Related papers: Principal subspaces for the quantum affine vertex …

Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno-Drinfeld Lie algebra t_n. We show that Gaudin subalgebras form a variety isomorphic to the moduli space of stable curves of genus zero…

The standard modules for an affine Lie algebra $\ga$ have natural subquotients called parafermionic spaces -- the underlying spaces for the so-called parafermionic conformal field theories associated with $\ga.$ We study the case $\ga =…

An elementary proof is given for the existence of infinite dimensional abelian subalgebras in quantum W-algebras. In suitable realizations these subalgebras define the conserved charges of various quantum integrable systems. We consider all…

We develop the theory of $q$-characters for quantum affine superalgebras of type $A$ in connection with deformed Cartan matrices. To achieve this, we establish a Khoroshkin-Tolstoy-type multiplicative formula of the universal $R$-matrix of…

We first prove that the K-theoretic Hall algebra of a preprojective algebra of affine type is isomorphic to the positive half of a quantum toroidal quantum group. An essential step consists to deform the K-theoretic Hall algebra so that the…

We discuss quiver gauge models with bi-fundamental and fundamental matter obtained from F-theory compactified on ALE spaces over a four dimensional base space. We focus on the base geometry which consists of intersecting F0=CP1xCP1…

We classify principal bundles over anti-affine schemes with affine and commutative structural group. We show that this yields the classification of quasi-abelian varieties over a field k (i.e., group k-schemes with no non constant global…

Let $C$ be a symmetrizable generalized Cartan Matrix, and $q$ an indeterminate. ${\fg}(C)$ is the Kac-Moody Lie algebra and $U=U_q({\fg}(C))$ the associated quantum enveloping algebra over $ k={\Bbb Q}(q)$. The quantum function algebra…

In this paper we construct combinatorial bases of Feigin-Stoyanovsky's type subspaces of standard modules for level $k$ affine Lie algebra $C_\ell^{(1)}$. We prove spanning by using annihilating field $x_\theta (z)^{k+1}$ of standard…

In this paper, we associate the quantum toroidal algebra $\mathcal{E}_N$ of type $\mathfrak{gl}_N$ with quantum vertex algebra through equivariant $\phi$-coordinated quasi modules. More precisely, for every $\ell\in \mathbb{C}$, by…

We present a generalization the G. Letzter's theory of quantum symmetric pairs of semisimple Lie algebras for the case of quantum affine algebras. We then study solutions of the reflection equation for the quantum affine algebras sl(2) and…

Feigin-Stoyanovsky's type subspace $W(\Lambda)$ of a standard $\tilde{{\mathfrak g}}$-module $L(\Lambda)$ is a $\tilde{{\mathfrak g}}_1$-submodule of $L(\Lambda)$ generated by the highest-weight vector $v_\Lambda$, where $\tilde{{\mathfrak…

In this short note, we compute the classical limits of the quantum toroidal and the affine Yangian algebras of sl(n) by generalizing our arguments for gl(1) from arXiv:1404.5240. These results were mentioned as motivation in the recent…

We construct level one dominant representations of the affine Kac-Moody algebra $\widehat{\mathfrak{gl}}_k$ on the equivariant cohomology groups of moduli spaces of rank one framed sheaves on the orbifold compactification of the minimal…

We introduce quantum super-spherical pairs as coideal subalgebras in general linear and orthosymplectic quantum supergroups. These subalgebras play a role of isotropy subgroups for matrices solving $\mathbb{Z}_2$-graded reflection equation.…

Let $\gtl$ be an affine Lie algebra of type $D_{\ell}^{(1)}$ and $L(\Lambda)$ its standard module with a highest weight vector $v_{\Lambda}$. For a given $\Z$-gradation $\gtl = \gtl_{-1} + \gtl_0 + \gtl_1$, we define Feigin-Stoyanovsky's…

Let Uq(g) be the quantum affine superalgebra associated with an affine Kac-Moody superalgebra g which belongs to the three series osp(1|2n)^(1),sl(1|2n)^(2) and osp(2|2n)^(2). We develop vertex operator constructions for the level 1…

Attention is focused on antisymmetrised versions of quantum spaces that are of particular importance in physics, i.e. Manin plane, q-deformed Euclidean space in three or four dimensions as well as q-deformed Minkowski space. For each of…

We discover a large class of simple affine vertex algebras $V_{k} (\mathfrak g)$, associated to basic Lie superalgebras $\mathfrak g$ at non-admissible collapsing levels $k$, having exactly one irreducible $\mathfrak g$-locally finite…

In this survey, we shall be concerned with the category of finite-dimensional representations of the untwisted quantum affine algebras when the quantum parameter q is not a root of unity. We review the foundational results of the subject,…