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Shifted Yangians and polynomial R-matrices

David Hernandez, Huafeng Zhang

Abstract

We study the category O of representations over a shifted Yangian. This category has a tensor product structure and contains distinguished modules, the positive prefundamental modules and the negative prefundamental modules. Motivated by the representation theory of the Borel subalgebra of a quantum affine algebra and by the relevance of quantum integrable systems in this context, we prove that tensor products of prefundamental modules with irreducible modules are either cyclic or co-cyclic. This implies the existence and uniqueness of morphisms, the R-matrices, for such tensor products. We prove the R-matrices are polynomial in the spectral parameter, and we establish functional relations for the R-matrices. As applications, we prove the Jordan--Hölder property in the category O. We also obtain a proof, uniform for any finite type, that any irreducible module factorizes through a truncated shifted Yangian.

Shifted Yangians and polynomial R-matrices

Abstract

We study the category O of representations over a shifted Yangian. This category has a tensor product structure and contains distinguished modules, the positive prefundamental modules and the negative prefundamental modules. Motivated by the representation theory of the Borel subalgebra of a quantum affine algebra and by the relevance of quantum integrable systems in this context, we prove that tensor products of prefundamental modules with irreducible modules are either cyclic or co-cyclic. This implies the existence and uniqueness of morphisms, the R-matrices, for such tensor products. We prove the R-matrices are polynomial in the spectral parameter, and we establish functional relations for the R-matrices. As applications, we prove the Jordan--Hölder property in the category O. We also obtain a proof, uniform for any finite type, that any irreducible module factorizes through a truncated shifted Yangian.

Paper Structure

This paper contains 23 sections, 39 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Shifted Yangians
  3. Definition and structure
  4. The ordinary Yangian
  5. First coproduct estimation
  6. Truncated shifted Yangians
  7. Representations of shifted Yangians
  8. Verma modules
  9. Finite-dimensional irreducible modules
  10. Category $\mathcal{O}^{sh}$ and rationality
  11. Examples in the $sl_2$-case
  12. Tensor products of prefundamental modules
  13. One-dimensional modules
  14. Cyclicity and cocyclicity
  15. Weyl modules and standard modules
  16. ...and 8 more sections

Key Result

Theorem 2.2

coproduct All shift homomorphisms are injective. The multiplication map $Y_{\mu}^<(\mathfrak{g}) \otimes Y_{\mu}^=(\mathfrak{g}) \otimes Y_{\mu}^>(\mathfrak{g}) \longrightarrow Y_{\mu}(\mathfrak{g})$ is an isomorphism of vector spaces. $Y_{\mu}^{\pm}(\mathfrak{g})$ is the algebra generated by $x_{i

Theorems & Definitions (105)

  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • ...and 95 more