Correction to: Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras

  • Seok-Jin Kang
  • Masaki Kashiwara
  • Myungho KimEmail author

Mathematics Subject Classification

81R50 16G 16T25 17B37 

1 Correction to: Invent. math. (2018) 211:591–685

We will freely make use of the notations from [1]. Let \(\mathcal T\) be a tensor category, and let \(\{P_i\}_{i\in I}\) be a family of objects in \(\mathcal T\). In the beginning of Appendix A.6, we say that \(\{(P_i,R_{P_i})_{i\in I}\}\) is a commuting family of central objects if it satisfies two conditions
  1. (a)
    for any \(i\in I\), \(R_{P_i}\) is an isomorphism functorial in \(X\in {\mathcal T}\) such that
  2. (b)

    \(R_{P_j}(P_i)\circ R_{P_i}(P_j)={\text {id}}_{P_i\mathop \otimes P_j}\) for any \(i,j\in I\) such that \(i \ne j\).

If it further satisfies the condition
$$\begin{aligned} R_{P_i}(P_i)={\text {id}}_{P_i\mathop \otimes P_i}\hbox { for any }i\in I, \end{aligned}$$
then it was called a strictly commuting family of central objects.
In the next pages, we asserted that if \(\{(P_i,R_{P_i})_{i\in I}\}\) is a commuting family of central objects, then the localization \(\widetilde{\mathcal T} = {\mathcal T}[P_i^{\mathop \otimes -1} \mid i \in I] \) of \(\mathcal T\) becomes a tensor category with a unit object \((\mathbf{{1}}, \vec {0})\). This is incorrect since the following gives a counterexample: Let \(\mathcal T\) be the category of vector spaces over a field \(\mathbf {k}\) and let \(P=\mathbf {k}^{\oplus 2}\). Let \(R_P(X) : P\mathop \otimes X \rightarrow X\mathop \otimes P\) be the map given by \(v \mathop \otimes x \mapsto x \mathop \otimes v\) for \(v \in P\) and \(x \in X\). Then \(\{(P, R_P)\}\) satisfy the conditions (a) and (b). Since the endomorphism ring
$$\begin{aligned} {\text {End}}_{\widetilde{\mathcal T}}\bigl ((\mathbf{{1}}, 0)\bigr )&=\varinjlim \limits _{\gamma \in {\mathbb Z}_{\ge 0}} {\text {End}}_{\mathcal T}(\mathbf{{1}}\mathop \otimes P^{\gamma })\\&\simeq \varinjlim \limits _{\gamma \in {\mathbb Z}_{\ge 0}} {\text {End}}_{\mathbf {k}}(P^{\gamma }) \simeq \varinjlim \limits _{\gamma \in {\mathbb Z}_{\ge 0}} \mathrm{Mat}_{2^\gamma \times 2^\gamma }(\mathbf {k}) \end{aligned}$$
is not commutative, the object \((\mathbf{{1}}, 0)\) cannot be a unit object of a tensor category ([2, Exercise 4.11 (ii)]).

The problem occurs since we do not assume that the family satisfy condition (A.4). As the simplest remedy, we include condition (A.4) in the definition of commuting family of central objects from the beginning and remove the definition of strictly commuting family of central objects in the paper. Under the new definition of commuting family of central objects, we can construct the category \({\mathcal T}[P_i^{\mathop \otimes -1} \mid i \in I]\) as well as the category \({\mathcal T}[P_i\simeq \mathbf{{1}}\mid i\in I]\) in the graded case (Appendix A.7). All the arguments in the main body of our paper also work without any problem.



  1. 1.
    Kang, S.-J., Kashiwara, M., Kim, M.: Symmetric quiver Hecke algebras and \(R\)-matrices of quantum affine algebras. Invent. Math. 211, 591–685 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kashiwara, M., Schapira, P.: Categories and Sheaves, Grundlehren der mathematischen Wissenschaften 332. Springer, Berlin (2006)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUnited Arab Emirates UniversityAl Ain, Abu DhabiUnited Arab Emirates
  2. 2.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  3. 3.Korea Institute for Advanced StudySeoulKorea
  4. 4.Department of MathematicsKyung Hee UniversitySeoulKorea

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