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iQuantum groups and iHopf algebras I: foundation

Jiayi Chen, Ming Lu, Xiaolong Pan, Shiquan Ruan, Weiqiang Wang

TL;DR

This work develops iHopf algebras as a unifying framework for iquantum groups, defining $H^\imath$ from a Hopf algebra $H$ with Hopf pairing $\varphi$ and showing how it realizes quasi-split universal iquantum groups, with the Drinfeld double emerging from a diagonal-type iHopf construction. It articulates precise connections between Lusztig’s braid group action on quantum groups and the relative ibraid actions on iquantum groups, grounding these symmetries in the iHopf formalism and enabling a structural approach to coideal subalgebras via twists and Satake diagrams. The paper then demonstrates that universal iquantum groups can be realized as iHopf algebras on Borel subalgebras, establishing isomorphisms between $ig(\widehat{\mathbf B}^\imath_\tau\big)$ and $\widehat{\mathbf U}^\imath$, and between $\big(\widetilde{\mathbf B}^\imath_\tau\big)$ and $\widetilde{\mathbf U}^\imath$ within a commuting diagram framework, with integral forms. Finally, it develops recursive root-vector constructions and shows, through a detailed case analysis of $c_{i,\tau i}=2,0,-1$, that iHopf-based ibraid actions reproduce Lusztig braid symmetries on $\vartheta_j$, linking two braid-action frameworks at the level of root data and setting the stage for dual canonical bases in future work.

Abstract

We introduce the notion of iHopf algebra, a new associative algebra structure defined on a Hopf algebra equipped with a Hopf pairing. The iHopf algebra on a Borel quantum group endowed with a $τ$-twisted Hopf pairing is shown to be a quasi-split universal iquantum group. In particular, the Drinfeld double quantum group is realized as the iHopf algebra on the double Borel. This iHopf approach allows us to develop connections between Lusztig's braid group action and ibraid group action. It will further lead to the construction of dual canonical basis in a sequel.

iQuantum groups and iHopf algebras I: foundation

TL;DR

This work develops iHopf algebras as a unifying framework for iquantum groups, defining from a Hopf algebra with Hopf pairing and showing how it realizes quasi-split universal iquantum groups, with the Drinfeld double emerging from a diagonal-type iHopf construction. It articulates precise connections between Lusztig’s braid group action on quantum groups and the relative ibraid actions on iquantum groups, grounding these symmetries in the iHopf formalism and enabling a structural approach to coideal subalgebras via twists and Satake diagrams. The paper then demonstrates that universal iquantum groups can be realized as iHopf algebras on Borel subalgebras, establishing isomorphisms between and , and between and within a commuting diagram framework, with integral forms. Finally, it develops recursive root-vector constructions and shows, through a detailed case analysis of , that iHopf-based ibraid actions reproduce Lusztig braid symmetries on , linking two braid-action frameworks at the level of root data and setting the stage for dual canonical bases in future work.

Abstract

We introduce the notion of iHopf algebra, a new associative algebra structure defined on a Hopf algebra equipped with a Hopf pairing. The iHopf algebra on a Borel quantum group endowed with a -twisted Hopf pairing is shown to be a quasi-split universal iquantum group. In particular, the Drinfeld double quantum group is realized as the iHopf algebra on the double Borel. This iHopf approach allows us to develop connections between Lusztig's braid group action and ibraid group action. It will further lead to the construction of dual canonical basis in a sequel.

Paper Structure

This paper contains 18 sections, 44 theorems, 185 equations, 1 table.

Table of Contents

  1. Introduction
  2. iHopf algebras
  3. Definition
  4. iHopf algebras of diagonal type
  5. Connection to Drinfeld doubles
  6. A coideal subalgebra via twists
  7. Quantum groups and $\mathrm{i}$quantum groups
  8. Quantum groups
  9. iQuantum groups
  10. Relative braid group symmetries
  11. $\mathrm{i}$Quantum groups as $\mathrm{i}$Hopf algebras on the Borel
  12. iHopf algebra defined on $\tB$
  13. A recursive formula
  14. $\mathrm{i}$Braid group symmetries on $\widetilde{{\mathbf U}}^\imath$ via $\mathrm{i}$Hopf algebra
  15. Connecting 2 braid group actions via iHopf
  16. ...and 3 more sections

Key Result

Theorem A

There exist natural algebra homomorphisms $\xi_{\tau,\chi}: H^\imath_\tau\rightarrow (H\otimes H)^\imath$ and $\Psi: H_\tau^\imath\rightarrow H^\imath_\tau\otimes (H\otimes H)^\imath$.

Theorems & Definitions (86)

  • Theorem A: Propositions \ref{['prop:iHopf']}, \ref{['prop:embedding-iHopf-diagonal']} and \ref{['prop:coideal']}
  • Theorem B: Lemmas \ref{['lem:U=iHopfBB']} and \ref{['lem:iHembed']}, Theorem \ref{['thm:iH=iQG']}
  • Theorem C: Theorems \ref{['thm:braidiHopf']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • ...and 76 more