Related papers: On quantum vertex algebras and their modules
In this paper, we study in the context of quantum vertex algebras a certain Clifford-like algebra introduced by Jing and Nie. We establish bases of PBW type and classify its $\mathbb N$-graded irreducible modules by using a notion of Verma…
Varchenko's approach to quantum groups, from the theory of arrangements of hyperplanes, can be usefully applied to q-algebras in general, of which quantum groups and quantum (super) Kac-Moody algebras are special cases. New results are…
The connection between q-analogs of special functions and representations of quantum algebras has been developed recently. It has led to advances in the theory of q-special functions that we here review.
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large…
In this article we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang-Baxter and boundary…
We introduce the concept of linear topological modules over vertex algebras and apply it to representations of $\beta-\gamma$ system and affine Kac-Moody algebras.
The notions of vertex Lie algebra and vertex Poisson algebra are presented and connections among vertex Lie algebras, vertex Poisson algebras and vertex algebras are discussed.
By using the ideas of Feigin and Stoyanovsky and Calinescu, Lepowsky and Milas we introduce and study the principal subspaces associated with the Etingof-Kazhdan quantum affine vertex algebra of integer level $k\geqslant 1$ and type…
The K\"ahler-Yang-Mills equations are coupled equations for a K\"ahler metric on a compact complex manifold and a connection on a complex vector bundle over it. After briefly reviewing the main aspects of the geometry of the…
Using the formalism of discrete quantum group gauge theory, one can construct the quantum algebras of observables for the Hamiltonian Chern-Simons model. The resulting moduli algebras provide quantizations of the algebra of functions on the…
We develop the quantum cluster algebra approach recently introduced by Sun and Yagi to investigate the tetrahedron equation, a three-dimensional generalization of the Yang-Baxter equation. In the case of square quiver, we devise a new…
Following the formuation of Borcherds, we develop the theory of (quantum) vertex algebras, including several concrete examples. We also investigate the relationship between the vertex algebra and the chiral algebra due to Beilinson and…
We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, $D^*$, which hold on vertex coalgebras. The…
We establish a relationship between the modern theory of Yangians and the classical construction of the Gelfand-Zetlin bases for the complex Lie algebra $\gn$. Our approach allows us to produce the $q$-analogues of the Gelfand-Zetlin…
Quantum universal enveloping algebras, quantum elliptic algebras and double (deformed) Yangians provide fundamental algebraic structures relevant for many integrable systems. They are described in the FRT formalism by R-matrices which are…
As an analog of the quantum TKK algebra, a twisted quantum toroidal algebra of type A_1 is introduced. Explicit realization of the new quantum TKK algebra is constructed with the help of twisted quantum vertex operators over a Fock space.
We describe a connection between finite--dimensional representations of quantum affine algebras and affine Hecke algebras.
A general theory of the Frolicher-Nijenhuis and Schouten-Nijenhuis brackets in the category of modules over a commutative algebra is described. Some related structures and (co)homology invariants are discussed, as well as applications to…
A concept of quantum triad and its solution is introduced. It represents a common framework for several situations where we have a quantale with a right module and a left module, provided with a bilinear inner product. Examples include Van…
Let $\Uq$ be a quantum group. Regarding a (noncommutative) space with $\Uq$-symmetry as a $\Uq$-module algebra $A$, we may think of equivariant vector bundles on $A$ as projective $A$-modules with compatible $\Uq$-action. We construct an…