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

root multiplicities;representation theory;Kac-Moody algebras;Lie algebras

Williams, Vicky Lynn ; Dr. Kailash Misra, Committee Chair,Dr. Ernest Stitzinger, Committee Member,Dr. Naihuan Jing, Committee Member,Dr. Jacqueline Hughes-Oliver, Committee Member,Williams, Vicky Lynn ; Dr. Kailash Misra ; Committee Chair ; Dr. Ernest Stitzinger ; Committee Member ; Dr. Naihuan Jing ; Committee Member ; Dr. Jacqueline Hughes-Oliver ; Committee Member

University:North Carolina State University

关键词: root multiplicities;    representation theory;    Kac-Moody algebras;    Lie algebras;   

Others  :  https://repository.lib.ncsu.edu/bitstream/handle/1840.16/5422/etd.pdf?sequence=1&isAllowed=y

美国|英语

来源: null

【 摘 要 】

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 suitableg₀-modules.From this viewpoint, root spaces of HC[subscript n]⁽¹⁾ become weight spaces ofcertain 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 indefiniteKac-Moody algebras HC[subscript n]⁽¹⁾.