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On the theory of $q$-characters for quantum affine superalgebras of type $A$

Sin-Myung Lee

Abstract

We develop the theory of $q$-characters for quantum affine superalgebras of type $A$ in connection with deformed Cartan matrices. To achieve this, we establish a Khoroshkin-Tolstoy-type multiplicative formula of the universal $R$-matrix of the associated generalized quantum group, from which one can read off a 2-parameter deformation of Cartan matrices of super type $A$. We also propose a Frenkel-Mukhin-type algorithm for $q$-characters of finite-dimensional simple modules with integral highest $\ell$-weights.

On the theory of $q$-characters for quantum affine superalgebras of type $A$

Abstract

We develop the theory of -characters for quantum affine superalgebras of type in connection with deformed Cartan matrices. To achieve this, we establish a Khoroshkin-Tolstoy-type multiplicative formula of the universal -matrix of the associated generalized quantum group, from which one can read off a 2-parameter deformation of Cartan matrices of super type . We also propose a Frenkel-Mukhin-type algorithm for -characters of finite-dimensional simple modules with integral highest -weights.

Paper Structure

This paper contains 33 sections, 77 theorems, 273 equations.

Table of Contents

  1. Introduction
  2. Quantum affine superalgebras and generalized quantum groups
  3. Main results
  4. Braid action and a multiplicative formula for the universal $R$-matrix
  5. The $q$-character map and its transfer matrix construction
  6. Applications
  7. Deformed Cartan matrices for $\mathfrak{sl}_{M|N}$
  8. Organization
  9. Generalized quantum groups of type $A$
  10. Generalized quantum groups and a braid action
  11. Lusztig's algebra $\mathbf{f}(\epsilon)$ and a bilinear form
  12. Drinfeld-type presentation of $\mathcal{U}(\epsilon)$
  13. A multiplicative formula for the universal $R$-matrix
  14. Hopf pairing and universal $R$-matrix
  15. Affine root vectors, PBW basis and Levendorskii--Soibelman formula
  16. ...and 18 more sections

Key Result

Theorem 1

If $\epsilon$ is such that $(\epsilon_{i}+\epsilon_{i+1})(\epsilon_{i+1}+\epsilon_{i+2})=0$ for $1\leq i\leq n-2$, then the universal $R$-matrix $\mathcal{R}$ for $\mathcal{U}(\epsilon)$ has the following decomposition: where $\widetilde{C}^{r}=(\widetilde{C}_{ij}^{r})_{i,j}$ is the inverse matrix of

Theorems & Definitions (135)

  • Theorem
  • Definition 2.1: KOSMachida
  • Remark 2.2
  • Theorem 2.3: MachidaYu
  • Remark 2.4
  • Lemma 2.5
  • Proposition 2.6: cf. L
  • proof
  • Proposition 2.7: cf. Lb
  • Proposition 2.8: KL
  • ...and 125 more