Some Researches On Artin-Schelter Regular Algebras And Piecewise-Koszul Algebras

Artin-Schelter regular algebras may be thought of as homogeneous coordinate rings of quantum Pn. They were introduced by Artin and Schelter in 1987. After then, search-ing and classifying Artin-Schelter regular algebras become a major project in the field of noncommutative projective geometry. This thesis considers Artin-Schelter algebras via deformation and piecewise-Koszul algebras by A∞-theory. It mainly consists of the following four parts.Firstly, by using of Grobner bases, the graphs of PBW algebras and quantum bi-nomial algebra theories, we discuss the sufficient conditions, with which a class of quadratic algebras are binomial skew polynomial rings.Secondly, we construct a class of quadratic algebras A by parameterizing the en-veloping algebra U(η)) of a graded Lie algebra η. The cases that the algebras A being not Artin-Schelter regular algebra are excluded with the consideration of Grobner bases and n-chains. With the aid of proper regular element, we turn the algebras A to be Artin-Schelter regular algebras of low dimension, then prove the algebras A are Artin-Schelter regular under some restrictions on parameters.Furthermore, we discuss the A∞-structures of the Ext-algebras E(A) for the Artin-Schelter regular algebras A obtained by parameterizing the enveloping algebra U(η), and recover A from E(A). With endowing proper Z4-gradation on the bases of E(A), we can find out the relations among the coefficients of the.A∞-algebra E(A) by considering the Stasheff identities. In particular, we give one kind of A∞-structure of the enveloping algebra U(η).Finally, in accordance with the symmetry property of Gorenstein, the Artin-Schelter regular algebras obtained by parameterizing U(η) are also piecewise-Koszul algebras. After seeking out minimal resolutions from the Anick resolutions, we can obtain more piecewise-Koszul algebras in the quadratic algebras A. Ax-theories bring more infor-mation for non-Koszul algebras. For connected graded algebras B, we deduce the dual theorem for piecewise-Koszul algebras B, by defining suitable reduced conditions on the A∞-algebra structures on the Koszul duals E(B). At last, we take a concrete exam-ple to illustrate the dual theorem of piecewise-Koszul algebras.