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Geometric Approach to I-Quantum Group of Affine Type D

Quanyong Chen, Zhaobing Fan

Abstract

In this paper, we study the structures of Schur algebra and Lusztig algebra associated to partial flag varieties of affine type D. We show that there is a subalgebra of Lusztig algebra and the quantum groups arising from this subalgebras via stabilization procedures is a coideal subalgebra of quantum group of affine $\mathfrak{sl}$ type. We construct monomial and canonical bases of the idempotented quantum algebra and establish the positivity properties of the canonical basis with respect to multiplication and the bilinear pairing.

Geometric Approach to I-Quantum Group of Affine Type D

Abstract

In this paper, we study the structures of Schur algebra and Lusztig algebra associated to partial flag varieties of affine type D. We show that there is a subalgebra of Lusztig algebra and the quantum groups arising from this subalgebras via stabilization procedures is a coideal subalgebra of quantum group of affine type. We construct monomial and canonical bases of the idempotented quantum algebra and establish the positivity properties of the canonical basis with respect to multiplication and the bilinear pairing.
Paper Structure (15 sections, 37 theorems, 160 equations, 1 figure)

This paper contains 15 sections, 37 theorems, 160 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Convolution algebra and Schur Duality
  3. Lattice presentation of affine flag varieties
  4. The double centralizer
  5. Canonical bases of $\mathbf S_{n,d;\mathcal{A}}^\mathfrak d$
  6. Multiplication formulas of Chevalley generators
  7. The leading term
  8. The Lusztig algebra $\mathbf U_{n,d}^\mathfrak d$ and its basis
  9. The Lusztig algebra $\mathbf U_{n,d}^\mathfrak d$
  10. A raw comultiplication
  11. Canonical basis of $\mathbf U_{n,d}^\mathfrak d$
  12. Corresponding Quantum symmetric pair
  13. The algebras $\mathbf U_n^\mathfrak d$ and $\dot {\mathbf U}_n^\mathfrak d$
  14. Bilinear form on $\dot{\mathbf U}_n^\mathfrak d$
  15. The quantum symmetric pair

Key Result

Theorem A

The algebra $\mathbf U_{n,d}^\mathfrak d$ admits a monomial basis $\{\zeta_\mathfrak a \mid \mathfrak a \in \Xi_\mathfrak d^{ap}\}$. Similarly, it possesses a canonical basis $\{\{\mathfrak a\}_d \mid \mathfrak a \in \Xi_\mathfrak d^{ap}\}$, which is compatible with the corresponding bases in $\math

Figures (1)

  • Figure 1: Dynkin diagram of type $\mathrm A^{(1)}_{2r-1}$ with involution of type.

Theorems & Definitions (52)

  • Theorem A: Theorem \ref{['1222']}
  • Theorem B: Proposition \ref{['1400']}, Proposition \ref{['1900']}
  • Theorem C: Theorem \ref{['1500']},Theorem \ref{['1501']}
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4
  • Proposition 2.5
  • Lemma 2.6
  • proof
  • ...and 42 more