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The restricted quantum double of the Yangian

Curtis Wendlandt

Abstract

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with associated Yangian $Y_\hbar\mathfrak{g}$. In the mid-1990s, Khoroshkin and Tolstoy formulated a conjecture which asserts that the algebra $\mathrm{D}Y_\hbar\mathfrak{g}$ obtained by doubling the generators of $Y_\hbar\mathfrak{g}$, called the Yangian double, provides a realization of the quantum double of the Yangian. We provide a uniform proof of this conjecture over $\mathbb{C}[\![\hbar]\!]$ which is compatible with the theory of quantized enveloping algebras. As a byproduct, we identify the universal $R$-matrix of the Yangian with the canonical element defined by the pairing between the Yangian and its restricted dual.

The restricted quantum double of the Yangian

Abstract

Let be a complex semisimple Lie algebra with associated Yangian . In the mid-1990s, Khoroshkin and Tolstoy formulated a conjecture which asserts that the algebra obtained by doubling the generators of , called the Yangian double, provides a realization of the quantum double of the Yangian. We provide a uniform proof of this conjecture over which is compatible with the theory of quantized enveloping algebras. As a byproduct, we identify the universal -matrix of the Yangian with the canonical element defined by the pairing between the Yangian and its restricted dual.
Paper Structure (50 sections, 45 theorems, 379 equations)

This paper contains 50 sections, 45 theorems, 379 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Motivation
  4. Remarks
  5. Outline
  6. Acknowledgments
  7. Homogeneous quantizations
  8. Topological modules
  9. Graded topological modules
  10. The restricted dual
  11. Homogeneous quantizations
  12. The Yangian Manin triple
  13. The Yangian revisited
  14. The Lie algebra g
  15. The Yangian
  16. ...and 35 more sections

Key Result

Theorem I

There is a unique $\mathbb{Z}$-graded Hopf algebra structure on $\mathrm{D}Y_\hbar\mathfrak{g}$ over $\mathbb{C}[\![\hbar]\!]$ such that the formal shift operator intertwines the Hopf structures on $\mathrm{D}Y_\hbar\mathfrak{g}$ and $Y_\hbar\mathfrak{g}$. Moreover:

Theorems & Definitions (108)

  • Theorem I
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 98 more