On quantum commutation relations for the coordinate algebras of coefficients of binary forms

The purpose of this thesis is to take the first step in developing quantum analogs of the classical invariant theory of binary forms. The classical theory begins by considering the linear transformation of a 2 x 1 vector by a 2 x 2 matrix. This transformation induces a linear transformation of the coefficients of all the binary forms. The classical invariant theory is concerned with functions of these coefficients that are invariant under the induced linear transformation for all 2 x 2 matrices.

In the classical theory all elements commute. In this thesis a nonclassical or quantum invariant theory of binary forms will be considered in which elements do not commute. Two types of quantization will be introduced. Each quantization is a deformation into noncommutative or quantum theory and is represented by deformation parameters. By setting the deformation parameters equal to the appropriate values (usually zero or unity) one retrieves the commutative or classical theory. The main goal of this thesis is to find commutation relations for the coefficients of the quantum binary forms. The quantum binary forms considered will be of degree less than or equal to three. It is required that these commutation relations be invariant in form under the transformations induced by the quantum 2 x 2 matrix groups and it is hoped that these commutation relations will be such that the algebras of the coefficients satisfy PBW property. The determination of these commutation relations is the preliminary step in the development of the invariant theory of quantum binary forms.