• DOI:10.1007/s00029-021-00686-7
• Corpus ID: 209405145

Remarks on the derived center of small quantum groups

@article{Lachowska2019RemarksOT,
  title={Remarks on the derived center of small quantum groups},
  author={Anna Lachowska and You Qi},
  journal={Selecta Mathematica},
  year={2019},
  volume={27}
}

• A. Lachowska, You Qi
• Published 18 December 2019
• Mathematics
• Selecta Mathematica

Let uq(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {u}}_q(\mathfrak {g})$$\end{document} be the small quantum group associated with a complex semisimple Lie algebra g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {g}$$\end{document} and a primitive root of unity q, satisfying certain restrictions. We establish the equivalence between three different actions of g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {g}$$\end{document} on the center of uq(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {u}}_q(\mathfrak {g})$$\end{document} and on the higher derived center of uq(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {u}}_q(\mathfrak {g})$$\end{document}. Based on the triviality of this action for g=sl2,sl3,sl4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {g}= \mathfrak {sl}_2, \mathfrak {sl}_3, \mathfrak {sl}_4$$\end{document}, we conjecture that, in finite type A, central elements of the small quantum group uq(sln)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {u}}_q(\mathfrak {sl}_n)$$\end{document} arise as the restriction of central elements in the big quantum group Uq(sln)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {U}}_q(\mathfrak {sl}_n)$$\end{document}. We also study the role of an ideal zHig\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {z}}_\mathrm{Hig}$$\end{document} known as the Higman ideal in the center of uq(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {u}}_q(\mathfrak {g})$$\end{document}. We show that it coincides with the intersection of the Harish-Chandra center and its Fourier transform, and compute the dimension of zHig\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {z}}_\mathrm{Hig}$$\end{document} in type A. As an illustration we provide a detailed explicit description of the derived center of uq(sl2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {u}}_q(\mathfrak {sl}_2)$$\end{document} and its various symmetries.