Realization of $U_q({\mathfrak{sp}}_{2n})$ within the Differential Algebra on Quantum Symplectic Space

TL;DR

This paper constructs a realization of the quantum symplectic algebra $U_q({\mathfrak{sp}}_{2n})$ as quantum differential operators on a quantum symplectic space, demonstrating its module algebra structure and root vector actions.

Contribution

It provides a new algebraic realization of $U_q({\mathfrak{sp}}_{2n})$ within quantum differential operators and details the structure of its module algebra and root vectors.

Findings

Realization of $U_q({\mathfrak{sp}}_{2n})$ as quantum differential operators.

Quantum symplectic space as a $U_q({\mathfrak{sp}}_{2n})$-module algebra.

Explicit description of root vectors via Lusztig's braid automorphisms.

Abstract

We realize the Hopf algebra $U_q({\mathfrak {sp}}_{2n})$ as an algebra of quantum differential operators on the quantum symplectic space $\mathcal{X}(f_s;\mathrm{R})$ and prove that $\mathcal{X}(f_s;\mathrm{R})$ is a $U_q({\mathfrak{sp}}_{2n})$-module algebra whose irreducible summands are just its homogeneous subspaces. We give a coherence realization for all the positive root vectors under the actions of Lusztig's braid automorphisms of $U_q({\mathfrak {sp}}_{2n})$.