### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 084, 21 pages      arXiv:1609.05270      https://doi.org/10.3842/SIGMA.2017.084

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

Jiao Zhang a and Naihong Hu b
a) Department of Mathematics, Shanghai University, Baoshan Campus, Shangda Road 99, Shanghai 200444, P.R. China
b) Department of Mathematics, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Minhang Campus, Dong Chuan Road 500, Shanghai 200241, P.R. China

Received April 18, 2017, in final form October 20, 2017; Published online October 27, 2017

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 $U_q({\mathfrak {sp}}_{2n})$.

Key words: quantum symplectic group; quantum symplectic space; quantum differential operators; differential calculus; module algebra.

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