The Structure of $F^2$ as an Associative $F$-Algebra Based on Quadratic Forms and Some of its Properties

Abstract

 Abstract:

Let $F$ be a totally ordered field and $\omega$ be a solution to the equation $x^2=ax+b \in F[x]$, where $a$ and $b$ are fixed with $b \neq 0$. With this, we convert the $F$-vector space $F^2$ into an associative $F$-algebra. In particular, $F^2$ can even be converted to a field. Based on a quadratic form, we define an inner product on $F^2$ with values in $F$ and call it the $F$-inner product. The defined inner product is further studied for its properties. In the case that $F=\mathbb{R}$, we show $\mathbb{R}^2$ with the defined product satisfies the well-known inequalities such as the Cauchy–Schwarz and the triangle inequality. Under certain conditions, the reverse of recent inequalities is established. Let $SL(2, \mathbb{R})$ be the subgroup of $M(2, \mathbb{R})$ consisting of those matrices with determinant $1$ and let ${\mathbb{G} =SL(2, \mathbb{R}) \cap \mathbb{M}_{\mathbb{R}}}$. We then show that the coset space $\frac{SL(2, \mathbb{R})}{\mathbb{G}}$ with the quotient topology is homeomorphic to $H$ (the upper half-plane) with the usual topology. Finally, we determine some families of functions in $C(H, \mathbb{C})$, the ring consisting of complex-valued continuous functions on $H$; related to elements of $\mathbb{G}$ for which the functional equation $f o g=g o f$ is satisfied.