Quantum Hyperdeterminants and Hyper-Pfaffians

TL;DR

This paper introduces quantum hyperdeterminants and hyper-Pfaffians within a new quantum hyper-algebra framework, linking them to classical hyperdeterminants and expanding quantum algebra theory.

Contribution

It develops the concept of quantum hyperdeterminants and hyper-Pfaffians, extending quantum algebra structures to include these generalized determinants.

Findings

Quantum hyperdeterminant is a q-analog of Cayley's hyperdeterminant.

Quantum coordinate ring can be lifted to a quantum hyper-algebra.

Quantum hyperdeterminant and hyper-Pfaffian are realized within this framework.

Abstract

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.