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A general notion of a quasi-finite algebra is introduced as an algebra graded by the set of all integers equipped with topologies on the homogeneous subspaces satisfying certain properties. An analogue of the regular bimodule is introduced…
Affine quantum groups are certain pseudo-quasitriangular Hopf algebras that arise in mathematical physics in the context of integrable quantum field theory, integrable quantum spin chains, and solvable lattice models. They provide the…
We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra $\hat{\mathfrak{g}}$, from $\mathfrak{g}$-module homomorphisms. When…
We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define…
This is a paper in a series systematically to study toroidal vertex algebras. Previously, a theory of toroidal vertex algebras and modules was developed and toroidal vertex algebras were explicitly associated to toroidal Lie algebras. In…
We consider C-graded vertex algebras, which are vertex algebras V with a C-grading such that V is an admissible V-module generated by 'lowest weight vectors'. We show that such vertex algebras have a 'good' representation theory in the…
We establish rigid tensor category structure on finitely-generated weight modules for the subregular $W$-algebras of $\mathfrak{sl}_n$ at levels $ - n + \frac{n}{n+1}$ (the $\mathcal{B}_{n+1}$-algebras of Creutzig-Ridout-Wood) and at levels…
We incorporate a category of certain modules for an affine Lie algebra, of a certain fixed non-positive-integral level, considered by Kazhdan and Lusztig, into the representation theory of vertex operator algebras, by using the logarithmic…
We represent Feigin's construction [22] of lattice W algebras and give some simple results: lattice Virasoro and $W_3$ algebras. For simplest case $g=sl(2)$ we introduce whole $U_q(sl(2))$ quantum group on this lattice. We find simplest…
We consider boundary scattering for a semi-infinite one-dimensional deformed Hubbard chain with boundary conditions of the same type as for the Y=0 giant graviton in the AdS/CFT correspondence. We show that the recently constructed quantum…
Given a collection of modules of a vertex algebra parametrized by an abelian group, together with one dimensional spaces of composable intertwining operators, we assign a canonical element of the cohomology of an Eilenberg-Mac Lane space.…
We study graded nonlocal $\underline{\mathsf{q}}$-vertex algebras and we prove that they can be generated by certain sets of vertex operators. As an application, we consider the family of graded nonlocal $\underline{\mathsf{q}}$-vertex…
We give a $K$-theoretic realization of all affine Hecke algebras with two unequal parameters including exceptional types. This extends the celebrated work of Kazhdan and Lusztig, who gave a $K$-theoretic realization of affine Hecke algebras…
A theory of quasi modules at infinity for (weak) quantum vertex algebras including vertex algebras was previously developed in \cite{li-infinity}. In this current paper, quasi modules at infinity for vertex algebras are revisited. Among the…
We introduce a cohomology theory of grading-restricted vertex algebras. To construct the {\it correct} cohomologies, we consider linear maps from tensor powers of a grading-restricted vertex algebra to "rational functions valued in the…
In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the lambda-bracket. In Section 2 we construct, in the most general framework, the Zhu…
Within the quantum affine algebra representation theory we construct linear covariant operators that generate the Askey-Wilson algebra. It has the property of a coideal subalgebra, which can be interpreted as the boundary symmetry algebra…
A notion of vertex bialgebra and a notion of module nonlocal vertex algebra for a vertex bialgebra are studied and then a smash product construction of nonlocal vertex algebras is presented. For every nonlocal vertex algebra $V$ satisfying…
We study $\N$-graded $\phi$-coordinated modules for a general quantum vertex algebra $V$ of a certain type in terms of an associative algebra $\widetilde{A}(V)$ introduced by Y.-Z. Huang. Among the main results, we establish a bijection…
We define a natural quantum analogue for the ${\cal Z}$ algebra, and which we refer to as the ${\cal Z}_q$ algebra, by modding out the Heisenberg algebra from the quantum affine algebra $U_q(\hat{sl(2)})$ with level $k$. We discuss the…