Related papers: Perfect Crystals and q-deformed Fock Spaces

We propose a new non-perturbative method for studying UV complete unitary quantum field theories (QFTs) with a mass gap in general number of spacetime dimensions. The method relies on unitarity formulated as positive semi-definiteness of…

We give a general construction for finite dimensional representations of $U_q(\hat{\G})$ where $\hat{\G}$ is a non-twisted affine Kac-Moody algebra with no derivation and zero central charge. At $q=1$ this is trivial because…

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce…

This paper proposes a different interpretation on the renormalization program of an interacting Klein-Gordon field in curved space-times. Rather than being just another renormalization program we argue that it that it makes the QFT…

Increasing tensor powers of the $k\times k$ matrices $M_k({\mathbb{C}})$ are known to give rise to a continuous bundle of $C^*$-algebras over $I=\{0\}\cup 1/\mathbb{N}\subset[0,1]$ with fibers $A_{1/N}=M_k({\mathbb{C}})^{\otimes N}$ and…

We construct level-0 modules of the quantum affine algebra $\Uq$, as the $q$-deformed version of the Lie algebra loop module construction. We give necessary and sufficient conditions for the modules to be irreducible. We construct the…

We develop a graded version of the theory of cyclotomic q-Schur algebras, in the spirit of the work of Brundan-Kleshchev on Hecke algebras and of Ariki on q-Schur algebras. As an application, we identify the coefficients of the canonical…

This paper is about establishing a natural connection of quantum affine algebras with quantum vertex algebras. Among the main results, we establish $\hbar$-adic versions of the smash product construction of quantum vertex algebras and their…

We study the structure of finite quandles in terms of subquandles. Every finite quandle $Q$ decomposes in a natural way as a union of disjoint $Q$-complemented subquandles; this decomposition coincides with the usual orbit decomposition of…

We construct the Fock space representations for the quantum affine algebra of type $C_2^{(1)}$ in terms of Young walls. Using this construction, we give a generalized Lascoux-Leclerc-Thibon algorithm for computing the global bases of the…

Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner product space $V$. The reproducing kernel for the Fock space of square integrable holomorphic functions on $V$ relative to the Gaussian measure…

We introduce a notion of formally \'etale $\mathbb{E}_{\infty}$-coalgebras and show that they admit essentially unique, functorial lifts along square zero extensions of $\mathbb{E}_{\infty}$-rings. Using this, we show that for a perfect…

Thinking back the long history of physics, we see that the calculation used by physicists was nothing but the ordinary calculus. Another word, physicists have never wrote theories beyond the basic axioms of the calculus. This is not to…

When the $q$-deformed creation and annihilation operators are used in a second quantization procedure, the algebra satisfied by basis vectors (orthogonal complete set) should be also deformed such as a field operator remains invariant under…

We consider deformations of bounded complexes of modules for a profinite group G over a field of positive characteristic. We prove a finiteness theorem which provides some sufficient conditions for the versal deformation of such a complex…

Let $U_q(\mathfrak{g})$ be the quantum supergroup of $\mathfrak{gl}_{m|n}$ or the modified quantum supergroup of $osp_{m|2n}$ over the field of rational functions in $q$, and let $V_q$ be the natural module for $U_q(\mathfrak{g})$. There…

We give a realization of the quantum affine Lie algebras $\uqa$ and $\uqc$ in terms of anyons defined on a one-dimensional chain (or on a two-dimensional lattice), the deformation parameter $q$ being related to the statistical parameter…

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D4,E6,E7,E8). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to…

We establish a degeneration isomorphism between quantum toroidal algebras and untwisted affine Yangians, valid for all untwisted affine Kac-Moody Lie algebras. Specifically, we prove that the affine Yangian $Y_\hbar(\mathfrak{g})$ is…

A new approach is demonstrated that QFTs can be UV finite if they are viewed as the low energy effective theories of a fundamental underlying theory (that is complete and well-defined in all respects) according to the nowaday's standard…