Dynamical quantum determinants and Pfaffians

Quantum Pfaffians and Hyper-Pfaffians

The concept of the quantum Pfaffian is rigorously examined and refurbished using the new method of quantum exterior algebras. We derive a complete family of Plücker 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. Finally the quantum hyper-Pfaffian is introduced, and we prove a similar result of expressing quantum determinants in terms of quantum …

Pfaffian-行列式恒等式的量子类比：q-斜对称情形下的代数与几何研究

A Quantum Analogue of the Pfaffian-Determinant Identity, An Algebraic and Geometric Study in the q-Skew-Symmetric Case

本文研究q-斜对称矩阵中Pfaffian-行列式恒等式的量子形变，通过FRT构造和量子外代数定义量子Pfaffian与量子行列式，验证低维情形下的量子恒等式，并探讨其在量子辛几何及非交换几何中的意义。

Multiparameter quantum Pfaffians

The multiparameter quantum Pfaffian of the $(p, λ)$-quantum group is introduced and studied together with the quantum determinant, and an identity relating the two invariants is given. Generalization to the multiparameter hyper-Pfaffian and relationship with the quantum minors are also considered.

Sklyanin行列式的次要恒等式

Minor identities for Sklyanin determinants

本文利用R矩阵技术研究正交与辛型量子对称空间的不变理论，建立了量子行列式、Sklyanin行列式与量子Pfaffian的联系，并将多种q-恒等式推广至Sklyanin子式，同时给出其准行列式表达。

Quantum permanents and Hafnians via Pfaffians

Quantum determinants and Pfaffians or permanents and Hafnians are introduced on the two parameter quantum general linear group. Fundamental identities among quantum Pf, Hf, and det are proved in the general setting. We show that there are two special quantum algebras among the quantum groups, where the quantum Pfaffians have integral Laurent polynomials as coefficients. As a consequence, the quantum Hafnian is computed by a closely related quantum permanent and identical to the quantum Pfaffian on this special quantum algebra.

Quantum Hyperdeterminants and Hyper-Pfaffians

The notion of generalized quantum monoids is introduced. It is proved that the quantum coordinate ring of the monoid can be lifted to a quantum hyper-algebra, in which the quantum determinant and quantum Pfaffian are sent to the quantum hyperdeterminant and quantum hyper-Pfaffian respectively. The quantum hyperdeterminant in even dimension is shown to be a $q$-analog of Cayley's first hyperdeterminant.