A vector space basis of the quantum symplectic sphere

TL;DR

This paper constructs a candidate vector space basis for the algebra of the quantum symplectic sphere, using the Diamond Lemma and computer experiments to support the conjecture for all n ≥ 1.

Contribution

It provides a new candidate basis for the quantum symplectic sphere algebra, extending understanding of its algebraic structure for all dimensions.

Findings

Candidate basis constructed using the Diamond Lemma

Supported by computer experiments for n=1 to 8

Conjecture holds for all n ≥ 1

Abstract

We present a candidate of a vector space basis for the algebra $\mathcal{O}(S_q^{4n-1})$ of the quantum symplectic sphere for every $n\geq 1$. The algebra $\mathcal{O}(S_q^{4n-1})$ is defined as a certain subalgebra of the quantum symplectic group $\mathcal{O}(SP_q(2n))$. A non-trivial application of the Diamond Lemma is used to construct the vector space basis and the conjecture is supported by computer experiments for $n=1,2,...,8$.