The structure of F2 as an associative algebra via quadratic forms

Abstract

Let F be a totally ordered field and ? ? F (a field extension of F) be a solution to the equation x2 = ax + b ? F[x], where a and b are fixed with b ? 0. With the help of this idea, we convert the F-vector space F2 into an associative F-algebra. As far as F2 can even be converted into a field. In the next step, based on a quadratic form, we define an inner product on F2 with values in F and call it the F-inner product. The defined inner product is mostly studied for its various properties. In particular, when F = R, we show that R2 with the defined product satisfies well-known inequalities such as the Cauchy-Schwarz and the triangle inequality. Under certain conditions, the reverse of recent inequalities is established. Some interesting properties of quadratic forms on F2 such as the invariant property are presented. In the sequel, we let SL(2,R) denote the subgroup of M(2,R) that consists of matrices with determinant 1 and set G = SL(2,R) ?MR, where MR is the matrix representation of R2. We then verify the coset space SL(2,R)/G with the quotient topology is homeomorphic to H (the upper-half complex plane) with the usual topology. Finally, we determine some families of functions in C(H,C), the ring consisting of complex-valued continuous functions onH; related to elements of G for which the functional equation f ?1 = 1? f is satisfied.