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Analogue of Feigin's map on $\imath$quantum group of split type

Ming Lu, Shiquan Ruan, Haicheng Zhang

TL;DR

The paper constructs an $\imath$analogue of Feigin's map for universal $\imath$quantum groups of split type by embedding them into a quantum torus associated with a valued quiver $Q$. It defines an explicit algebra homomorphism $\varphi_{\mathbf{i}}$ mapping generators $B_i$ and $\Bbbk_i$ to concrete expressions in the quantum torus, and proves that this map preserves the $\imath$Serre relations by developing closed formulas for $\imath$divided powers and their expansions, which are then reduced to delicate combinatorial identities. A key simplification comes from first handling the case of quivers without $2$-cycles, followed by a rank-two reduction argument to extend to general cases; the result yields a cluster-realization type embedding of $\widetilde{{\mathbf U}}^\imath$ into a quantum torus and relates to integration/cluster character frameworks. The work also establishes corollaries for the standard $\imath$quantum group and the $q$-Onsager/Affine $\mathfrak{sl}_2$ setting, highlighting the broader applicability of Feigin-type integration maps in the $\imath$quantum context.

Abstract

The (universal) $\imath$quantum groups are as a vast generalization of (Drinfeld double) quantum groups. We establish an algebra homomorphism from universal $\imath$quantum group of split type to a certain quantum torus, which can be viewed as an $\imath$analogue of Feigin's map on the quantum group.

Analogue of Feigin's map on $\imath$quantum group of split type

TL;DR

The paper constructs an analogue of Feigin's map for universal quantum groups of split type by embedding them into a quantum torus associated with a valued quiver . It defines an explicit algebra homomorphism mapping generators and to concrete expressions in the quantum torus, and proves that this map preserves the Serre relations by developing closed formulas for divided powers and their expansions, which are then reduced to delicate combinatorial identities. A key simplification comes from first handling the case of quivers without -cycles, followed by a rank-two reduction argument to extend to general cases; the result yields a cluster-realization type embedding of into a quantum torus and relates to integration/cluster character frameworks. The work also establishes corollaries for the standard quantum group and the -Onsager/Affine setting, highlighting the broader applicability of Feigin-type integration maps in the quantum context.

Abstract

The (universal) quantum groups are as a vast generalization of (Drinfeld double) quantum groups. We establish an algebra homomorphism from universal quantum group of split type to a certain quantum torus, which can be viewed as an analogue of Feigin's map on the quantum group.

Paper Structure

This paper contains 27 sections, 22 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Feigin's map and Hall algebras
  3. Feigin's map and cluster algebras
  4. Quantum symmetric pairs
  5. Goal
  6. Strategy
  7. Organization
  8. Main result
  9. Universal $\imath$quantum groups of split type
  10. Quantum torus
  11. Main result
  12. $\imath$Divided powers in the quantum torus
  13. $\imath$Divided powers in quantum torus
  14. The expansion formulas
  15. $\imath$Serre relations in the quantum torus
  16. ...and 12 more sections

Key Result

Theorem 2.1

Fix $\overline{p}_i\in \mathbb Z/2\mathbb Z$ for each $i\in \mathbb{I}$. The universal $\imath$quantum group $\widetilde{{\mathbf U}}^\imath$ has a presentation with generators $B_i, \Bbbk_i$$(i\in \mathbb{I})$ and the relations for any $i,j\in\mathbb{I}$:

Theorems & Definitions (36)

  • Theorem 2.1: LW20a; see also CLW18
  • Proposition 2.2: CLW21
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 26 more