Quantum queer supergroups via v-differential operators

TL;DR

This paper constructs a new super representation of the quantum queer supergroup U_v(q_n) using quantum differential operators, introduces new bases and presentations, and extends classical quantum gl_n results to the queer supergroup setting.

Contribution

It develops a novel approach to represent and present the quantum queer supergroup using v-differential operators, bases, and explicit multiplication formulas.

Findings

Constructed a super representation of U_v(q_n) via quantum differential operators.

Established new bases M and L for the supergroup and its algebra.

Extended classical quantum gl_n constructions to the queer supergroup context.

Abstract

By using certain quantum differential operators, we construct a super representation for the quantum queer supergroup U_v(q_n). The underlying space of this representation is a deformed polynomial superalgebra in 2n^2 variables whose homogeneous components can be used as the underlying spaces of queer q-Schur superalgebras. We then extend the representation to its formal power series algebra which contains a (super) submodule isomorphic to the regular representation of U_v(q_n). A monomial basis M for U_v(q_n) plays a key role in proving the isomorphism. In this way, we may present the quantum queer supergroup U_v(q_n) by another new basis L together with some explicit multiplication formulas by the generators. As an application, similar presentations are obtained for queer q-Schur superalgebras via the above mentioned homogeneous components. The existence of the bases M and L and the new presentation show that the seminal construction of quantum gl_n established by Beilinson-Lusztig-MacPherson thirty years ago extends to this "queer" quantum supergroup via a completely different approach.