Serre Relation and Higher Grade Generators of the AdS/CFT Yangian Symmetry

TL;DR

This paper proves that the evaluation representation in the AdS/CFT Yangian symmetry model satisfies the Serre relation, introduces an alternative construction for higher grade generators, and simplifies the proof of the algebraic structure.

Contribution

It provides a complete proof that the evaluation representation satisfies the Serre relation and proposes a new construction for higher grade generators in the Yangian algebra.

Findings

Evaluation representation satisfies the Serre relation.

Alternative construction for higher grade generators is proposed.

Proof extends to the exceptional superalgebra.

Abstract

It was shown that the spin chain model coming from AdS/CFT correspondence satisfies the Yangian symmetry if we assume evaluation representation, though so far there is no explicit proof that the evaluation representation satisfies the Serre relation, which is one of the defining equations of the Yangian algebra imposing constraints on the whole algebraic structure. We prove completely that the evaluation representation adopted in the model satisfies the Serre relation by introducing a three-dimensional gamma matrix. After studying the Serre relation, we proceed to the whole Yangian algebraic structure, where we find that the conventional construction of higher grade generators is singular and we propose an alternative construction. In the discussion of the higher grade generators, a great simplification for the proof of the Serre relation is found. Using this expression, we further show that the proof is lifted to the exceptional superalgebra, which is a non-degenerate deformation of the original superalgebra.