Related papers: Crystal bases of the Fock space representations an…

We develop a rigorous framework for constructing Fock representations of quantum fields obeying generalized statistics associated with certain solutions of the spectral quantum Yang-Baxter equation. The main features of these…

We study cosets of the type $H_l/U(1)^r$, where $H$ is any Lie algebra at level $l$ and rank $r$. These theories are parafermionic and their characters are related to the string functions, which are generating functions for the…

We give a construction which produces a positive energy representation of the affine Lie algebra of type A_n from the Stokes data of a solution of the tt*-Toda equations of type A_n. The construction appears to play a role in conformal…

We establish an explicit embedding of a quantum affine $\mathfrak{sl}_n$ into a quantum affine $\mathfrak{sl}_{n+1}$. This embedding serves as a common generalization of two natural, but seemingly unrelated, embeddings, one on the quantum…

We study the category $\mathcal{C}$ generated by extremal weight modules over $U_q(\mathfrak{gl}_{>0})$. We show that $\mathcal{C}$ is a tensor category, and give an explicit description of the socle filtration of tensor product of any two…

In this paper, we will fully describe the representations of the crystallographic rank two affine Hecke algebras using elementary methods, for all possible values of q. The focus is on the case when q is a root of unity of small order.

We study generating functions for Lusztig's $t$-analog of weight multiplicities associated to integrable highest weight representations of the simplest affine Lie algebra $A_1^{(1)}$. At $t=1$, these reduce to the {\em string functions} of…

The loop quantum gravity technique is applied to the free bosonic string. A Hilbert space similar to loop space in loop quantum gravity as well as representations of diffeomorphism and hamiltonian constraints on it are constructed. The…

We classify the simple infinite dimensional integrable modules with finite dimensional weight spaces over the quantized enveloping algebra of an untwisted affine algebra. We prove that these are either highest (lowest) weight integrable…

In \cite{W}, there is a graphic description of any irreducible, finite dimensional $\mathfrak{sl}(3)$ module. This construction, called diamond representation is very simple and can be easily extended to the space of irreducible finite…

Let $\tilde{\mathfrak g}$ be an affine Lie algebra of type $A_\ell^{(1)}$. Suppose we're given a $\mathbb Z$-gradation of the corresponding simple finite-dimensional Lie algebra ${\mathfrak g}={\mathfrak g}_{-1}\oplus{\mathfrak g}_0 \oplus…

A weight basis for each finite-dimensional irreducible representation of the orthogonal Lie algebra o(2n) is constructed. The basis vectors are parametrized by the D-type Gelfand--Tsetlin patterns. Explicit formulas for the matrix elements…

We consider a special category of Hopf algebras, depending on parameters $\Sigma$ which possess properties similar to the category of representations of simple Lie group with highest weight $\lambda$. We connect quantum groups to minimal…

The quantum supergroup ${\rm{U}}_q({\mathfrak {osp}}(1|2n))$ admits a finite dimensional spinor representation, which does not have a classical limit. We construct a realisation of this representation on the Fock space of $q$-fermions. We…

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$. We give a polyhedral realization of the crystal basis for the extremal weight module of extremal weight $\lambda$, where $\lambda$ is an integral weight whose Weyl group…

We consider Fock representations of the $Q$-deformed commutation relations $$\partial_s\partial^\dag_t=Q(s,t)\partial_t^\dag\partial_s+\delta(s,t), \quad s,t\in T.$$ Here $T:=\mathbb R^d$ (or more generally $T$ is a locally compact Polish…

Kac and Wakimoto introduced the admissible highest weight representations in order to classify all modular invariant representations of the Kac--Moody algebras. For the Kac--Moody algebra $A_1^{(1)}$ the string functions of admissible…

We present a general vertex operator construction based on the Fock space for an affine Lie algebras of type $A$. This construction allows us to give a unified treatment for both the homogeneous and principle realizations of the affine Lie…

We propose a new approach to studying hyperbolic Kac-Moody algebras, focussing on the rank-3 algebra $\mathfrak{F}$ first investigated by Feingold and Frenkel. Our approach is based on the concrete realization of this Lie algebra in terms…

We review recent advancements in understanding nucleon structure within the Basis Light-Front Quantization (BLFQ) framework--a fully relativistic, nonperturbative approach to solving quantum field theories. In its initial phase, we start…