Artin-Schelter Regular Algebras Based On Combinatorial Methods

Artin-Schelter regular algebras may be thought of as homogeneous coordinate rings of quantum P". They were introduced by Artin and Schelter in1987. After then, searching and classifying Artin-Schelter regular algebras become a major project in the realm of noncommutative projective geometry. This thesis contributes to this project by using some combinatorial theories and methods. It mainly consists of the following two parts.In the part I, we consider the construction problem of high dimensional Artin-Schelter regular algebras. For this, we introduce graded algebras that have Lyndon presentations. The class of such algebras includes universal enveloping algebras of graded Lie algebras. Firstly, with the help of the combinatorial features of Lyndon words, by depicting the graph of chains on an antichain of Lyndon words, we characterize some elementary invariants of such algebras in terms of closed sets of Lyndon words. Then, through a detailed analysis of a variation of the Lie commutators on free algebras, we obtain an Artin-Schelter regularity criterion for such algebras. The discussion that taken for the criterion also indicates how tc construct Artin-Schelter regular algebras via closed sets of Lyndon words and the variation of Lie commutators. Lastly, we exploit the criterion to some concrete constructions of Artin-Schelter regular algebras.In the part II, we consider the classification problem of5-dimensional Artin-Schelter regular algebras. Taking advantage of a truncated version of the diamond lemma, by comparing Hilbert series, we completely classify the class of5-dimensional Artin-Schelter regular, properly Z2-graded algebras which are generated by two elements and have some natural extra conditions. The results show that all such algebras fall into totally16fam-ilies, they are all strongly noetherian, Auslander regular and Cohen-Macaulay, and one family of them also provides a positive answer to Fl(?)ystad-Vatne’s existence question. Moreover, The results also show that all such algebras admit Lyndon presentaions. This observation gives us a motivation to study Artin-Schelter regular algebras that have Lyn-don presentations.