On the quantum symplectic sphere

Abstract: The algebra of the quantum symplectic $(4n-1)$-sphere $\mathcal{O}(S_q^{4n-1})$ is defined as a subalgebra of the quantum symplectic group by Faddeev, Reshetikhin and Takhtajan. Recently D'Andrea and Landi investigated faithfull irreducible $*$-representations of $\mathcal{O}(S_q^{4n-1})$. They proved that the first $n-1$ generators of its enveloping $C^*$-algebra $C(S_q^{4n-1})$ are all zero. The result is a generalisation of the case where $n=2$ which was shown by Mikkelsen and Szymański. In this talk, I will first present how $C(S_q^{4n-1})$ can be described as a graph $C^*$-algebra, from which it follows that $C(S_q^{4n-1})$ is isomorphic to the quantum $(2n+1)$-sphere by Vaksman and Soibelman. Then, I present a candidate of a vector space basis for $\mathcal{O}(S_q^{4n-1})$ which is constructed by a nontrivial application of the Diamond lemma. The conjecture is supported by computer experiments for $n=1,...,8$. By finding a vector space basis we can moreover conclude that the $n-1$ generators are non-zero inside the algebra $\mathcal{O}(S_q^{4n-1})$.

operator algebrasquantum algebra

Audience: researchers in the topic

Quantum Groups Seminar [QGS]

Series comments: This seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

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