【作者】 张健；

【导师】 景乃桓；

【作者基本信息】 华南理工大学 ， 应用数学， 2015， 博士

【摘要】 量子群是李群和李代数经过形变后得到的一类特殊的Hopf代数。它起源于理论物理,最早是由Drinfeld [6]和.Jimbo[16]分别提出来的。1988年,Faddeev, Reshetikhin和Takhtajan利用R矩阵给出了量子群的另外一种实现方式[7],并说明量子行列式是A(Mat。(n))的中心。随后Hashimoto, Hayashi, Taft和Towber将行列式的很多性质推广到量子行列式[11,37]。1996年,Strickland [36]在研究量子不变理论的时候引入了量子Pfaffian,但是从中我们并不清楚量子Pfaffian和量子行列式之间有什么联系。Ray, Jing[15]和Noumi[32]在量子反对称矩阵上给出量子Pfaffian新的定义。在[15]中,Ray和Jing利用量子代数的表示论证明：通过一些变量替换后量子行列式等于量子Pfaffiano本文主要内容是用量子外代数得到量子行列式和量子Pfaffian之间的关系,并推广到更一般的形式。在第3章,我们用量子外代数研究量子Pfaffian得到了一系列的Plucker关系,并利用Pliicker关系得到定义量子Pfaffian所必须的关系,我们称之为Maya关系。其次我们得到量子行列式和量子Pfaffian之间的关系从而证明任何一个量子行列式都可以表示成量子Pfaffiano在最后一节,我们将Luque-Thibon的结果推广到量子hyper-Pfaffiano一个非常有趣的现象是只有当q为一些特殊的单位根的时,量子行列式可以表示成量子hyper-Pfaffiano行列式是定义在矩阵上的函数,如果将行列式所有的符号改为+1就得到permanent[31]。Permanent和行列式有许多相同的性质,他们都是immanant的特例,分别对应n的两个partition (n)和(1n)。在本文第4章,我们引入一个新的量子群,在上面定义了量子permanent。我们证明了量子permanent和量子行列式相等。并得到行列式相关结论的fermionic形式：任何一个量子permanent都能表示成量子Hafniano在数学和物理中,人们通常将一般的矩阵推广到高维的矩阵。在1843年,Cayley[5]给出了三种hyperdeterminant的定义。本文第5章将要推广Cay ley的第一类hyperdeterminant。我们引入高维的量子矩阵,将Mata(n)推广到高维情形,并对任意m维量子矩阵定义了高维量子行列式。m为偶数时,高维量子行列式是Cayley第一类hyperdeterminant的q形变。此外,我们将量子Pfaffian推广到Matsumoto的高维情形,得到DeRahm复形的量子化。我们利用外代数得到任何一个高维的量子行列式都能表示成量子hyper-Pfaffiano更多还原

【Abstract】 Quantum groups are certain deformation of Lie groups and Lie groups. They are a particular class of Hopf algebras which first appeared in theoretical physics and then formalized by Drinfeld and Jimbo. In 1988, Faddeev, Reshetikhin and Takhtajan gave another realization of A(Matq(n)) using R-matrix,the quantum determinant (cf. [14,33]) serves as a distinguished central element in the quantum group and it is well-known that lots of properties of the determinant can be generalized to the quantum case (cf. [11,37]).In [36] a notion of a quantum Pfaffian was introduced in the context of quantum invariant theory, but it was not clear how it was related to the quantum determinant from the context. In [15,32] the invariant theory of certain quantum symplectic group was used to define the notion of quantum anti-symmetric matrices, where the quantum Pfaffian can be realized on the quantum coordinate ring as well, and Ray and Jing proved that the quantum Pfaffian is equal to the quantum determinant after change of variables. This relation between the quantum determinant and quantum Pfaffian was derived by making use of the representation theory of the quantum algebra.In chapter 3 we study quantum Pfaffian using quantum exterior algebras and derive a complete family of Pliicker relations for the quantum linear transformations, and then use them to give an optimal set of relations required for the quantum Pfaffian. We then give the formula between the quantum determinant and the quantum Pfaffian and prove that any quantum determinant can be expressed as a quantum Pfaffian. In the last section we introduce the notion of quantum hyper-pfaffian generalizing Luque-Thibon’s work in the classical case [27]. An interesting new phenomenon appears that the quantum hyper-Pfaffian satisfies the non-trivial identity only for the modular case.It is known that the permanent [26] has some properties similar to the determinant. It sometimes referred as the positive determinant (cf. [31]) and is defined by changing all the signs to+1 in the definition of determinant. In chapter 4 a quantum group is introduced and on which the quantum determinant is shown to be equal to the quantum permanent. This identity would be a bosonic version of the identity between the quantum Pfaffian and quantum determinant proved in [18] if the latter is taken as the fermionic case. Similarly the quantum Pfaffian is proved to be identical to the quantum Hafnian on the quantum algebra.In mathematics and physics, one is often lead to consider m-dimensional hyper-matrices A= (ai1...im) indexed by multi-indices, which generalize the usual rectangular matrices [8,13,30]. In chapter 5, we introduce the quantum hyper-monoid. It is proved that the quantum coordinate ring of the monoid can be lifted to a quantum hvper-algebra, in which the Quantum determinant and the Quantum Pfaman are lifted to the quantum hyperdeterminant and quantum hyper-Pfaffian respectively. The quan-tum hyperdeterminant is defined for any dimension, while the even case is shown to be a q-analog of Cayley’s first hyperdeterminant [5]. Any quantum hyperdetermianni can be expressed as the quantum analog of Matsumoto’s hyper-Pfaffian.更多还原