In this paper, we introduce a motivic version of To¨en’s derived Hall algebra. Then we point out that the two kinds of Hall algebras in the sense of To¨en and Kontsevich–Soibelman, respectively, are Drinfeld dual pairs, not only in the classical case (by counting over finite fields) but also in the motivic version. Consequently they are canonically isomorphic. All proofs, including that for the most important associative property, are deduced in a self-contained way by analyzing the symmetry properties around the octahedral axiom, a method we used previously

Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a general framework. We give a new method to construct graphical quantum error correcting codes on quantized graphs and characterize all optimal ones. We establish a further connection to geometric group theory and construct quantum low-density parity-check stabilizer codes on the Cayley graphs of groups. Their logical qubits can be encoded by the ground states of newly constructed exactly solvable models with translation-invariant local Hamiltonians. Moreover, the Hamiltonian is gapped in the large limit when the underlying group is infinite.

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.

The subject for investigation in this note is concerned with strongly homotopy (SH) Lie algebras, also known as $L_\infty$-algebras, or $L_\infty[1]$-algebras. We extract an intrinsic character, the Atiyah class, when a smaller $L_\infty[1]$-algebra $A$ is extended to a bigger one $L$. In fact, we establish that given such an SH Lie pair $(L,A)$, and any $A$-module $E$, there associates a canonical cohomology class, the Atiyah class $[\alpha^E]$, living in $H^2_{\mathrm{CE}}(A, A^\perp\otimes End(E))$, which generalizes earlier known Atiyah classes out of Lie algebra pairs. In particular, the quotient space $L/A$ is a canonical $A$-module and there associates the Atiyah class $[\alpha^{L/A}]\in H^2_{\mathrm{CE}}(A, Hom(L/A\otimes L/A,L/A))$. The nature of such classes is that they induce on $H^\bullet_{\mathrm{CE}}(A,L/A[-2])$ a Lie algebra structure, and on $H^\bullet_{\mathrm{CE}}(A,E[-2])$ a Lie algebra module structure. An alternative point of view is that the process of taking Atiyah classes defines a functor from the category of $A$-modules to the category of $H^\bullet_{\mathrm{CE}}(A,L/A[-2])$-modules. Moreover, Atiyah classes are invariant under gauge equivalent $A$-compatible infinitesimal deformations of $L$.

Let O_25 be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge 25 whose composition factors are the irreducible quotients of reducible Verma modules. We show that O_25 is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian 3-cocycle twist of the category of finite-dimensional sl_2-modules. We also show that this sl_2-type subcategory is braid-reversed tensor equivalent to a similar category for the Virasoro algebra at central charge 1. As an application, we construct a simple conformal vertex algebra which contains the Virasoro vertex operator algebra of central charge 25 as a PSL_2(C)-orbifold. We also use our results to study Arakawa's chiral universal centralizer algebra of SL_2 at level -1, showing that it has a symmetric tensor category of representations equivalent to Rep PSL_2(C). This algebra is an extension of the tensor product of Virasoro vertex operator algebras of central charges 1 and 25, analogous to the modified regular representations of the Virasoro algebra constructed earlier for generic central charges by I. Frenkel-Styrkas and I. Frenkel-M. Zhu.