Universal Capelli identities and quantum immanants for the queer Lie superalgebra

TL;DR

This paper constructs quantum immanants for the queer Lie superalgebra using Sergeev superalgebra idempotents, proves Capelli identities, and connects these to factorial Schur Q-polynomials.

Contribution

It introduces a new method to define quantum immanants for ${rak q}_N$ and establishes Capelli identities that relate these to differential operators and factorial Schur Q-polynomials.

Findings

Quantum immanants are central elements in the universal enveloping algebra of ${rak q}_N$.

Universal Capelli identities for ${rak q}_N$ are proven.

Harish-Chandra images of quantum immanants match factorial Schur Q-polynomials.

Abstract

We apply the recently introduced idempotents for the Sergeev superalgebra to construct quantum immanants for the queer Lie superalgebra ${\mathfrak q}_N$ as central elements of its universal enveloping algebra. We prove universal odd and even Capelli identities for ${\mathfrak q}_N$ and use them to calculate the images of the quantum immanants under the action of ${\mathfrak q}_N$ in differential operators. We show that the Harish-Chandra images of the quantum immanants coincide with the factorial Schur $Q$-polynomials.