Classification of marked elliptic root systems with non-reduced affine quotient

The class of root systems, called elliptic root systems, were introduced in 1985 by K. Saito, for his studies on a normal surface singularity which contains a regular elliptic curve in its minimal resolution. He also classified such root systems when they admit a reduced affine quotient, as root system. In this note, we provide the classification of elliptic root systems that admit a non-reduced affine quotient, thus complete the classification of such root systems.Comment: This is a short annoucement. A detailed (long) version is in preparatio

Notes on highest weight modules of the elliptic algebra Aq,p(sl^2){\cal A}_{q,p}\left(\widehat{sl}_2\right)

We discuss a construction of highest weight modules for the recently defined elliptic algebra ${\cal A}_{q,p}(\widehat{sl}_2)$, and make several conjectures concerning them. The modules are generated by the action of the components of the operator $L$ on the highest weight vectors. We introduce the vertex operators $\Phi$ and $\Psi^*$ through their commutation relations with the $L$-operator. We present ordering rules for the $L$- and $\Phi$-operators and find an upper bound for the number of linearly independent vectors generated by them, which agrees with the known characters of $\widehat{sl}_2$-modules.Comment: Nonstandard macro package eliminate

The elliptic quantum algebra Uq,p(slN^)U_{q,p}(\hat{sl_N}) and its vertex operators

We construct a realization of the elliptic quantum algebra $U_{q,p}(\hat{sl_N})$ for any given level $k$ in terms of free boson fields and their twisted partners. It can be considered as the elliptic deformation of the Wakimoto realization of the quantum affine algebra $U_{q}(\hat{sl_N})$. We also construct a family of screening currents, which commute with the currents of $U_{q,p}(\hat{sl_N})$ up to total q-differences. And we give explicit twisted expressions for the type $I$ and the type $II$ vertex operators of $U_{q,p}(\hat{sl_N})$ by twisting the known results of the type $I$ vertex operators of the quantum affine algebra $U_{q}(\hat{sl_N})$ and the new results of the type $II$ vertex operators of $U_{q}(\hat{sl_N})$ we obtained in this paper.Comment: 28 page

Super Yangian Double DY(gl(1∣1))DY( gl(1|1)) and Its Gauss Decomposition

We extend Yangian double to super (or graded) case and give its Drinfel'd generators realization by Gauss decomposition.Comment: 6 pages, Latex, no figure

The Virasoro vertex algebra and factorization algebras on Riemann surfaces

This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello-Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as are the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta-gamma system using the method of effective BV quantization

Differential-difference system related to toroidal Lie algebra

We present a novel differential-difference system in (2+1)-dimensional space-time (one discrete, two continuum), arisen from the Bogoyavlensky's (2+1)-dimensional KdV hierarchy. Our method is based on the bilinear identity of the hierarchy, which is related to the vertex operator representation of the toroidal Lie algebra \sl_2^{tor}.Comment: 10 pages, 4 figures, pLaTeX2e, uses amsmath, amssymb, amsthm, graphic

On quantum vertex algebras and their modules

We give a survey on the developments in a certain theory of quantum vertex algebras, including a conceptual construction of quantum vertex algebras and their modules and a connection of double Yangians and Zamolodchikov-Faddeev algebras with quantum vertex algebras.Comment: 18 pages; contribution to the proceedings of the conference in honor of Professor Geoffrey Maso