Tame symmetric algebras of period four

Abstract

In this paper, we are concerned with the structure of tame symmetric algebras Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} of period four (TSP4 algebras for short). For a tame algebra, the number of arrows starting or ending at a given vertex cannot be large. Here we will mostly focus on the case when the Gabriel quiver of Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} is biserial, that is, there are at most two arrows ending and at most two arrows starting at each vertex. We present a range of properties (with relatively short proofs) which must hold for the Gabriel quiver of such an algebra. In particular, we show that triangles (and squares) appear naturally, so as for weighted surface algebras (Erdmann and Skowroński in J Algebra 505:490–558, 2018, J Algebra 544:170–227, 2020, J Algebra 569:875–889, 2021). Furthermore, we prove results on the minimal relations defining the ideal I for an admissible presentation of Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} in the form KQ/I. This will be the input for the classification of all TSP4 algebras with biserial Gabriel quiver.