Minor identities for Sklyanin determinants
Naihuan Jing, Jian Zhang
TL;DR
The paper develops a quantum invariant theory for orthogonal and symplectic quantum spaces using R-matrix methods, introducing the Sklyanin determinant sdet_q(X) and the quantum Pfaffian Pf_q(X) as central elements and relating them to det_q(T) through embeddings into A_q(Mat_N). It extends classical determinant identities to the Sklyanin setting by proving Jacobi-, Cayley-, Sylvester-, and Muir-type minor identities, along with q↔q^{-1} duality relations for these central invariants. The work further provides quasideterminant expressions for sdet_q(X) and Pf_q(X) and generalizes factorization formulas in the spirit of Krob–Leclerc, connecting these constructions with twisted Yangians and invariant theory on quantum symmetric spaces. Overall, the results unify and extend determinant and Pfaffian-like structures from the quantum GL setting to orthogonal and symplectic quantum spaces, with concrete algebraic control via coideal subalgebras and comatrix relations.
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.