{'fraction_non_alphanumeric': 0.05232558139534884, 'fraction_numerical': 0.007751937984496124, 'mean_word_length': 3.575221238938053, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 1, '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}}
Category O and its basic properties

1 Definitions Let g denote a semisimple Lie algebra over C with fixed Cartan and Borel subalgebras h ⊂ b ⊂ g. Define n = ⊕ α>0 gα, n− = ⊕ α<0 gα. As we know b = n ⊕ h. Also there is a partial ordering on the set of weights defined as follows: λ ≥ μ ⇔ λ − μ is equal to a Z-linear combination of positive roots. We start with the definition of BGG category O: Definition 1. O is the full subcategory in the category U(g)-mod such that its objects satisfy the following properties: