Algebro-geometric characterizations of commuting differential operators in semi-graded rings (2023)
In this thesis, we study algebro-geometric characterizations of commuting differential operators in families of semi-graded rings. First, we present some ring-theoretical notions of semi-graded rings that are necessary throughout the thesis. We include a non-exhaustive list of noncommutative rings that are particular examples of these rings. Second, to motivate the study of commuting differential operators beloging to noncommutative algebras, and hence to develop a possible Burchnall-Chaundy (BC) theory for them, we review algebraic and matrix results appearing in the literature on the theory of these operators in some families of semi-graded rings. Third, we introduce the notion of pseudo-multidegree function as a generalization of pseudo-degree function, and hence we establish a criterion to determine whether the centralizer of an element has finite dimension over a noncommutative ring having PBW basis. In this way, we formulate a BC theorem for rings having pseudo-multidegree functions. We illustrate our results with families of algebras appearing in ring theory and noncommutative geometry. Fourth, we develop a first approach to the BC theory for quadratic algebras having PBW bases defined by Golovashkin and Maksimov. We prove combinatorial properties on products of elements in these algebras, and then consider the notions of Sylvester matrix and resultant for quadratic algebras with the purpose of exploring common right factors. Then, by using the concept of determinant polynomial, we formulate the version of BC theory for these algebras. We present illustrative examples of the assertions about these algebras. Finally, we establish some bridging ideas with the aim of extending results on centralizers for graded rings to the setting of semi-graded rings.