[Paper Review] Affine Super Yangian

Mamoru Ueda|arXiv (Cornell University)|Nov 15, 2019

Algebraic structures and combinatorial models23 references4 citations

TL;DR

This paper introduces the affine super Yangian $Y_{ ho_1, ho_2}( ext{sl}(m|n))$ with a well-defined coproduct, establishing a new algebraic structure for affine super Yang-Mills theories. It constructs an evaluation homomorphism from this affine super Yangian to the completion of the universal enveloping algebra of $\widehat{\mathfrak{gl}}(m|n)$, providing a bridge between quantum symmetry algebras and affine Lie superalgebras.

ABSTRACT

In this paper, we define the affine super Yangian $Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}}(m|n))$ with a coproduct structure. We also obtain an evaluation homomorphism, that is, an algebra homomorphism from $Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}}(m|n))$ to the completion of the universal enveloping algebra of $\widehat{\mathfrak{gl}}(m|n)$.

Motivation & Objective

• To define a new quantum algebraic structure, the affine super Yangian, for the affine Lie superalgebra $\widehat{\mathfrak{sl}}(m|n)$.
• To equip this algebra with a coproduct structure, enabling its use in representation theory and integrable systems.
• To establish a canonical evaluation homomorphism from the affine super Yangian to the completion of the universal enveloping algebra of $\widehat{\mathfrak{gl}}(m|n)$.
• To generalize the framework of affine Yangians to the superalgebra setting, extending known results in quantum integrability.

Proposed method

• The affine super Yangian $Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}}(m|n))$ is defined via generators and relations, generalizing the affine Yangian construction to superalgebras.
• A coproduct map is explicitly constructed on the affine super Yangian, ensuring compatibility with its algebraic relations.
• The evaluation homomorphism is defined as a map from the affine super Yangian to the completion of $U(\widehat{\mathfrak{gl}}(m|n))$.
• The homomorphism is shown to preserve algebraic relations, establishing a consistent algebraic embedding.
• The construction relies on the use of spectral parameters and grading structures inherent in affine superalgebras.

Experimental results

Research questions

• RQ1 How can the affine Yangian construction be generalized to the setting of Lie superalgebras?
• RQ2 What is the appropriate coproduct structure for the affine super Yangian $Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}}(m|n))$?
• RQ3 Does there exist a well-defined evaluation homomorphism from the affine super Yangian to the universal enveloping algebra of $\widehat{\mathfrak{gl}}(m|n)$?
• RQ4 How does the evaluation map interact with the coproduct and algebraic relations in the affine super Yangian?

Key findings

• The affine super Yangian $Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}}(m|n))$ is successfully defined with a consistent coproduct structure.
• An evaluation homomorphism from $Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}}(m|n))$ to the completion of $U(\widehat{\mathfrak{gl}}(m|n))$ is constructed and verified as an algebra homomorphism.
• The coproduct structure on the affine super Yangian is compatible with the algebraic relations and grading of the underlying superalgebra.
• The evaluation map provides a concrete realization of the affine super Yangian in terms of a larger universal enveloping algebra, enabling representation-theoretic applications.