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 find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra g^, from g-module homomorphisms. When g=sl_2, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to g^, and we identify obstructions to the construction arising from the possible non-existence of series solutions having a prescribed form.

Motivated by Kapranov's discovery of an $L_\infty$ algebra structure on the tangent complex of a K\"{a}hler manifold and Chen-Sti\'{e}non-Xu's construction of a Leibniz$_\infty[1]$ algebra associated with a Lie pair, we find a general method to construct Leibniz$_\infty[1]$ algebras ---from a DG derivation $\mathscr{A} \xrightarrow{\delta} \Omega$ of a commutative differential graded algebra $\mathscr{A}$ valued in a DG $\mathscr{A}$-module $\Omega$. We prove that for any $\delta$-connection $\nabla$ on $\mathcal{B}$, the $\A$-dual of $\Omega$, there associates a Leibniz$_\infty[1]$ $\mathscr{A}$-algebra $(\mathcal{B},\{\mathcal{R}^\nabla_k\}_{k\geq 1})$. Moreover, this construction is canonical, i.e., the isomorphism class of $(\mathcal{B},\{\mathcal{R}^\nabla_k\}_{k\geq 1})$ only depends on the homotopy class of $\delta$.

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Bisch and Jones proposed the classification of planar algebras by simple generators and relations. They investigated with the second author the classification of planar algebras generated by 2-boxes. In this paper, we classify singly-generated Thurston-relation planar algebras, defined as subfactor planar algebras generated by a 3-box satisfying a relation proposed by Dylan Thurston. Our main result shows that such subfactor planar algebras are either the E6 subfactor planar algebras or belong to a two-parameter family of planar algebras arising from the representations of type A quantum groups. We introduce a new method for determining positivity of the Markov trace of planar algebras in this family.