A vector space basis of the quantum symplectic sphere

We give a candidate of a vector space basis for the algebra O(S 4n−1 q ) of the quantum symplectic sphere for every n ≥ 1. The construction follows by a nontrivial application of the Diamond Lemma. The conjecture is supported by computer experiments for n = 1, 2, ..., 8. The work is motivated by a result of Landi and D’Andrea, who proved that the first n − 1 generators of the C ∗ -algebra C(S 4n−1 q ), n ≥ 2 are zero. By finding a vector space basis, we can conclude that these generators are different from zero in the corresponding algebra O(S 4n−1 q ).