Quantum Symmetric Spaces and Quantum Symplectic Invariants

quantum groups;quantum symmetric spaces;quantum antisymmetric matrices;quantum symplectic invariants

Ray, Robert ; Naihuan Jing, Committee Chair,Ernest Stitzinger, Committee Member,Tom Lada, Committee Member,Kailash Misra, Committee Member,Ray, Robert ; Naihuan Jing ; Committee Chair ; Ernest Stitzinger ; Committee Member ; Tom Lada ; Committee Member ; Kailash Misra ; Committee Member

University:North Carolina State University

关键词: quantum groups;    quantum symmetric spaces;    quantum antisymmetric matrices;    quantum symplectic invariants;   

Others  :  https://repository.lib.ncsu.edu/bitstream/handle/1840.16/3916/etd.pdf?sequence=1&isAllowed=y

美国|英语

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【 摘 要 】

Although subgroups of the general linear group are well understood, properties of quantum analogs of these subgroupshave been a little more elusive.This is due in part to the fact that these sets are not groups and there does not appear to be a natural way to embed them in a quantum GL(n). With respect to the matrix multiplication, these sets generally fail to be closed, and the elements do not all have inverses.However, the associated quantized regular functions and quantized universal enveloping Lie algebras still retain Hopf algebra structures.It is this structure that was used by Jing and Yamada in 1994 toconstruct q-analogs of the orthogonal group and the associated q-orthogonal invariants (quantum symmetric algebra). The first two chapters of this text provide the basic background information for the main thesis topic.In chapter 1, we review some basic definitions and properties of Hopf algebras, first discussing algebras and coalgebras (drawing mostly from material by Montgomery and Kassel).Here, the definitions of the regular functions of GLn are recalled and other examples of Hopf algebras are given to illustrate some of the properties.In chapter 2,quantum versions of these Hopf algebras are presented.In their paper, Jing and Yamada use a differential method of defining q-orthogonal invariants of the action of the quantum orthogonal group on Aq(X).In other words, the q-orthogonal invariant subspace is defined as the subspace of Aq(X) that is annihilatad by a q-analog of U(so(n)).In the third chapter,a q-analog of U(sp(n,C)) is constructed and theq-symplectic invariants in Aq(X) are defined relative to the left and right action of this quantum universal symplectic Lie algebra, in a differential fashion similar to Jing and Yamada, where we require n be even.The space of these q-symplectic invariants is then decomposed into right and left irreducible modules and several properties are discussed and we show how these q-symplectic invariants define quantum antisymmetric matrices.