Publication

Fusion and Specialization for Type ADE Shuffle Algebras

2025

Abstract

Root vectors in quantum groups (of finite type) generalize to fused currents in quantum loop groups [J. Ding, S. Khoroshkin, Transform. Groups , 5 , No. 1, 35–59 (2000)]. In the present paper, we construct fused currents as duals to specialization maps of the corresponding shuffle algebras [B. Enriquez, Transform. Groups , 5 , No. 2, 111–120 (2000), B. Enriquez, J. Lie Theory , 13 , No. 1, 21–64 (2003), and B. Feigin, A. Odesskii, NATO Sci., Ser. II, Math. Phys. Chem. , 35 (2001)] in types ADE; an approach, which has a potential for generalization to arbitrary Kac–Moody types. Both root vectors and fused currents depend on a convex order of the positive roots and the choice we make in the present paper is that of the Auslander–Reiten order [C. Ringel, J. reine und angew. Math. , 470 , 51–88 (1996)] corresponding to the orientation of the ADE-type Dynkin diagram.

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2025

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/25293d06-786e-4ab5-9f0c-94c6ed07f651

About this result

Related concepts (32)

Group of Lie type

In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted precise definition, but the important collection of finite simple groups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simple groups.

Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied.

Lie theory

In mathematics, the mathematician Sophus Lie (liː ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan. The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence.

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