The structure of F2 as an associative algebra via quadratic forms

Amir Veisi ; Ali Delbaznasab

Filomat, Tome 37 (2023) no. 14, p. 4671

Résumé

Let F be a totally ordered field and ω ∈ F (a field extension of F) be a solution to the equation x 2 = 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 F 2 into an associative F-algebra. As far as F 2 can even be converted into a field. In the next step, 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 mostly studied for its various properties. In particular, when F = R, we show that R 2 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 F 2 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) ∩ M R , where M R is the matrix representation of R 2. 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 on H; related to elements of G for which the functional equation f • = • f is satisfied.

Détail

DOI : 10.2298/FIL2314671V

Classification : 54B15, 15A63
Keywords: Associative algebra, totally ordered field, quadratic form, F-inner product, quotient topology