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Orthosymplectic Yangians

Rouven Frassek, Alexander Tsymbaliuk

TL;DR

The study provides a comprehensive Drinfeld realization for RTT orthosymplectic Yangians associated with arbitrary parity sequences, unifying non-super BCD and super A-type cases. It leverages Gauss decomposition of the RTT generator, rank-reduction embeddings, and explicit rank-1 and rank-2 analyses to derive currents, central elements, and higher order Serre relations, proving PBW-type results. By establishing isomorphisms between RTT and Drinfeld pictures across all parity configurations, the work advances the representation theory and integrable systems for Lie superalgebras in the orthosymplectic family. The framework has potential applications in superalgebra representations and quantum integrable models where parity choices play a crucial role.

Abstract

We study the RTT orthosymplectic super Yangians and present their Drinfeld realizations for any parity sequence, generalizing the results for non-super types BCD, a standard parity sequence, and super A-type.

Orthosymplectic Yangians

TL;DR

The study provides a comprehensive Drinfeld realization for RTT orthosymplectic Yangians associated with arbitrary parity sequences, unifying non-super BCD and super A-type cases. It leverages Gauss decomposition of the RTT generator, rank-reduction embeddings, and explicit rank-1 and rank-2 analyses to derive currents, central elements, and higher order Serre relations, proving PBW-type results. By establishing isomorphisms between RTT and Drinfeld pictures across all parity configurations, the work advances the representation theory and integrable systems for Lie superalgebras in the orthosymplectic family. The framework has potential applications in superalgebra representations and quantum integrable models where parity choices play a crucial role.

Abstract

We study the RTT orthosymplectic super Yangians and present their Drinfeld realizations for any parity sequence, generalizing the results for non-super types BCD, a standard parity sequence, and super A-type.
Paper Structure (39 sections, 49 theorems, 512 equations)

This paper contains 39 sections, 49 theorems, 512 equations.

Table of Contents

  1. Introduction
  2. Summary
  3. Outline
  4. Acknowledgement
  5. Orthosymplectic Lie superalgebras
  6. Setup and notations
  7. Orthosymplectic Lie superalgebras
  8. Dynkin diagrams with labels via parity sequences
  9. Chevalley-Serre type presentation
  10. RTT orthosymplectic Yangians
  11. RTT extended orthosymplectic super Yangian
  12. RTT orthosymplectic super Yangian
  13. Relation to Lie superalgebras and PBW theorem
  14. Gauss decomposition and rank reduction
  15. Useful lemma
  16. ...and 24 more sections

Key Result

Theorem 2.21

z The Lie superalgebra $\mathfrak{osp}(V)$ is generated by $\{e_i,f_i,h_i\}_{i=1}^{{\mathsf{r}}}$, with the $\mathbb{Z}_2$-grading subject to the quadratic Chevalley relations the standard Serre relations and the higher order Serre relations (eq:Lie-Serre-superA, eq:Lie-Serre-new3, eq:Lie-Serre-new6, eq:Lie-Serre-new7) that are described in details below.

Theorems & Definitions (115)

  • Remark 2.13
  • Remark 2.19
  • Theorem 2.21
  • Remark 2.28
  • Remark 2.29
  • Remark 2.36
  • Remark 2.37
  • Remark 3.8
  • Remark 3.10
  • Lemma 3.12
  • ...and 105 more