Classification on irreducible Whittaker modules over quantum group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${U_q}\left( {{\mathfrak{s}\mathfrak{l}_3},\,{\rm{\Lambda }}} \right)$$\end{document}

We define the Whittaker modules over the simply-connected quantum group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${U_q}\left( {{\mathfrak{s}\mathfrak{l}_3},\,{\rm{\Lambda }}} \right)$$\end{document}, where Λ is the weight lattice of Lie algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{s}\mathfrak{l}_3}$$\end{document}. Then we completely classify all those simple ones. Explicitly, a simple Whittaker module over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${U_q}\left( {{\mathfrak{s}\mathfrak{l}_3},\,{\rm{\Lambda }}} \right)$$\end{document} is either a highest weight module, or determined by two parameters z ∈ ℂ and γ ∈ ℂ* (up to a Hopf automorphism).