Minor identities for Sklyanin determinants

TL;DR

This paper investigates the invariant theory of quantum symmetric spaces of orthogonal and symplectic types, establishing new identities and expressions for Sklyanin determinants and quantum Pfaffians using R-matrix techniques.

Contribution

It extends classical identities like q-Jacobi and q-Cayley's to Sklyanin minors and quantum Pfaffians, providing new formulas and connections in quantum invariant theory.

Findings

Extended q-Jacobi, q-Cayley's, q-Sylvester identities to Sklyanin minors

Derived expressions for Sklyanin determinants and quantum Pfaffians via quasideterminants

Connected quantum determinants with Sklyanin determinants and quantum Pfaffians in symplectic and orthogonal cases

Abstract

We explore the invariant theory of quantum symmetric spaces of orthogonal and symplectic types by employing R-matrix techniques. Our focus involves establishing connections among the quantum determinant, Sklyanin determinants associated with the orthogonal and symplectic cases, and the quantum Pfaffians over the symplectic quantum space. Drawing inspiration from twisted Yangians, we not only demonstrate but also extend the applicability of q-Jacobi identities, q-Cayley's complementary identities, q-Sylvester identities, and Muir's theorem to Sklyanin minors in both orthogonal and symplectic types, along with q-Pfaffian analogs in the symplectic scenario. Furthermore, we present expressions for Sklyanin determinants and quantum Pfaffians in terms of quasideterminants.