An Explicit Description of the B(\infty) Crystal For Generalized Quantum Groups of a Family of Comet Quivers

TL;DR

This paper explicitly characterizes the crystal basis () for a family of generalized quantum groups associated with quivers with loops, providing a complete set of relations among Kashiwara operators.

Contribution

It offers a detailed description of the () crystal for generalized quantum groups of quivers with loops, extending previous theoretical frameworks.

Findings

Explicit relations among Kashiwara operators are established.

The characterization applies to a specific family of quivers with multiple loops.

Provides a foundation for further study of generalized crystals in quantum algebra.

Abstract

Tristan Bozec gave a definition of generalized quantum groups that extends the usual definition of quantum groups to finite quivers with loops at vertices, and he introduced a theory of generalized crystals for this new family of Hopf algebras. We explicitly characterize the generalized crystal $\mathcal{B}(\infty)$ associated to a certain family of quivers with multiple loops by providing a complete set of relations among the Kashiwara operators themselves.