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Drinfeld super Yangian of the exceptional Lie superalgebra $D(2,1;λ)$

Hongda Lin, Honglian Zhang

TL;DR

This work constructs the first rigorous Drinfeld presentation for the Drinfeld super Yangian of the exceptional Lie superalgebra $D(2,1;\lambda)$ with trivial central charge. It develops a current-type Drinfeld presentation, derives a PBW basis via degeneration from the quantum loop superalgebra, and proves a Hopf superalgebra structure with an explicit coproduct, counit, and antipode. The paper also connects the Yangian to its quantum-loop origin, analyzes two non-conjugate Dynkin diagrams, and establishes isomorphisms with the orthosymplectic case for special $\lambda$ values, alongside a conjecture relating odd reflections to isomorphisms of associated quantum affine and loop superalgebras. These results lay a foundation for the representation theory of an exceptional Lie superalgebra's Yangian and illuminate links to broader quantum-group settings and physical applications. All constructions use Drinfeld-type currents and degeneration techniques, providing a robust framework for $D(2,1;\lambda)$ in the Yangian context.

Abstract

In this paper, we establish the first rigorous framework for the Drinfeld super Yangian associated with an exceptional Lie superalgebra, which lacks a classical Lie algebraic counterpart. Specifically, we systematically investigate the Drinfeld presentation and structural properties of the super Yangian associated with the exceptional Lie superalgebra $D(2,1;λ)$. First, we introduce a Drinfeld presentation for the super Yangian associated with the exceptional Lie superalgebra $D(2,1;λ)$, explicitly constructing its current generators and defining relations. A key innovation is the construction of a Poincaré-Birkhoff-Witt (PBW) basis using degeneration techniques from the corresponding quantum loop superalgebra. Furthermore, we demonstrate that the super Yangian possesses a Hopf superalgebra structure, explicitly providing the coproduct, counit, and antipode.

Drinfeld super Yangian of the exceptional Lie superalgebra $D(2,1;λ)$

TL;DR

This work constructs the first rigorous Drinfeld presentation for the Drinfeld super Yangian of the exceptional Lie superalgebra with trivial central charge. It develops a current-type Drinfeld presentation, derives a PBW basis via degeneration from the quantum loop superalgebra, and proves a Hopf superalgebra structure with an explicit coproduct, counit, and antipode. The paper also connects the Yangian to its quantum-loop origin, analyzes two non-conjugate Dynkin diagrams, and establishes isomorphisms with the orthosymplectic case for special values, alongside a conjecture relating odd reflections to isomorphisms of associated quantum affine and loop superalgebras. These results lay a foundation for the representation theory of an exceptional Lie superalgebra's Yangian and illuminate links to broader quantum-group settings and physical applications. All constructions use Drinfeld-type currents and degeneration techniques, providing a robust framework for in the Yangian context.

Abstract

In this paper, we establish the first rigorous framework for the Drinfeld super Yangian associated with an exceptional Lie superalgebra, which lacks a classical Lie algebraic counterpart. Specifically, we systematically investigate the Drinfeld presentation and structural properties of the super Yangian associated with the exceptional Lie superalgebra . First, we introduce a Drinfeld presentation for the super Yangian associated with the exceptional Lie superalgebra , explicitly constructing its current generators and defining relations. A key innovation is the construction of a Poincaré-Birkhoff-Witt (PBW) basis using degeneration techniques from the corresponding quantum loop superalgebra. Furthermore, we demonstrate that the super Yangian possesses a Hopf superalgebra structure, explicitly providing the coproduct, counit, and antipode.
Paper Structure (16 sections, 26 theorems, 137 equations, 1 table)

This paper contains 16 sections, 26 theorems, 137 equations, 1 table.

Table of Contents

  1. Introduction
  2. Simple Lie superalgebras $D(2,1;\lambda)$ and its current algebras
  3. Non-conjugacy Dynkin diagrams
  4. Half Casimir element
  5. Affine Lie superalgebras and current Lie superalgebras
  6. Quantum affine(loop) superalgebras of $D(2,1;\lambda)$
  7. Drinfeld-Jimbo presentation $U_q(\widehat{\mathfrak{g}})$
  8. Drinfeld presentation $\mathcal{U}_q(\widehat{\mathfrak{g}})$
  9. Super Yangians of $D(2,1;\lambda)$
  10. Definition of the super Yangians
  11. From quantum loop superalgebras to super Yangians
  12. Isomorphisms of super Yangians
  13. Hopf superalgebra structure of $Y_{\hbar}(\mathfrak{g})$
  14. Minimalistic presentation
  15. Coproduct, counit and antipode
  16. ...and 1 more sections

Key Result

Lemma 2.1

For $\alpha,\beta\in\mathcal{R}^+$ and $z\in\mathfrak{g}_{\beta-\alpha}$, the following equation holds in $\mathfrak{g}\otimes \mathfrak{g}$,

Theorems & Definitions (48)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • ...and 38 more