Quantum Determinants And Quantum Pfaffians

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.