ON THE INTEGRAL FORM OF RANK 1 KAC–MOODY ALGEBRAS

Abstract

In this paper we shall prove that the ℤ-subalgebra generated by the divided powers of the Drinfeld generators xr±\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {x}_r^{\\pm } $$\\end{document} (r ∈ ℤ) of the Kac–Moody algebra of type A22\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\ extrm{A}}_2^{(2)} $$\\end{document} is an integral form (strictly smaller than Mitzman’s; see [Mi]) of the enveloping algebra, we shall exhibit a basis generalizing the one provided in [G] for the untwisted affine Kac–Moody algebras and we shall determine explicitly the commutation relations. Moreover, we prove that both in the untwisted and in the twisted case the positive (respectively negative) imaginary part of the integral form is an algebra of polynomials over ℤ.