# Dynamical centers for the elliptic quantum algebra ${\mathcal{B}}_{q,\lambda}(\hat{gl}_2)_c$

## TL;DR

This paper constructs dynamical central elements in the face-type elliptic quantum algebra ${\

## Contribution

It introduces quadratic trace-like functionals that serve as dynamical centers, revealing deformed centrality properties at specific central charge values.

## Key findings

- Identification of generating functionals satisfying dynamical exchange relations
- Construction of quadratic trace-like objects as dynamical centers
- Establishment of abelian subalgebras at specific central charges

## Abstract

We identify generating functionals that satisfy dynamical exchange relations with the Lax matrices defining the face-type elliptic quantum algebra ${\mathcal{B}}_{q,\lambda}(\hat{gl}_{2})_c$, when the central charge takes the two possible values $c =\pm 2$. These generating functionals are constructed as quadratic trace-like objects in terms of the Lax matrices. The obtained structures are characterized as "dynamical centers", i.e. the centrality property is deformed by dynamical shifts. For these values, the functionals define (genuine) abelian subalgebras of ${\mathcal{B}}_{q,\lambda}(\hat{gl}_{2})_c$.