In this paper, we associate quantum vertex algebras to a certain family of associative algebras $\widetilde{\A}(g)$ which are essentially Ding-Iohara algebras. To do this, we introduce another closely related family of associative algebras $\A(h)$. The associated quantum vertex algebras are based on the vacuum modules for $\A(h)$, whereas $\phi$-coordinated modules for these quantum vertex algebras are associated to $\widetilde{A}(g)$-modules. Furthermore, we classify their irreducible $\phi$-coordinated modules.

We study isometric Lie group actions on symmetric spaces admitting a section, i.e., a submanifold that meets all orbits orthogonally at every intersection point. We classify such actions on the compact symmetric spaces with simple isometry group and rank greater than one. In particular, we show that these actions are hyperpolar, i.e., the sections are flat.

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