Braided Yangians

TL;DR

This paper introduces two new classes of Yangian-like algebras associated with current R-matrices, explores their properties, and constructs quantum determinants, revealing their centrality and algebraic identities.

Contribution

It defines braided Yangians and Yangians of RTT type, studies their structures, constructs evaluation morphisms, and introduces quantum determinants and identities.

Findings

Quantum determinants are always central in braided Yangians.

In RTT-type Yangians, quantum determinants are not always central.

Evaluation morphisms and bi-algebra structures are explicitly constructed.

Abstract

Yangian-like algebras, associated with current R-matrices, different from the Yang ones, are introduced. These algebras are of two types. The so-called braided Yangians are close to the Reflection Equation algebras, arising from involutive or Hecke symmetries. The Yangians of RTT type are close to the corresponding RTT algebras. Some properties of these two classes of the Yangians are studied. Thus, evaluation morphisms for them are constructed, their bi-algebra structures are described, and quantum analogs of certain symmetric polynomials, in particular, quantum determinants, are introduced. It is shown that in any braided Yangian this determinant is always central, whereas in the Yangians of RTT type it is not in general so. Analogs of the Cayley-Hamilton-Newton identity in the braided Yangians are exhibited. A bozonization of the braided Yangians is performed.