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Coproducts for affine super-Yangian and Weyl groupoid action

Vasiliy Volkov, Vladimir Stukopin

Abstract

For affine special linear superalgebra $\widehat{sl}(m|n, Π)$ defined by an arbitrary system of simple roots $Π$ we define the affine super Yangian $Y_{\hbar}(\widehat{sl}(m|n, Π))$ as Hopf superalgebra which is a quantization of superbialgebra $\widehat{sl}(m|n, Π)[t]$ and describe super Yangian in terms of minimalistic system of generators. We consider Drinfeld presentation for $Y^D_{\hbar}(\widehat{sl}(m|n, Π))$ and prove that these two presentations are isomorphic as associative superalgebras. We induce by means of this isomorphism a co-multiplication on the Drinfeld presentation $Y^D_{\hbar}(\widehat{sl}(m|n, Π))$ of the super Yangian. We introduce the action of Weyl groupoid by isomorphisms on super Yangians as an extension of its action on universal enveloping algebra and deformation of action on univesal enveloping superalgebra of current Lie superalgebra and prove that such extension exists and unique. As a consequence of this construction we obtain that super Yangians $Y_{\hbar}(\widehat{sl}(m|n, Π_1))$ and $Y_{\hbar}(\widehat{sl}(m|n, Π_2))$, defined by different simple root systems $ Π_1$ and $ Π_2$ are isomorphic as Hopf superalgebras.

Coproducts for affine super-Yangian and Weyl groupoid action

Abstract

For affine special linear superalgebra defined by an arbitrary system of simple roots we define the affine super Yangian as Hopf superalgebra which is a quantization of superbialgebra and describe super Yangian in terms of minimalistic system of generators. We consider Drinfeld presentation for and prove that these two presentations are isomorphic as associative superalgebras. We induce by means of this isomorphism a co-multiplication on the Drinfeld presentation of the super Yangian. We introduce the action of Weyl groupoid by isomorphisms on super Yangians as an extension of its action on universal enveloping algebra and deformation of action on univesal enveloping superalgebra of current Lie superalgebra and prove that such extension exists and unique. As a consequence of this construction we obtain that super Yangians and , defined by different simple root systems and are isomorphic as Hopf superalgebras.

Paper Structure

This paper contains 22 sections, 27 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Definitions
  4. Classical Lie superalgebras
  5. Special linear superalgebra
  6. Affine Kac-Moody superalgebras
  7. Current Lie superalgebras and superbialgebras
  8. Affine super Yangian
  9. Weyl groupoid
  10. Different coproducts for the affine super Yangian
  11. Coproducts for affine super Yangians defined different simple root systems
  12. Description of Hopf superalgebra structures and Weyl groupoid
  13. Drinfeld presentation of affine super Yangian
  14. Coproducts on Drinfeld affine super Yangian
  15. Proofs of the main results
  16. ...and 7 more sections

Key Result

Proposition 2.1

If $\mathfrak{g}$ is a classical simple Lie superalgebra and $\alpha,\beta,\alpha+\beta$ are roots of $\mathfrak{g}$, then $[\mathfrak{g}^{\alpha},\mathfrak{g}^{\beta}]=\mathfrak{g}^{\alpha+\beta}$.

Theorems & Definitions (39)

  • Proposition 2.1
  • Definition 2.1
  • Definition 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 5.3
  • Definition 6.1
  • Theorem 6.1
  • ...and 29 more