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.