{'fraction_non_alphanumeric': 0.022108843537414966, 'fraction_numerical': 0.0, 'mean_word_length': 4.4036697247706424, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Homological Expressions of the Tits Form of a Finite Dimensional Algebra

Let k be an algebraically closed field and A be a finite-dimensional associative basic k-algebra of the form A = kQ/I where Q is a quiver without oriented cycles or double arrows and I is an admissible ideal of kQ. Consider the Tits form q A and the corresponding (nonsymmetric) bilinear form In this note we show that for any two finite-dimensional A-modules X, Y the inequality holds. We apply this results to the structure of roots of the Tits form over particular classes of algebras, especially tame algebras.