Quantum Berezinian and the classical capelli identity

Abstract

A quantum Capelli identity is given on the multiparame-ter quantum general linear group based on the (pij, u)-condition. The multiparameter quantum Pfaffian of the (pij, u)-quantum group is also introduced and the transformation under the congruent action is given. Generalization to the multiparameter hyper-Pfaffian and relationship with the quantum minors are also investigated.

Gauss Decomposition of the Yangian $$Y({\mathfrak{gl}}_{m|n})$$

“…This was conjectured by Nazarov who proved that the quantum Berezinian was central [16]. A new proof of the centrality of the quantum Berezinian was also given in [9].…”

“…The quantum Berezinian was defined by Nazarov [16] as the following power series with coefficients in the Yangian Y (gl m|n ):…”

“…The Yangian Y (gl m|n ) was introduced in [16]. It has applications in mathematical physics because it describes symmetry in integrable models of Calogero-Sutherland systems [1,12], superstrings in AdS 5 × S 5 [11], and in the hierarchy of a form of the non-linear super-Schrödinger equation with m bosons and n fermions [4].…”

Yangians and their applications

“…2.10. The series z(u) was introduced by Nazarov [124]. The quantum Liouville formula is also due to him.…”

General eigenvalue formula for Casimir invariants of type I quantum superalgebras

“…Another application is to the construction of two variable link polynomials 8 and their multivariable extensions. 9 A general method for constructing Casimir invariants, corresponding to an arbitrary reference irrep, has been previously developed for both quantum algebras 10,11 and quantum superalgebras, 12,13 utilizing the universal R matrix. In the case of quantum algebras their eigenvalues have been explicitly determined, 11 based on techniques previously developed for simple Lie algebras.…”