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 $\delta$.

For each simple Lie algebra g, we construct an algebra embedding of the quantum group Uq(g) into certain quantum torus algebra Dg via the positive representations of split real quantum group. The quivers corresponding to Dg is obtained from an amalgamation of two basic quivers, each of which is mutation equivalent to one describing the cluster structure of the moduli space of framed G-local system on a disk with 3 marked points on its boundary when G is of classical type. We derive a factorization of the universal R-matrix into quantum dilogarithms of cluster monomials, and show that conjugation by the R-matrix corresponds to a sequence of quiver mutations which produces the half-Dehn twist rotating one puncture about the other in a twice punctured disk.

Let V⊆A be a conformal inclusion of vertex operator algebras and let C be a category of grading-restricted generalized V-modules that admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. We give conditions under which C inherits semisimplicity from the category of grading-restricted generalized A-modules in C, and vice versa. The most important condition is that A be a rigid V-module in C with non-zero categorical dimension, that is, we assume the index of V as a subalgebra of A is finite and non-zero. As a consequence, we show that if A is strongly rational, then V is also strongly rational under the following conditions: A contains V as a V-module direct summand, V is C_2-cofinite with a rigid tensor category of modules, and A has non-zero categorical dimension as a V-module. These results are vertex operator algebra interpretations of theorems proved for general commutative algebras in braided tensor categories. We also generalize these results to the case that A is a vertex operator superalgebra.

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