscript n]⁽¹⁾ become weight spaces of certain C[subscript n]⁽¹⁾-modules. Using a multiplicity formula due to Kang we reduce our problem to finding weight multiplicities in certain irreducible highest weight C[subscript n]⁽¹⁾-modules. We then use crystal basis theory for the affine Kac-Moody algebras C[subscript n]⁽¹⁾ to find these weight multiplicities. With this strategy we calculate the multiplicities of some roots of HC[subscript n]⁽¹⁾. In particular, we determine the multiplicities of the level two roots -2(alpha₋₁)-k(delta) of HC[subscript n]⁽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 HC[subscript n]⁽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 HC[subscript n]⁽¹⁾.
Date: 2003-06-27
Degree: PhD
Discipline: Mathematics
URI: http://www.lib.ncsu.edu/resolver/1840.16/5422


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